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Data structures

ContentsArticlesIntroduction 1

Data structure 1Linked data structure 3Succinct data structure 5Implicit data structure 7Compressed data structure 8Search data structure 9Persistent data structure 11Concurrent data structure 15

Abstract data types 18Abstract data type 18List 26Stack 29Queue 57Deque 60Priority queue 63Map 67Bidirectional map 70Multimap 71Set 72Tree 76

Arrays 79Array data structure 79Row-major order 84Dope vector 86Iliffe vector 87Dynamic array 88Hashed array tree 91Gap buffer 92Circular buffer 94Sparse array 109

Bit array 110Bitboard 115Parallel array 119Lookup table 121

Lists 127Linked list 127XOR linked list 143Unrolled linked list 145VList 147Skip list 149Self-organizing list 154

Binary trees 158Binary tree 158Binary search tree 166Self-balancing binary search tree 176Tree rotation 178Weight-balanced tree 181Threaded binary tree 182AVL tree 188Red-black tree 192AA tree 207Scapegoat tree 212Splay tree 216T-tree 230Rope 233Top Trees 238Tango Trees 242van Emde Boas tree 264Cartesian tree 268Treap 273

B-trees 276B-tree 276B+ tree 287Dancing tree 2912-3 tree 292

2-3-4 tree 293Queaps 295Fusion tree 299Bx-tree 299

Heaps 303Heap 303Binary heap 305Binomial heap 311Fibonacci heap 3162-3 heap 321Pairing heap 321Beap 324Leftist tree 325Skew heap 328Soft heap 331d-ary heap 333

Tries 335Trie 335Radix tree 342Suffix tree 344Suffix array 349Compressed suffix array 352FM-index 353Generalised suffix tree 353B-trie 355Judy array 355Directed acyclic word graph 357

Multiway trees 359Ternary search tree 359Andor tree 360(a,b)-tree 362Link/cut tree 363SPQR tree 363Spaghetti stack 366Disjoint-set data structure 367

Space-partitioning trees 371Space partitioning 371Binary space partitioning 372Segment tree 377Interval tree 380Range tree 385Bin 386k-d tree 388Implicit k-d tree 395min/max kd-tree 398Adaptive k-d tree 399Quadtree 399Octree 402Linear octrees 404Z-order 404UB-tree 409R-tree 409R+ tree 415R* tree 416Hilbert R-tree 419X-tree 426Metric tree 426vP-tree 427BK-tree 428

Hashes 429Hash table 429Hash function 442Open addressing 450Lazy deletion 453Linear probing 453Quadratic probing 454Double hashing 458Cuckoo hashing 459Coalesced hashing 463Perfect hash function 466Universal hashing 468

Linear hashing 473Extendible hashing 4742-choice hashing 480Pearson hashing 480FowlerNollVo hash function 481Bitstate hashing 483Bloom filter 483Locality preserving hashing 494Morton number 495Zobrist hashing 500Rolling hash 501Hash list 503Hash tree 504Prefix hash tree 506Hash trie 506Hash array mapped trie 507Distributed hash table 508Consistent hashing 513Stable hashing 515Koorde 515

Graphs 518Graph 518Adjacency list 520Adjacency matrix 522And-inverter graph 525Binary decision diagram 527Binary moment diagram 531Zero-suppressed decision diagram 533Propositional directed acyclic graph 534Graph-structured stack 535Scene graph 536

Appendix 541Big O notation 541Amortized analysis 551Locality of reference 553Standard Template Library 556

ReferencesArticle Sources and Contributors 566Image Sources, Licenses and Contributors 575

Article LicensesLicense 580

1Introduction

Data structureIn computer science, a data structure is a particular way of storing and organizing data in a computer so that it canbe used efficiently.[1] [2]

Different kinds of data structures are suited to different kinds of applications, and some are highly specialized tospecific tasks. For example, B-trees are particularly well-suited for implementation of databases, while compilerimplementations usually use hash tables to look up identifiers.Data structures are used in almost every program or software system. Data structures provide a means to managehuge amounts of data efficiently, such as large databases and internet indexing services. Usually, efficient datastructures are a key to designing efficient algorithms. Some formal design methods and programming languagesemphasize data structures, rather than algorithms, as the key organizing factor in software design.

Overview An array stores a number of elements of the same type in a specific order. They are accessed using an integer to

specify which element is required (although the elements may be of almost any type). Arrays may be fixed-lengthor expandable.

Record (also called tuple or struct) Records are among the simplest data structures. A record is a value thatcontains other values, typically in fixed number and sequence and typically indexed by names. The elements ofrecords are usually called fields or members.

A hash or dictionary or map is a more flexible variation on a record, in which name-value pairs can be added anddeleted freely.

Union. A union type definition will specify which of a number of permitted primitive types may be stored in itsinstances, e.g. "float or long integer". Contrast with a record, which could be defined to contain a float and aninteger; whereas, in a union, there is only one value at a time.

A tagged union (also called a variant, variant record, discriminated union, or disjoint union) contains an additionalfield indicating its current type, for enhanced type safety.

A set is an abstract data structure that can store certain values, without any particular order, and no repeatedvalues. Values themselves are not retrieved from sets, rather one tests a value for membership to obtain a boolean"in" or "not in".

An object contains a number of data fields, like a record, and also a number of program code fragments foraccessing or modifying them. Data structures not containing code, like those above, are called plain old datastructure.

Many others are possible, but they tend to be further variations and compounds of the above.

Basic principlesData structures are generally based on the ability of a computer to fetch and store data at any place in its memory,specified by an addressa bit string that can be itself stored in memory and manipulated by the program. Thus therecord and array data structures are based on computing the addresses of data items with arithmetic operations; whilethe linked data structures are based on storing addresses of data items within the structure itself. Many data structuresuse both principles, sometimes combined in non-trivial ways (as in XOR linking)

Data structure 2

The implementation of a data structure usually requires writing a set of procedures that create and manipulateinstances of that structure. The efficiency of a data structure cannot be analyzed separately from those operations.This observation motivates the theoretical concept of an abstract data type, a data structure that is defined indirectlyby the operations that may be performed on it, and the mathematical properties of those operations (including theirspace and time cost).

Language supportMost assembly languages and some low-level languages, such as BCPL, lack support for data structures. Manyhigh-level programming languages, and some higher-level assembly languages, such as MASM, on the other hand,have special syntax or other built-in support for certain data structures, such as vectors (one-dimensional arrays) inthe C language or multi-dimensional arrays in Pascal.Most programming languages feature some sorts of library mechanism that allows data structure implementations tobe reused by different programs. Modern languages usually come with standard libraries that implement the mostcommon data structures. Examples are the C++ Standard Template Library, the Java Collections Framework, andMicrosoft's .NET Framework.Modern languages also generally support modular programming, the separation between the interface of a librarymodule and its implementation. Some provide opaque data types that allow clients to hide implementation details.Object-oriented programming languages, such as C++, Java and .NET Framework use classes for this purpose.Many known data structures have concurrent versions that allow multiple computing threads to access the datastructure simultaneously.

References[1] Paul E. Black (ed.), entry for data structure in Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and

Technology. 15 December 2004. Online version (http:/ / www. itl. nist. gov/ div897/ sqg/ dads/ HTML/ datastructur. html) Accessed May 21,2009.

[2] Entry data structure in the Encyclopdia Britannica (2009) Online entry (http:/ / www. britannica. com/ EBchecked/ topic/ 152190/data-structure) accessed on May 21, 2009.

Further readings Peter Brass, Advanced Data Structures, Cambridge University Press, 2008. Donald Knuth, The Art of Computer Programming, vol. 1. Addison-Wesley, 3rd edition, 1997. Dinesh Mehta and Sartaj Sahni Handbook of Data Structures and Applications, Chapman and Hall/CRC Press,

2007. Niklaus Wirth, Algorithms and Data Structures, Prentice Hall, 1985.

External links UC Berkeley video course on data structures (http:/ / academicearth. org/ courses/ data-structures) Descriptions (http:/ / nist. gov/ dads/ ) from the Dictionary of Algorithms and Data Structures CSE.unr.edu (http:/ / www. cse. unr. edu/ ~bebis/ CS308/ ) Data structures course with animations (http:/ / www. cs. auckland. ac. nz/ software/ AlgAnim/ ds_ToC. html) Data structure tutorials with animations (http:/ / courses. cs. vt. edu/ ~csonline/ DataStructures/ Lessons/ index.

html) An Examination of Data Structures from .NET perspective (http:/ / msdn. microsoft. com/ en-us/ library/

aa289148(VS. 71). aspx) Schaffer, C. Data Structures and Algorithm Analysis (http:/ / people. cs. vt. edu/ ~shaffer/ Book/ C+ +

3e20110915. pdf)

