Shortest path algorithms, Java collections framework and AVL tree

8/10/2019 Shortest path algorithms, Java collections framework and AVL tree

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AVL TREE

Self balancing binary trees

aself-balancing (or height-balanced) binary search tree is anynode-

basedbinary search tree that automatically keeps its height (maximal number

of levels below the root) small in the face of arbitrary item insertions and

deletions

Since we know that on binary search trees (BST), most of the operations take

time in accordance with the height of the tree, hence we strive to keep its

height minimal with respect to the number of elements entered. Andoperations hence take less time to perform. So we dont have to worry about

the performance overhead in search, however insertion takes more time,

since we know that insertion is less performed operation than search, so this

approach is acceptable in such a scenario.

Avl tree.

Introduction:AVL tree is a self-balancing binary seach tree [BST], which was introduced

in1962 in a research paper (An algorithm for the organization of

information) by Soviet mathematicians, namely:

Georgy Adelson-Velsky

Evgenii Mikhailovich Landis

Time complexity

Worst case time complexity for search, delete and insert isO (log n).

where n= the total number of nodes in the tree.

However rebalancing after insertion and deletion may take some extra time

which has not been accounted for in this time complexity analysis.

Total time taken for adjustment and rotation is O(1).
http://en.wikipedia.org/wiki/Node_(computer_science)http://en.wikipedia.org/wiki/Binary_search_treehttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Binary_search_treehttp://en.wikipedia.org/wiki/Node_(computer_science)


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Working of AVL tree:

AVL tree is used in applications in which search is the most performed

operation. after each insertion or deletion, it is checked from the current place

of insertion to the root. if the braches are sagging more than one level from

the level of other branches, then the rotation is done.

Balance factor:

Balance factor of an AVL tree is calculated by following formula at each node:

Height of left subtree Height of right subtree

If the balancing factor is less than -2 or more than 2 then rotation is done in

case of mildly balanced tree in order to reduce the imbalance in the tree,

however in strictly balanced trees, height of both subtrees should be equal.

Rotation mechanism

Left rotation

Consider the tree shown in the diagram on the right,

we can see that it is unbalanced on node 4 accordingto the rule of balance factor, what we will have to

rotate the neighboring nodes so balance could be

achieved, and AVL tree will remain in its healthy

state, and does not degenerate into an unbalanced

tree.

Now, since we know that the AVL tree still workon the principle that:

left child < parent < right child

so after the rotation, we can see (in the picture on

the right) that node 5 has taken the place of parent


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instead of node 4 which has moved to the status of left child. And node 6 has

been moved to the place of right child.

Right rotation:

Now as indicated in the picture that an element is

added below the leaf X, so we know that the rotation is

compulsory since the tree will be unbalanced. hence to

balance the tree, we have to move node m to the right

hand side f the node n, such that node Z will become

the child of