Linked data structure 3

Linked data structureIn computer science, a linked data structure is a data structure which consists of a set of data records (nodes) linkedtogether and organized by references (links or pointers).In linked data structures, the links are usually treated as special data types that can only be dereferenced or comparedfor equality. Linked data structures are thus contrasted with arrays and other data structures that require performingarithmetic operations on pointers. This distinction holds even when the nodes are actually implemented as elementsof a single array, and the references are actually array indices: as long as no arithmetic is done on those indices, thedata structure is essentially a linked one.Linking can be done in two ways - Using dynamic allocation and using array index linking.Linked data structures include linked lists, search trees, expression trees, and many other widely used data structures.They are also key building blocks for many efficient algorithms, such as topological sort[1] and set union-find.[2]

Common Types of Linked Data Structures

1) Linked ListsA linked list is a collection of structures ordered not by their physical placement in memory but by logical links thatare stored as part of the data in the structure itself. It is not necessary that it should be stored in the adjacent memorylocations. Every structure has a data field and a address field. The Address field contains the address of its successor.Linked list can be singly, doubly or multiply linked and can either be linear or circular.Basic Properties

Objects, called nodes, are linked in a linear sequence A reference to the first node of the list is always kept. This is called the 'head' or 'front'.[3]

A linked list with three nodes contain two fields each: an integer value and a link to the next node

A linked list with a single node.

Example in Java

This is an example of the node class used to store integers in a Java implementation of a linked list.

public class IntNode {

public int value;

public IntNode link;

public IntNode(int v) { value = v; }

}


Example in c

This is an example of the node structure used for implementation of linked list in C.

struct node

{

int val;

struct node *next;

};

Note: A structure like this which contains a member that points to the same structure is called a self refrentialstructure.

2) Search TreesA search tree is a tree data structure in whose nodes data values can be stored from some ordered set, which is suchthat in an in-order traversal of the tree the nodes are visited in ascending order of the stored values.Basic Properties

Objects, called nodes, are stored in an ordered set. In-order traversal provides an ascending readout of the data in the tree Sub trees of the tree are in themselves, trees.

Advantages and disadvantages

Advantages Against ArraysCompared to arrays, linked data structures allow more flexibility in organizing the data and in allocating space for it.In arrays, the size of the array must be specified precisely at the beginning, this can be a potential waste of memory.A linked data structure is built dynamically and never needs to be bigger than the programmer requires. It alsorequires no guessing in terms of how much space you must allocate when using a linked data structure. This is afeature that is key in saving wasted memory.In array, the array elements have to be in contiguous(connected and sequential) portion of memory. But in linkeddata structure, the reference to each node gives us the information where to find out the next one. The nodes of alinked data structure can also be moved individually to different locations without affecting the logical connectionsbetween them, unlike arrays. With due care, a process can add or delete nodes to one part of a data structure evenwhile other processes are working on other parts.On the other hand, access to any particular node in a linked data structure requires following a chain of referencesthat stored in it. If the structure has n nodes, and each node contains at most b links, there will be some nodes thatcannot be reached in less than logb n steps. For many structures, some nodes may require worst case up to n1 steps.In contrast, many array data structures allow access to any element with a constant number of operations,independent of the number of entries.Broadly the implementation of these linked data structure is through dynamic data structures. It gives us the chanceto use particular space again. Memory can be utilized more efficiently by using this data structures. Memory isallocated as per the need and when memory is not further needed, deallocation is done.


General DisadvantagesLinked data structures may also incur in substantial memory allocation overhead (if nodes are allocated individually)and frustrate memory paging and processor caching algorithms (since they generally have poor locality of reference).In some cases, linked data structures may also use more memory (for the link fields) than competing array structures.This is because linked data structures are not contiguous. Instances of data can be found all over in memory, unlikearrays.In arrays, nth element can be accessed immediately, while in a linked data structure we have to follow multiplepointers so element access time varies according to where in the structure the element is.In some theoretical models of computation that enforce the constraints of linked structures, such as the pointermachine, many problems require more steps than in the unconstrained random access machine model.

References[1] Donald Knuth, The Art of Computer Programming[2] Bernard A. Galler and Michael J. Fischer. An improved equivalence algorithm. Communications of the ACM, Volume 7, Issue 5 (May 1964),

pages 301-303. The paper originating disjoint-set forests. ACM Digital Library (http:/ / portal. acm. org/ citation. cfm?doid=364099. 364331)[3] http:/ / www. cs. toronto. edu/ ~hojjat/ 148s07/ lectures/ week5/ 07linked. pdf

Succinct data structureIn computer science, a succinct data structure is data structure which uses an amount of space that is "close" to theinformation-theoretic lower bound, but (unlike other compressed representations) still allows for efficient queryoperations. The concept was originally introduced by Jacobson [1] to encode bit vectors, (unlabeled) trees, and planargraphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use themin-place, without decompressing them first. A related notion is that of a compressed data structure, in which the sizeof the data structure depends upon the particular data being represented.Suppose that is the information-theoretical optimal number of bits needed to store some data. A representation ofthis data is called

implicit if it takes bits of space, succinct if it takes bits of space, and compact if it takes bits of space.Implicit structures are thus usually reduced to storing information using some permutation of the input data; the mostwell-known example of this is the heap.

Succinct dictionariesSuccinct indexable dictionaries, also called rank/select dictionaries, form the basis of a number of succinctrepresentation techniques, including binary trees, -ary trees and multisets,[2] as well as suffix trees and arrays.[3]

The basic problem is to store a subset of a universe , usuallyrepresented as a bit array where iff . An indexable dictionary supports the usualmethods on dictionaries (queries, and insertions/deletions in the dynamic case) as well as the following operations:for .There is a simple representation [4] which uses bits of storage space (the original bit array and an auxiliary structure) and supports rank and select in constant time. It uses an idea similar to that for range-minimum

Succinct data structure 6

queries; there are a constant number of recursions before stopping at a subproblem of a limited size. The bit array ispartitioned into large blocks of size bits and small blocks of size bits. For each large block, the rank of its first bit isstored in a separate table ; each such entry takes bits for a total of bits of storage. Within a large block, anotherdirectory stores the rank of each of the small blocks it contains. The difference here is that it only needs bits for eachentry, since only only the differences from the rank of the first bit in the containing large block need to be stored.Thus, this table takes a total of bits. A lookup table can then be used that stores the answer to every possible rankquery on a bit string of length for ; this requires bits of storage space. Thus, since each of these auxiliary tables take space, this data structure supports rank queries in time and bits of space.To answer a query for in constant time, a constant time algorithm computes

In practice, the lookup table can be replaced by bitwise operations and smaller tables to perform find the numberof bits set in the small blocks. This is often beneficial, since succinct data structures find their uses in large data sets,in which case cache misses become much more frequent and the chances of the lookup table being evicted fromcloser CPU caches becomes higher.[5] Select queries can be easily supported by doing a binary search on the sameauxiliary structure used for rank; however, this takes time in the worst case. A more complicated structureusing bits of additional storage can be used to support select inconstant time.[6] In practice, many of these solutions have hidden constants in the notation which dominatebefore any asymptotic advantage becomes apparent; implementations using broadword operations and word-alignedblocks often perform better in practice.[7]

Entropy-compressed dictionaries

The space approach can be improved by noting that there are distinct -subsets of (or binarystrings of length with exactly 1s), and thus is an information theoretic lower boundon the number of bits needed to store . There is a succinct (static) dictionary which attains this bound, namelyusing space.[8] This structure can be extended to support rank and select queries andtakes space.[2] This bound can be reduced to a space/time tradeoff byreducing the storage space of the dictionary to with queries taking time.[9]

ExamplesWhen a sequence of variable-length items needs to be encoded, the items can simply be placed one after another,with no delimiters. A separate binary string consisting of 1s in the positions where an item begins, and 0s everywhere else is encoded along with it. Given this string, the function can quickly determine where each itembegins, given its index.[10]

Another example is the representation of a binary tree: an arbitrary binary tree on n nodes can be represented inbits while supporting a variety of operations on any node, which includes finding its parent, its left and

right child, and returning the size of its subtree, each in constant time. The number of different binary trees on nodes is . For large , this is about ; thus we need at least about bits toencode it. A succinct binary tree therefore would occupy only bits per node.

Succinct data structure 7

References[1] Jacobson, G. J (1988). Succinct static data structures.[2] Raman, R.; V. Raman, S. S Rao (2002). "Succinct indexable dictionaries with applications to encoding k-ary trees and multisets" (http:/ /

www. cs. cmu. edu/ afs/ cs. cmu. edu/ project/ aladdin/ wwwlocal/ hash/ RaRaRa02. pdf). Proceedings of the thirteenth annual ACM-SIAMsymposium on Discrete algorithms. pp.233242. ISBN089871513X. .

[3] Sadakane, K.; R. Grossi (2006). "Squeezing succinct data structures into entropy bounds" (http:/ / www. dmi. unisa. it/ people/ cerulli/ www/WSPages/ WSFiles/ Abs/ S3/ S33_abs_Grossi. pdf). Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm.pp.12301239. ISBN0898716055. .

[4] Jacobson, G. (1989). Space-efficient static trees and graphs (http:/ / www. cs. cmu. edu/ afs/ cs/ project/ aladdin/ wwwlocal/ compression/00063533. pdf). .

[5] Gonzlez, R.; S. Grabowski, V. Mkinen, G. Navarro (2005). "Practical implementation of rank and select queries" (http:/ / www. dcc. uchile.cl/ ~gnavarro/ algoritmos/ ps/ wea05. pdf). Poster Proceedings Volume of 4th Workshop on Efficient and Experimental Algorithms (WEA).pp.2738. .

[6] Clark, D. (1998). Compact pat trees (https:/ / uwspace. uwaterloo. ca/ bitstream/ 10012/ 64/ 1/ nq21335. pdf). .[7] Vigna, S. (2008). "Broadword implementation of rank/select queries" (http:/ / sux. dsi. unimi. it/ paper. pdf). Experimental Algorithms:

154168. .[8] Brodnik, A.; J. I Munro (1999). "Membership in constant time and almost-minimum space" (http:/ / www. cs. cmu. edu/ afs/ cs. cmu. edu/

project/ aladdin/ wwwlocal/ compression/ BM99. pdf). SIAM J. Comput. 28 (5): 16271640. doi:10.1137/S0097539795294165. .[9] Patrascu, M. (2008). "Succincter" (http:/ / people. csail. mit. edu/ mip/ papers/ succinct/ succinct. pdf). Foundations of Computer Science,

2008. FOCS'08. IEEE 49th Annual IEEE Symposium on. pp.305313. .[10] Belazzougui, Djamal. "Hash, displace, and compress" (http:/ / cmph. sourceforge. net/ papers/ esa09. pdf). .

Implicit data structureIn computer science, an implicit data structure is a data structure that uses very little memory besides the actualdata elements i.e. very little information other than main data is stored in these structures. These are storage schemeswhich retain no pointers and represent the file of n k-key records as a simple n by k array n thus retrieve faster. Inimplicit data structures the only structural information to be given is to allow the array to grow and shrink as n. Noextra information is required. It is called "implicit" because most of the structure of the elements is expressedimplicitly by their order. Another term used interchangeably is space efficient. Definitions of very little is vagueand can mean from O(1) to O(log n) extra space. Implicit data structure encodes data efficiently,so that it does notneed to be decoded to be used. Everything is accessed in-place, by reading bits at various position in data. Toachieve optimal coding, we use bits instead of bytes. Implicit data structures are frequently also succinct datastructures.Although one may argue that disk space is no longer a problem and we should not concern ourselves with improvingspace utilization, the issue that implicit data structures are designed to improve is main memory utilization. Harddisks, or any other means of large data capacity, I/O devices, are orders of magnitudes slower than main memory.Hence, the higher percentage of a task can fit in buffers in main memory the less dependence is on slow I/O devices.Hence, if a larger chunk of an implicit data structure fits in main memory the operations performed on it can befaster even if the asymptotic running time is not as good as its space-oblivious counterpart. Furthermore, since theCPU-cache is usually much smaller than main-memory, implicit data structures can improve cache-efficiency andthus running speed, especially if the method used improves locality. Keys are scanned very efficiently. Downloadingindexed data in mobiles becomes easier.

Implicit data structure 8

Implicit data structure for weighted elementFor presentation of elements with different weight several data structures are required.The structure uses one morelocation besides required for values of elements.The first structure supports worst case search time in terms of rankof weight of elements w.r.t multi-set of weights.If the elements are drawn from uniform distribution, then variationof this structure takes average time.The same result obtain for the data structures in which the intervals betweenconsecutive values have access probabilities.

ExamplesExamples of implicit data structures include Binary heap Beap

Further reading See publications of Herv Brnnimann [1], J. Ian Munro [2], Greg Frederickson [3]

References[1] http:/ / photon. poly. edu/ ~hbr/[2] http:/ / www. cs. uwaterloo. ca/ ~imunro/[3] http:/ / www. cs. purdue. edu/ people/ gnf

Compressed data structureThe term compressed data structure arises in the computer science subfields of algorithms, data structures, andtheoretical computer science. It refers to a data structure whose operations are roughly as fast as those of aconventional data structure for the problem, but whose size can be substantially smaller. The size of the compresseddata structure is typically highly dependent upon the entropy of the data being represented.Important examples of compressed data structures include the compressed suffix array[1] [2] and the FM-index,[3]

both of which can represent an arbitrary text of characters T for pattern matching. Given any input pattern P, theysupport the operation of finding if and where P appears in T. The search time is proportional to the sum of the lengthof pattern P, a very slow-growing function of the length of the text T, and the number of reported matches. The spacethey occupy is roughly equal to the size of the text T in entropy-compressed form, such as that obtained byPrediction by Partial Matching or gzip. Moreover, both data structures are self-indexing, in that they can reconstructthe text T in a random access manner, and thus the underlying text T can be discarded. In other words, theysimultaneously provide a compressed and quickly searchable representation of the text T. They represent asubstantial space improvement over the conventional suffix tree and suffix array, which occupy many times morespace than the size of T. They also support searching for arbitrary patterns, as opposed to the inverted index, whichcan support only word-based searches. In addition, inverted indexes do not have the self-indexing feature.An important related notion is that of a succinct data structure, which uses space roughly equal to theinformation-theoretic minimum, which is a worst-case notion of the space needed to represent the data. In contrast,the size of a compressed data structure depends upon the particular data being represented. When the data arecompressible, as is often the case in practice for natural language text, the compressed data structure can occupysubstantially less space than the information-theoretic minimum.

Compressed data structure 9

References[1] R. Grossi and J. S. Vitter, Compressed Suffix Arrays and Suffix Trees with Applications to Text Indexing and String Matching], Proceedings

of the 32nd ACM Symposium on Theory of Computing, May 2000, 397-406. Journal version in SIAM Journal on Computing, 35(2), 2005,378-407.

[2] R. Grossi, A. Gupta, and J. S. Vitter, High-Order Entropy-Compressed Text Indexes, Proceedings of the 14th Annual SIAM/ACM Symposiumon Discrete Algorithms, January 2003, 841-850.

[3] P. Ferragina and G. Manzini, Opportunistic Data Structures with Applications, Proceedings of the 41st IEEE Symposium on Foundations ofComputer Science, November 2000, 390-398. Journal version in Indexing Compressed Text, Journal of the ACM, 52(4), 2005, 552-581.

Search data structureIn computer science, a search data structure is any data structure that allows the efficient retrieval of specific itemsfrom a set of items, such as a specific record from a database.The simplest, most general, and least efficient search structure is merely an unordered sequential list of all the items.Locating the desired item in such a list, by the linear search method, inevitably requires a number of operationsproportional to the number n of items, in the worst case as well as in the average case. Useful search data structuresallow faster retrieval; however, they are limited to queries of some specific kind. Moreover, since the cost ofbuilding such structures is at least proportional to n, they only pay off if several queries are to be performed on thesame database (or on a database that changes little between queries).Static search structures are designed for answering many queries on a fixed database; dynamic structures also allowinsertion, deletion, or modification of items between successive queries. In the dynamic case, one must also considerthe cost of fixing the search structure to account for the changes in the database.

ClassificationThe simplest kind of query is to locate a record that has a specific field (the key) equal to a specified value v. Othercommon kinds of query are "find the item with smallest (or largest) key value", "find the item with largest key valuenot exceeding v", "find all items with key values between specified bounds vmin and vmax".In certain databases the key values may be points in some multi-dimensional space. For example, the key may be ageographic position (latitude and longitude) on the Earth. In that case, common kinds of queries are find the recordwith a key closest to a given point v", or "find all items whose key lies at a given distance from v", or "find all itemswithin a specified region R of the space".A common special case of the latter are simultaneous range queries on two or more simple keys, such as "find allemployee records with salary between 50,000 and 100,000 and hired between 1995 and 2007".

Single ordered keys Array if the key values span a moderately compact interval. Priority-sorted list; see linear search Key-sorted array; see binary search Self-balancing binary search tree Hash table

Search data structure 10

Finding the smallest element Heap

Asymptotic amortized worst-case analysis

In this table, the asymptotic notation O(f(n)) means "not exceeding some fixed multiple of f(n) in the worst case."

Insert Delete Search Find maximum Space usage

Unsorted array O(1) O(1) O(n) O(n) O(n)

Value-indexed array O(1) O(1) O(1) O(n) O(n)

Sorted array O(n) O(n) O(log n) O(1) O(n)

Unsorted linked list O(1)* O(1)* O(n) O(n) O(n)

Sorted linked list O(1)* O(1)* O(n) O(1) O(n)

Self-balancing binary tree O(log n) O(log n) O(log n) O(log n) O(n)

Heap O(log n) O(log n)** O(n) O(1) O(n)

Hash table O(1) O(1) O(1) O(n) O(n)

* The cost to add or delete an element into a known location in the list (i.e. if you have an iterator to the location) is O(1). If you don't know the

location, then you need to traverse the list to the location of deletion/insertion, which takes O(n) time. ** The deletion cost is O(log n) for theminimum or maximum, O(n) for an arbitrary element.

This table is only an approximate summary; for each data structure there are special situations and variants that maylead to different costs. Also two or more data structures can be combined to obtain lower costs.

Footnotes

Persistent data structure 11

Persistent data structureIn computing, a persistent data structure is a data structure which always preserves the previous version of itselfwhen it is modified; such data structures are effectively immutable, as their operations do not (visibly) update thestructure in-place, but instead always yield a new updated structure. (A persistent data structure is not a datastructure committed to persistent storage, such as a disk; this is a different and unrelated sense of the word"persistent.")A data structure is partially persistent if all versions can be accessed but only the newest version can be modified.The data structure is fully persistent if every version can be both accessed and modified. If there is also a meld ormerge operation that can create a new version from two previous versions, the data structure is called confluentlypersistent. Structures that are not persistent are called ephemeral.[1]

These types of data structures are particularly common in logical and functional programming, and in a purelyfunctional program all data is immutable, so all data structures are automatically fully persistent.[1] Persistent datastructures can also be created using in-place updating of data and these may, in general, use less time or storagespace than their purely functional counterparts.While persistence can be achieved by simple copying, this is inefficient in time and space, because most operationsmake only small changes to a data structure. A better method is to exploit the similarity between the new and oldversions to share structure between them, such as using the same subtree in a number of tree structures. However,because it rapidly becomes infeasible to determine how many previous versions share which parts of the structure,and because it is often desirable to discard old versions, this necessitates an environment with garbage collection.

Examples of persistent data structuresPerhaps the simplest persistent data structure is the singly linked list or cons-based list, a simple list of objectsformed by each carrying a reference to the next in the list. This is persistent because we can take a tail of the list,meaning the last k items for some k, and add new nodes on to the front of it. The tail will not be duplicated, insteadbecoming shared between both the old list and the new list. So long as the contents of the tail are immutable, thissharing will be invisible to the program.Many common reference-based data structures, such as red-black trees,[2] and queues,[3] can easily be adapted tocreate a persistent version. Some other like Stack, Double-ended queues (dequeue), Min-Dequeue (which haveadditional operation min returning minimal element in constant time without incurring additional complexity onstandard operations of queuing and dequeuing on both ends), Random access list (with constant cons/head as singlelinked list, but with additional operation of random access with sub-linear, most often logarithmic, complexity),Random access queue, Random access double-ended queue and Random access stack (as well Random accessMin-List, Min-Queue, Min-Dequeue, Min-Stack) needs slightly more effort.There exists also persistent data structures which uses destructible operations (thus impossible to implementefficiently in the purely functional languages like Haskell, however possible in languages like C, Java), they arehowever not needed, as most data structures are currently available in pure versions which are often simpler toimplement, and often behaves better in multi-threaded environments.


Linked listsThis example is taken from Okasaki. See the bibliography.

Singly linked lists are the bread-and-butter data structure in functional languages. In ML-derived languages andHaskell, they are purely functional because once a node in the list has been allocated, it cannot be modified, onlycopied or destroyed. Note that ML itself is not purely functional.Consider the two lists:

xs = [0, 1, 2]

ys = [3, 4, 5]

These would be represented in memory by:

where a circle indicates a node in the list (the arrow out showing the second element of the node which is a pointer toanother node).Now concatenating the two lists:

zs = xs ++ ys

results in the following memory structure:


Notice that the nodes in list xs have been copied, but the nodes in ys are shared. As a result, the original lists (xsand ys) persist and have not been modified.The reason for the copy is that the last node in xs (the node containing the original value 2) cannot be modified topoint to the start of ys, because that would change the value of xs.

TreesThis example is taken from Okasaki. See the bibliography.

Consider a binary tree used for fast searching, where every node has the recursive invariant that subnodes on the leftare less than the node, and subnodes on the right are greater than the node.For instance, the set of data

xs = [a, b, c, d, f, g, h]

might be represented by the following binary search tree:

A function which inserts data into the binary tree and maintains the invariant is:

fun insert (x, E) = T (E, x, E)

| insert (x, s as T (a, y, b)) =

if x < y then T (insert (x, a), y, b)

else if x > y then T (a, y, insert (x, b))

else s

After executing

ys = insert ("e", xs)

we end up with the following:


Notice two points: Firstly the original tree (xs) persists. Secondly many common nodes are shared between the oldtree and the new tree. Such persistence and sharing is difficult to manage without some form of garbage collection(GC) to automatically free up nodes which have no live references, and this is why GC is a feature commonly foundin functional programming languages.

Reference cyclesSince every value in a purely functional computation is built up out of existing values, it would seem that it isimpossible to create a cycle of references. In that case, the reference graph (the graph of the references from object toobject) could only be a directed acyclic graph. However, in most functional languages, functions can be definedrecursively; this capability allows recursive structures using functional suspensions. In lazy languages, such asHaskell, all data structures are represented as implicitly suspended thunks; in these languages any data structure canbe recursive because a value can be defined in terms of itself. Some other languages, such as Objective Caml, allowthe explicit definition of recursive values.

References[1] Kaplan, Haim (2001). "Persistent data structures" (http:/ / www. math. tau. ac. il/ ~haimk/ papers/ persistent-survey. ps). Handbook on Data

Structures and Applications (CRC Press). .[2] Neil Sarnak, Robert E. Tarjan (1986). "Planar Point Location Using Persistent Search Trees" (http:/ / www. link. cs. cmu. edu/ 15859-f07/

papers/ point-location. pdf). Communications of the ACM 29 (7): 669679. doi:10.1145/6138.6151. .[3] Chris Okasaki. Purely Functional Data Structures (thesis) (http:/ / www. cs. cmu. edu/ ~rwh/ theses/ okasaki. pdf). .

Further reading Persistent Data Structures and Managed References (http:/ / www. infoq. com/ presentations/

Value-Identity-State-Rich-Hickey) - video presentation by Rich Hickey on Clojure's use of persistent datastructures and how they support concurrency


Making Data Structures Persistent (http:/ / www. cs. cmu. edu/ ~sleator/ papers/ Persistence. htm) by James R.Driscoll, Neil Sarnak, Daniel D. Sleator, Robert E. Tarjan

Fully persistent arrays for efficient incremental updates and voluminous reads (http:/ / citeseerx. ist. psu. edu/viewdoc/ summary?doi=10. 1. 1. 34. 1317)

Real-Time Deques, Multihead Turing Machines, and Purely Functional Programming (http:/ / citeseerx. ist. psu.edu/ viewdoc/ summary?doi=10. 1. 1. 51. 2895)

Purely functional data structures by Chris Okasaki, Cambridge University Press, 1998, ISBN 0-521-66350-4. Purely Functional Data Structures (http:/ / www. cs. cmu. edu/ ~rwh/ theses/ okasaki. pdf) thesis by Chris Okasaki

(PDF format) Fully Persistent Lists with Catenation (http:/ / www. cs. cmu. edu/ ~sleator/ papers/ fully-persistent-lists. pdf) by

James R. Driscoll, Daniel D. Sleator, Robert E. Tarjan (PDF) Persistent Data Structures (http:/ / ocw. mit. edu/ courses/ electrical-engineering-and-computer-science/

6-854j-advanced-algorithms-fall-2005/ lecture-notes/ persistent. pdf) from MIT open course AdvancedAlgorithms (http:/ / ocw. mit. edu/ courses/ electrical-engineering-and-computer-science/6-854j-advanced-algorithms-fall-2005)

External links Lightweight Java implementation of Persistent Red-Black Trees (http:/ / wiki. edinburghhacklab. com/

PersistentRedBlackTreeSet)

Concurrent data structureIn computer science, a concurrent data structure is a particular way of storing and organizing data for access bymultiple computing threads (or processes) on a computer.Historically, such data structures were used on uniprocessor machines with operating systems that supportedmultiple computing threads (or processes). The term concurrency captured the multiplexing/interleaving of thethreads' operations on the data by the operating system, even though the processors never issued two operations thataccessed the data simultaneously.Today, as multiprocessor computer architectures that provide parallelism become the dominant computing platform(through the proliferation of multi-core processors), the term has come to stand mainly for data structures that can beaccessed by multiple threads which may actually access the data simultaneously because they run on differentprocessors that communicate with one another. The concurrent data structure (sometimes also called a shared datastructure) is usually considered to reside in an abstract storage environment called shared memory, though thismemory may be physically implemented as either a "tightly coupled" or a distributed collection of storage modules.

Basic principlesConcurrent data structures, intended for use in parallel or distributed computing environments, differ from"sequential" data structures, intended for use on a processor machine, in several ways .[1] Most notably, in asequential environment one specifies the data structure's properties and checks that they are implemented correctly,by providing safety properties. In a concurrent environment, the specification must also describe livenessproperties which an implementation must provide. Safety properties usually state that something bad never happens,while liveness properties state that something good keeps happening. These properties can be expressed, forexample, using Linear Temporal Logic.The type of liveness requirements tend to define the data structure. The method calls can be blocking or non-blocking. Data structures are not restricted to one type or the other, and can allow combinations where some

Concurrent data structure 16

method calls are blocking and others are non-blocking (examples can be found in the Java concurrency softwarelibrary).The safety properties of concurrent data structures must capture their behavior given the many possible interleavingsof methods called by different threads. It is quite intuitive to specify how abstract data structures behave in asequential setting in which there are no interleavings. Therefore, many mainstream approaches for arguing the safetyproperties of a concurrent data structure (such as serializability, linearizability, sequential consistency, and quiescentconsistency [1] ) specify the structures properties sequentially, and map its concurrent executions to a collection ofsequential ones.In order to guarantee the safety and liveness properties, concurrent data structures must typically (though not always)allow threads to reach consensus as to the results of their simultaneous data access and modification requests. Tosupport such agreement, concurrent data structures are implemented using special primitive synchronizationoperations (see synchronization primitives) available on modern multiprocessor machines that allow multiple threadsto reach consensus. This consensus can be reached achieved in a blocking manner by using locks, or without locks,in which case it is non-blocking. There is a wide body of theory on the design of concurrent data structures (seebibliographical references).

Design and ImplementationConcurrent data structures are significantly more difficult to design and to verify as being correct than theirsequential counterparts.The primary source of this additional difficulty is concurrency, exacerbated by the fact that threads must be thoughtof as being completely asynchronous: they are subject to operating system preemption, page faults, interrupts, and soon.On today's machines, the layout of processors and memory, the layout of data in memory, the communication loadon the various elements of the multiprocessor architecture all influence performance. Furthermore, there is a tensionbetween correctness and performance: algorithmic enhancements that seek to improve performance often make itmore difficult to design and verify a correct data structure implementation.A key measure for performance is scalability, captured by the speedup of the implementation. Speedup is a measureof how effectively the application is utilizing the machine it is running on. On a machine with P processors, thespeedup is the ratio of the structures execution time on a single processor to its execution time on T processors.Ideally, we want linear speedup: we would like to achieve a speedup of P when using P processors. Data structureswhose speedup grows with P are called scalable. The extent to which one can scale the performance of a concurrentdata structure is captured by a formula known as Amdahl's law and more refined versions of it such as Gustafson'slaw.A key issue with the performance of concurrent data structures is the level of memory contention: the overhead intraffic to and from memory as a result of multiple threads concurrently attempting to access the same locations inmemory. This issue is most acute with blocking implementations in which locks control access to memory. In orderto acquire a lock, a thread must repeatedly attempt to modify that location. On a cache-coherent multiprocessor (onein which processors have local caches that are updated by hardware in order to keep them consistent with the latestvalues stored) this results in long waiting times for each attempt to modify the location, and is exacerbated by theadditional memory traffic associated with unsuccessful attempts to acquire the lock.

Concurrent data structure 17

References[1] Mark Moir and Nir Shavit (2007). " Concurrent Data Structures (http:/ / www. cs. tau. ac. il/ ~shanir/ concurrent-data-structures. pdf)". In

Dinesh Metha and Sartaj Sahni. 'Handbook of Data Structures and Applications' (1st ed.). Chapman and Hall/CRC Press. pp.47-14 47-30.

Further reading Nancy Lynch "Distributed Computing" Hagit Attiya and Jennifer Welch "Distributed Computing: Fundamentals, Simulations And Advanced Topics, 2nd

Ed" Doug Lea, "Concurrent Programming in Java: Design Principles and Patterns" Maurice Herlihy and Nir Shavit, "The Art of Multiprocessor Programming" Mattson, Sanders, and Massingil "Patterns for Parallel Programming"

External links Multithreaded data structures for parallel computing, Part 1 (http:/ / www. ibm. com/ developerworks/ aix/

library/ au-multithreaded_structures1/ index. html) (Designing concurrent data structures) by Arpan Sen Multithreaded data structures for parallel computing: Part 2 (http:/ / www. ibm. com/ developerworks/ aix/

library/ au-multithreaded_structures2/ index. html) (Designing concurrent data structures without mutexes) byArpan Sen

18

Abstract data types

Abstract data typeIn computing, an abstract data type (ADT) is a mathematical model for a certain class of data structures that havesimilar behavior; or for certain data types of one or more programming languages that have similar semantics. Anabstract data type is defined indirectly, only by the operations that may be performed on it and by mathematicalconstraints on the effects (and possibly cost) of those operations.[1]

For example, an abstract stack data structure could be defined by three operations: push, that inserts some data itemonto the structure, pop, that extracts an item from it (with the constraint that each pop always returns the mostrecently pushed item that has not been popped yet), and peek, that allows data on top of the structure to beexamined without removal. When analyzing the efficiency of algorithms that use stacks, one may also specify thatall operations take the same time no matter how many items have been pushed into the stack, and that the stack usesa constant amount of storage for each element.Abstract data types are purely theoretical entities, used (among other things) to simplify the description of abstractalgorithms, to classify and evaluate data structures, and to formally describe the type systems of programminglanguages. However, an ADT may be implemented by specific data types or data structures, in many ways and inmany programming languages; or described in a formal specification language. ADTs are often implemented asmodules: the module's interface declares procedures that correspond to the ADT operations, sometimes withcomments that describe the constraints. This information hiding strategy allows the implementation of the module tobe changed without disturbing the client programs.The term abstract data type can also be regarded as a generalised approach of a number of algebraic structures,such as lattices, groups, and rings.[2] This can be treated as part of subject area of Artificial intelligence. The notionof abstract data types is related to the concept of data abstraction, important in object-oriented programming anddesign by contract methodologies for software development .

Defining an abstract data type (ADT)An Abstract Data type is defined as a mathematical model of the data objects that make up a data type as well as thefunctions that operate on these objects. There are no standard conventions for defining them. A broad division maybe drawn between "imperative" and "functional" definition styles.

Imperative abstract data type definitionsIn the "imperative" view, which is closer to the philosophy of imperative programming languages, an abstract datastructure is conceived as an entity that is mutable meaning that it may be in different states at different times.Some operations may change the state of the ADT; therefore, the order in which operations are evaluated isimportant, and the same operation on the same entities may have different effects if executed at different times just like the instructions of a computer, or the commands and procedures of an imperative language. To underscorethis view, it is customary to say that the operations are executed or applied, rather than evaluated. The imperativestyle is often used when describing abstract algorithms. This is described by Donald E. Knuth and can be referencedfrom here The Art of Computer Programming.

Abstract data type 19

Abstract variable

Imperative ADT definitions often depend on the concept of an abstract variable, which may be regarded as thesimplest non-trivial ADT. An abstract variable V is a mutable entity that admits two operations: store(V,x) where x is a value of unspecified nature; and fetch(V), that yields a value;with the constraint that fetch(V) always returns the value x used in the most recent store(V,x) operation on the same variable V.As in so many programming languages, the operation store(V,x) is often written V x (or some similar notation),and fetch(V) is implied whenever a variable V is used in a context where a value is required. Thus, for example, V V + 1 is commonly understood to be a shorthand for store(V,fetch(V) + 1).In this definition, it is implicitly assumed that storing a value into a variable U has no effect on the state of a distinctvariable V. To make this assumption explicit, one could add the constraint that if U and V are distinct variables, the sequence { store(U,x); store(V,y) } is equivalent to { store(V,y);store(U,x) }.

More generally, ADT definitions often assume that any operation that changes the state of one ADT instance has noeffect on the state of any other instance (including other instances of the same ADT) unless the ADT axiomsimply that the two instances are connected (aliased) in that sense. For example, when extending the definition ofabstract variable to include abstract records, the operation that selects a field from a record variable R must yield avariable V that is aliased to that part of R.The definition of an abstract variable V may also restrict the stored values x to members of a specific set X, called therange or type of V. As in programming languages, such restrictions may simplify the description and analysis ofalgorithms, and improve their readability.Note that this definition does not imply anything about the result of evaluating fetch(V) when V is un-initialized,that is, before performing any store operation on V. An algorithm that does so is usually considered invalid,because its effect is not defined. (However, there are some important algorithms whose efficiency strongly dependson the assumption that such a fetch is legal, and returns some arbitrary value in the variable's range.)

Instance creation

Some algorithms need to create new instances of some ADT (such as new variables, or new stacks). To describesuch algorithms, one usually includes in the ADT definition a create() operation that yields an instance of theADT, usually with axioms equivalent to the result of create() is distinct from any instance S in use by the algorithm.This axiom may be strengthened to exclude also partial aliasing with other instances. On the other hand, this axiomstill allows implementations of create() to yield a previously created instance that has become inaccessible to theprogram.


Preconditions, postconditions, and invariants

In imperative-style definitions, the axioms are often expressed by preconditions, that specify when an operation maybe executed; postconditions, that relate the states of the ADT before and after the execution of each operation; andinvariants, that specify properties of the ADT that are not changed by the operations.

Example: abstract stack (imperative)

As another example, an imperative definition of an abstract stack could specify that the state of a stack S can bemodified only by the operations push(S,x), where x is some value of unspecified nature; and pop(S), that yields a value as a result;with the constraint that For any value x and any abstract variable V, the sequence of operations { push(S,x); V pop(S) } is equivalent

to { V x };Since the assignment { V x }, by definition, cannot change the state of S, this condition implies that { V pop(S) } restores S to the state it had before the { push(S,x) }. From this condition and from the properties ofabstract variables, it follows, for example, that the sequence

{ push(S,x); push(S,y); U pop(S); push(S,z); V pop(S); W pop(S); }where x,y, and z are any values, and U, V, W are pairwise distinct variables, is equivalent to

{ U y; V z; W x }Here it is implicitly assumed that operations on a stack instance do not modify the state of any other ADT instance,including other stacks; that is, For any values x,y, and any distinct stacks S and T, the sequence { push(S,x); push(T,y) } is equivalent to {push(T,y); push(S,x) }.

A stack ADT definition usually includes also a Boolean-valued function empty(S) and a create() operation thatreturns a stack instance, with axioms equivalent to create() S for any stack S (a newly created stack is distinct from all previous stacks) empty(create()) (a newly created stack is empty) not empty(push(S,x)) (pushing something into a stack makes it non-empty)

Single-instance style

Sometimes an ADT is defined as if only one instance of it existed during the execution of the algorithm, and alloperations were applied to that instance, which is not explicitly notated. For example, the abstract stack above couldhave been defined with operations push(x) and pop(), that operate on "the" only existing stack. ADT definitions inthis style can be easily rewritten to admit multiple coexisting instances of the ADT, by adding an explicit instanceparameter (like S in the previous example) to every operation that uses or modifies the implicit instance.On the other hand, some ADTs cannot be meaningfully defined without assuming multiple instances. This is the casewhen a single operation takes two distinct instances of the ADT as parameters. For an example, consider augmentingthe definition of the stack ADT with an operation compare(S,T) that checks whether the stacks S and T contain thesame items in the same order.


Functional ADT definitionsAnother way to define an ADT, closer to the spirit of functional programming, is to consider each state of thestructure as a separate entity. In this view, any operation that modifies the ADT is modeled as a mathematicalfunction that takes the old state as an argument, and returns the new state as part of the result. Unlike the"imperative" operations, these functions have no side effects. Therefore, the order in which they are evaluated isimmaterial, and the same operation applied to the same arguments (including the same input states) will alwaysreturn the same results (and output states).In the functional view, in particular, there is no way (or need) to define an "abstract variable" with the semantics ofimperative variables (namely, with fetch and store operations). Instead of storing values into variables, onepasses them as arguments to functions.

Example: abstract stack (functional)

For example, a complete functional-style definition of a stack ADT could use the three operations: push: takes a stack state and an arbitrary value, returns a stack state; top: takes a stack state, returns a value; pop: takes a stack state, returns a stack state;with the following axioms: top(push(s,x)) = x (pushing an item onto a stack leaves it at the top) pop(push(s,x)) = s (pop undoes the effect of push)In a functional-style definition there is no need for a create operation. Indeed, there is no notion of "stackinstance". The stack states can be thought of as being potential states of a single stack structure, and two stack statesthat contain the same values in the same order are considered to be identical states. This view actually mirrors thebehavior of some concrete implementations, such as linked lists with hash cons.Instead of create(), a functional definition of a stack ADT may assume the existence of a special stack state, theempty stack, designated by a special symbol like or "()"; or define a bottom() operation that takes no agumentsand returns this special stack state. Note that the axioms imply that push(,x) In a functional definition of a stack one does not need an empty predicate: instead, one can test whether a stack isempty by testing whether it is equal to .Note that these axioms do not define the effect of top(s) or pop(s), unless s is a stack state returned by a push.Since push leaves the stack non-empty, those two operations are undefined (hence invalid) when s = . On theother hand, the axioms (and the lack of side effects) imply that push(s,x) = push(t,y) if and only if x = y and s = t.As in some other branches of mathematics, it is customary to assume also that the stack states are only those whoseexistence can be proved from the axioms in a finite number of steps. In the stack ADT example above, this rulemeans that every stack is a finite sequence of values, that becomes the empty stack () after a finite number of pops.By themselves, the axioms above do not exclude the existence of infinite stacks (that can be poped forever, eachtime yielding a different state) or circular stacks (that return to the same state after a finite number of pops). Inparticular, they do not exclude states s such that pop(s) = s or push(s,x) = s for some x. However, since one cannotobtain such stack states with the given operations, they are assumed "not to exist".


Advantages of abstract data typing EncapsulationAbstraction provides a promise that any implementation of the ADT has certain properties and abilities; knowingthese is all that is required to make use of an ADT object. The user does not need any technical knowledge of howthe implementation works to use the ADT. In this way, the implementation may be complex but will be encapsulatedin a simple interface when it is actually used. Localization of changeCode that uses an ADT object will not need to be edited if the implementation of the ADT is changed. Since anychanges to the implementation must still comply with the interface, and since code using an ADT may only refer toproperties and abilities specified in the interface, changes may be made to the implementation without requiring anychanges in code where the ADT is used. FlexibilityDifferent implementations of an ADT, having all the same properties and abilities, are equivalent and may be usedsomewhat interchangeably in code that uses the ADT. This gives a great deal of flexibility when using ADT objectsin different situations. For example, different implementations of an ADT may be more efficient in differentsituations; it is possible to use each in the situation where they are preferable, thus increasing overall efficiency.

Typical operationsSome operations that are often specified for ADTs (possibly under other names) are compare(s,t), that tests whether two structures are equivalent in some sense; hash(s), that computes some standard hash function from the instance's state; print(s) or show(s), that produces a human-readable representation of the structure's state.In imperative-style ADT definitions, one often finds also create(), that yields a new instance of the ADT; initialize(s), that prepares a newly-created instance s for further operations, or resets it to some "initial

state"; copy(s,t), that puts instance s in a state equivalent to that of t; clone(t), that performs s new(), copy(s,t), and returns s; free(s) or destroy(s), that reclaims the memory and other resources used by s;The free operation is not normally relevant or meaningful, since ADTs are theoretical entities that do not "usememory". However, it may be necessary when one needs to analyze the storage used by an algorithm that uses theADT. In that case one needs additional axioms that specify how much memory each ADT instance uses, as afunction of its state, and how much of it is returned to the pool by free.

ExamplesSome common ADTs, which have proved useful in a great variety of applications, are Container Deque List Map Multimap Multiset Priority queue Queue


Set Stack String TreeEach of these ADTs may be defined in many ways and variants, not necessarily equivalent. For example, a stackADT may or may not have a count operation that tells how many items have been pushed and not yet popped.This choice makes a difference not only for its clients but also for the implementation.

ImplementationImplementing an ADT means providing one procedure or function for each abstract operation. The ADT instancesare represented by some concrete data structure that is manipulated by those procedures, according to the ADT'sspecifications.Usually there are many ways to implement the same ADT, using several different concrete data structures. Thus, forexample, an abstract stack can be implemented by a linked list or by an array.An ADT implementation is often packaged as one or more modules, whose interface contains only the signature(number and types of the parameters and results) of the operations. The implementation of the module namely,the bodies of the procedures and the concrete data structure used can then be hidden from most clients of themodule. This makes it possible to change the implementation without affecting the clients.When implementing an ADT, each instance (in imperative-style definitions) or each state (in functional-styledefinitions) is usually represented by a handle of some sort.[3]

Modern object-oriented languages, such as C++ and Java, support a form of abstract data types. When a class is usedas a type, it is a abstract type that refers to a hidden representation. In this model an ADT is typically implemented asclass, and each instance of the ADT is an object of that class. The module's interface typically declares theconstructors as ordinary procedures, and most of the other ADT operations as methods of that class. However, suchan approach does not easily encapsulate multiple representational variants found in an ADT. It also can underminethe extensibility of object-oriented programs. In a pure object-oriented program that uses interfaces as types, typesrefer to behaviors not representations.

Example: implementation of the stack ADTAs an example, here is an implementation of the stack ADT above in the C programming language.Imperative-style interface

An imperative-style interface might be:

typedef struct stack_Rep stack_Rep; /* Type: instance

representation (an opaque record). */

typedef stack_Rep *stack_T; /* Type: handle to a stack

instance (an opaque pointer). */

typedef void *stack_Item; /* Type: value that can be

stored in stack (arbitrary address). */

stack_T stack_create(void); /* Create new stack

instance, initially empty. */

void stack_push(stack_T s, stack_Item e); /* Add an item at the top of

the stack. */

stack_Item stack_pop(stack_T s); /* Remove the top item from

the stack and return it . */


int stack_empty(stack_T ts); /* Check whether stack is

empty. */

This implementation could be used in the following manner:

#include /* Include the stack interface. */

stack_T t = stack_create(); /* Create a stack instance. */

int foo = 17; /* An arbitrary datum. */

t = stack_push(t, &foo); /* Push the address of 'foo' onto the

stack. */

void *e = stack_pop(t); /* Get the top item and delete it from

the stack. */

if (stack_empty(t)) { } /* Do something if stack is empty. */

This interface can be implemented in many ways. The implementation may be arbitrarily inefficient, since the formaldefinition of the ADT, above, does not specify how much space the stack may use, nor how long each operationshould take. It also does not specify whether the stack state t continues to exist after a call s pop(t).In practice the formal definition should specify that the space is proportional to the number of items pushed and notyet popped; and that every one of the operations above must finish in a constant amount of time, independently ofthat number. To comply with these additional specifications, the implementation could use a linked list, or an array(with dynamic resizing) together with two integers (an item count and the array size)

Functional-style interface

Functional-style ADT definitions are more appropriate for functional programming languages, and vice-versa.However, one can provide a functional style interface even in an imperative language like C. For example:

typedef struct stack_Rep stack_Rep; /* Type: stack state

representation (an opaque record). */

typedef stack_Rep *stack_T; /* Type: handle to a stack

state (an opaque pointer). */

typedef void *stack_Item; /* Type: item (arbitrary

address). */

stack_T stack_empty(void); /* Returns the empty stack

state. */

stack_T stack_push(stack_T s, stack_Item x); /* Adds x at the top of s,

returns the resulting state. */

stack_Item stack_top(stack_T s); /* Returns the item

currently at the top of s. */

stack_T stack_pop(stack_T s); /* Remove the top item

from s, returns the resulting state. */

The main problem is that C lacks garbage collection, and this makes this style of programming impractical;moreover, memory allocation routines in C are slower than allocation in a typical garbage collector, thus theperformance impact of so many allocations is even greater.


ADT librariesMany modern programming languages,such as C++ and Java, come with standard libraries that implement severalcommon ADTs, such as those listed above.

Built-in abstract data typesThe specification of some programming languages is intentionally vague about the representation of certain built-indata types, defining only the operations that can be done on them. Therefore, those types can be viewed as "built-inADTs". Examples are the arrays in many scripting languages, such as Awk, Lua, and Perl, which can be regarded asan implementation of the Map or Table ADT.

References[1] Barbara Liskov, Programming with Abstract Data Types, in Proceedings of the ACM SIGPLAN Symposium on Very High Level Languages,

pp. 50--59, 1974, Santa Monica, California[2] Rudolf Lidl (2004). Abrtract Algebra. Springer. ISBN81-8128-149-7., Chapter 7,section 40.[3] Robert Sedgewick (1998). Algorithms in C. Addison/Wesley. ISBN0-201-31452-5., definition 4.4.

Further Mitchell, John C.; Plotkin, Gordon (July 1988). "Abstract Types Have Existential Type" (http:/ / theory. stanford.

edu/ ~jcm/ papers/ mitch-plotkin-88. pdf). ACM Transactions on Programming Languages and Systems 10 (3).

External links Abstract data type (http:/ / www. nist. gov/ dads/ HTML/ abstractDataType. html) in NIST Dictionary of

Algorithms and Data Structures

List 26

ListIn computer science, a list or sequence is an abstract data structure that implements an ordered collection of values,where the same value may occur more than once. An instance of a list is a computer representation of themathematical concept of a finite sequence, that is, a tuple. Each instance of a value in the list is usually called anitem, entry, or element of the list; if the same value occurs multiple times, each occurrence is considered a distinctitem.

A singly linked list structure, implementing a listwith 3 integer elements.

The name list is also used for several concrete data structures that canbe used to implement abstract lists, especially linked lists.

The so-called static list structures allow only inspection andenumeration of the values. A mutable or dynamic list may allowitems to be inserted, replaced, or deleted during the list's existence.

Many programming languages provide support for list data types, and have special syntax and semantics for listsand list operations. A list can often be constructed by writing the items in sequence, separated by commas,semicolons, or spaces, within a pair of delimiters such as parentheses '()', brackets, '[]', braces '{}', or angle brackets''. Some languages may allow list types to be indexed or sliced like array types. In object-oriented programminglanguages, lists are usually provided as instances of subclasses of a generic "list" class. List data types are oftenimplemented using arrays or linked lists of some sort, but other data structures may be more appropriate for someapplications. In some contexts, such as in Lisp programming, the term list may refer specifically to a linked listrather than an array.In type theory and functional programming, abstract lists are usually defined inductively by four operations: nil thatyelds the empty list, cons, which adds an item at the beginning of a list, first, that returns the first element of a list,and rest that returns a list minus its first element. Formally, Peano's natural numbers can be defined as abstract listswith elements of unit type.

OperationsImplementation of the list data structure may provide some of the following operations: a constructor for creating an empty list; an operation for testing whether or not a list is empty; an operation for prepending an entity to a list an operation for appending an entity to a list an operation for determining the first component (or the "head") of a list an operation for referring to the list consisting of all the components of a list except for its first (this is called the

"tail" of the list.)

CharacteristicsLists have the following properties: The size of lists. It indicates how many elements there are in the list. Equality of lists:

In mathematics, sometimes equality of lists is defined simply in terms of object identity: two lists are equal ifand only if they are the same object.

In modern programming languages, equality of lists is normally defined in terms of structural equality of thecorresponding entries, except that if the lists are typed, then the list types may also be relevant.

List 27

Lists may be typed. This implies that the entries in a list must have types that are compatible with the list's type.It is common that lists are typed when they are implemented using arrays.

Each element in the list has an index. The first element commonly has index 0 or 1 (or some other predefinedinteger). Subsequent elements have indices that are 1 higher than the previous element. The last element has index + 1. It is possible to retrieve the element at a particular index. It is possible to traverse the list in the order of increasing index. It is possible to change the element at a particular index to a different value, without affecting any other

elements. It is possible to insert an element at a particular index. The indices of higher elements at that are increased by

1. It is possible to remove an element at a particular index. The indices of higher elements at that are decreased

by 1.

ImplementationsLists are typically implemented either as linked lists (either singly or doubly linked) or as arrays, usually variablelength or dynamic arrays.The standard way of implementing lists, originating with the programming language Lisp, is to have each element ofthe list contain both its value and a pointer indicating the location of the next element in the list. This results in eithera linked list or a tree, depending on whether the list has nested sublists. Some older Lisp implementations (such asthe Lisp implementation of the Symbolics 3600) also supported "compressed lists" (using CDR coding) which had aspecial internal representation (invisible to the user). Lists can be manipulated using iteration or recursion. Theformer is often preferred in imperative programming languages, while the latter is the norm in functional languages.Lists can be implemented as self-balancing binary search trees holding index-value pairs, providing equal-timeaccess to any element (e.g. all residing in the fringe, and internal nodes storing the right-most child's index, used toguide the search), taking the time logarithmic in the list's size, but as long as it doesn't change much will provide theillusion of random access and enable swap, prefix and append operations in logarithmic time as well.

Programming language supportSome languages do not offer a list data structure, but offer the use of associative arrays or some kind of table toemulate lists. For example, Lua provides tables. Although Lua stores lists that have numerical indices as arraysinternally, they still appear as hash tables.In Lisp, lists are the fundamental data type and can represent both program code and data. In most dialects, the list ofthe first three prime numbers could be written as (list 2 3 5). In several dialects of Lisp, including Scheme, alist is a collection of pairs, consisting of a value and a pointer to the next pair (or null value), making a singly linkedlist.

ApplicationsAs the name implies, lists can be used to store a list of records. The items in a list can be sorted for the purpose offast search (binary search).Because in computing, lists are easier to realize than sets, a finite set in mathematical sense can be realized as a listwith additional restrictions, that is, duplicate elements are disallowed and such that order is irrelevant. If the list issorted, it speeds up determining if a given item is already in the set but in order to ensure the order, it requires moretime to add new entry to the list. In efficient implementations, however, sets are implemented using self-balancingbinary search trees or hash tables, rather than a list.

List 28

Abstract definitionThe abstract list type L with elements of some type E (a monomorphic list) is defined by the following functions:

nil: () Lcons: E L Lfirst: L Erest: L L

with the axiomsfirst (cons (e, l)) = erest (cons (e, l)) = l

for any element e and any list l. It is implicit thatcons (e, l) lcons (e, l) econs (e1, l1) = cons (e2, l2) if e1 = e2 and l1 = l2

Note that first (nil ()) and rest (nil ()) are not defined.These axioms are equivalent to those of the abstract stack data type.In type theory, the above definition is more simply regarded as an inductive type defined in terms of constructors: niland cons. In algebraic terms, this can be represented as the transformation 1 + E L L. first and rest are thenobtained by pattern matching on the cons constructor and separately handling the nil case.

The list monadThe list type forms a monad with the following functions (using E* rather than L to represent monomorphic lists withelements of type E):

where append is defined as:

Alternatively, the monad may be defined in terms of operations return, fmap and join, with:

Note that fmap, join, append and bind are well-defined, since they're applied to progressively deeper arguments ateach recursive call.The list type is an additive monad, with nil as the monadic zero and append as monadic sum.Lists form a monoid under the append operation. The identity element of the monoid is the empty list, nil. In fact,this is the free monoid over the set of list elements.

Stack 29

Stack

Simple representation of a stack

In computer science, a stack is a last in, first out (LIFO) abstractdata type and data structure. A stack can have any abstract datatype as an element, but is characterized by only three fundamentaloperations: push, pop and stack top. The push operation adds anew item to the top of the stack, or initializes the stack if it isempty. If the stack is full and does not contain enough space toaccept the given item, the stack is then considered to be in anoverflow state. The pop operation removes an item from the top ofthe stack. A pop either reveals previously concealed items, orresults in an empty stack, but if the stack is empty then it goes intounderflow state (It means no items are present in stack to beremoved). The stack top operation gets the data from the top-most position and returns it to the user without deletingit. The same underflow state can also occur in stack top operation if stack is empty.

A stack is a restricted data structure, because only a small number of operations are performed on it. The nature ofthe pop and push operations also means that stack elements have a natural order. Elements are removed from thestack in the reverse order to the order of their addition: therefore, the lower elements are those that have been on thestack the longest.[1]

HistoryThe stack was first proposed in 1955, and then patented in 1957, by the German Friedrich L. Bauer.[2] The sameconcept was developed independently, at around the same time, by the Australian Charles Leonard Hamblin.

Abstract definitionA stack is a fundamental computer science data structure and can be defined in an abstract, implementation-freemanner, or it can be generally defined as, Stack is a linear list of items in which all additions and deletion arerestricted to one end that is Top.This is a VDM (Vienna Development Method) description of a stack:[3]

Function signatures:

init: -> Stack

push: N x Stack -> Stack

top: Stack -> (N U ERROR)

remove: Stack -> Stack

isempty: Stack -> Boolean

(where N indicates an element (natural numbers in this case), and U indicates set union)Semantics:

top(init()) = ERROR

top(push(i,s)) = i

remove(init()) = init()

remove(push(i, s)) = s

isempty(init()) = true

isempty(push(i, s)) = false

Stack 30

Inessential operationsIn modern computer languages, the stack is usually implemented with more operations than just "push","pop" and"Stack Top". Some implementations have a function which returns the current number of items on the stack.Alternatively, some implementations have a function that just returns if the stack is empty. Another typical helperoperation stack top[4] (also known as peek) can return the current top element of the stack without removing it.

Software stacks

ImplementationIn most high level languages, a stack can be easily implemented either through an array or a linked list. Whatidentifies the data structure as a stack in either case is not the implementation but the interface: the user is onlyallowed to pop or push items onto the array or linked list, with few other helper operations. The following willdemonstrate both implementations, using C.

Array

The array implementation aims to create an array where the first element (usually at the zero-offset) is the bottom.That is, array[0] is the first element pushed onto the stack and the last element popped off. The program mustkeep track of the size, or the length of the stack. The stack itself can therefore be effectively implemented as atwo-element structure in C:

typedef struct {

size_t size;

int items[STACKSIZE];

} STACK;

The push() operation is used both to initialize the stack, and to store values to it. It is responsible for inserting(copying) the value into the ps->items[] array and for incrementing the element counter (ps->size). In aresponsible C implementation, it is also necessary to check whether the array is already full to prevent an overrun.

void push(STACK *ps, int x)

{

if (ps->size == STACKSIZE) {

fputs("Error: stack overflow\n", stderr);

abort();

} else

ps->items[ps->size++] = x;

}

The pop() operation is responsible for removing a value from the stack, and decrementing the value ofps->size. A responsible C implementation will also need to check that the array is not already empty.

int pop(STACK *ps)

{

if (ps->size == 0){

fputs("Error: stack underflow\n", stderr);

abort();

} else

return ps->items[--ps->size];

}

Stack 31

If we use a dynamic array, then we can implement a stack that can grow or shrink as much as needed. The size of thestack is simply the size of the dynamic array. A dynamic array is a very efficient implementation of a stack, sinceadding items to or removing items from the end of a dynamic array is amortized O(1) time.

Linked list

The linked-list implementation is equally simple and straightforward. In fact, a simple singly linked list is sufficientto implement a stackit only requires that the head node or element can be removed, or popped, and a node canonly be inserted by becoming the new head node.Unlike the array implementation, our structure typedef corresponds not to the entire stack structure, but to a singlenode:

typedef struct stack {

int data;

struct stack *next;

} STACK;

Such a node is identical to a typical singly linked list node, at least to those that are implemented in C.The push() operation both initializes an empty stack, and adds a new node to a non-empty one. It works byreceiving a data value to push onto the stack, along with a target stack, creating a new node by allocating memory forit, and then inserting it into a linked list as the new head:

void push(STACK **head, int value)

{

STACK *node = malloc(sizeof(STACK)); /* create a new node */

if (node == NULL){

fputs("Error: no space available for node\n", stderr);

abort();

} else { /* initialize node */

node->data = value;

node->next = empty(*head) ? NULL : *head; /* insert new head if

any */

*head = node;

}

}

A pop() operation removes the head from the linked list, and assigns the pointer to the head to the previous secondnode. It checks whether the list is empty before popping from it:

int pop(STACK **head)

{

if (empty(*head)) { /* stack is empty */

fputs("Error: stack underflow\n", stderr);

abort();

} else { /* pop a node */

STACK *top = *head;

int value = top->data;

*head = top->next;

free(top);

Stack 32

return value;

}

}

Stacks and programming languagesSome languages, like LISP and Python, do not call for stack implementations, since push and pop functions areavailable for any list. All Forth-like languages (such as Adobe PostScript) are also designed around language-definedstacks that are directly visible to and manipulated by the programmer.C++'s Standard Template Library provides a "stack" templated class which is restricted to only push/popoperations. Java's library contains a Stack class that is a specialization of Vector---this could be considered adesign flaw, since the inherited get() method from Vector ignores the LIFO constraint of the Stack. PHP has anSplStack [5] class.

Hardware stacksA common use of stacks at the architecture level is as a means of allocating and accessing memory.

Basic architecture of a stack

Stack 33

A typical stack, storing local data and call information for nested procedure calls (notnecessarily nested procedures!). This stack grows downward from its origin. Thestack pointer points to the current topmost datum on the stack. A push operation

decrements the pointer and copies the data to the stack; a pop operation copies datafrom the stack and then increments the pointer. Each procedure called in the program

stores procedure return information (in yellow) and local data (in other colors) bypushing them onto the stack. This type of stack implementation is extremely

common, but it is vulnerable to buffer overflow attacks (see the text).

A typical stack is an area of computermemory with a fixed origin and a variablesize. Initially the size of the stack is zero.A stack pointer, usually in the form of ahardware register, points to the mostrecently referenced location on the stack;when the stack has a size of zero, the stackpointer points to the origin of the stack.

The two operations applicable to all stacksare: a push operation, in which a data item

is placed at the location pointed to bythe stack pointer, and the address in thestack pointer is adjusted by the size ofthe data item;

a pop or pull operation: a data item atthe current location pointed to by thestack pointer is removed, and the stackpointer is adjusted by the size of thedata item.

There are many variations on the basicprinciple of stack operations. Every st

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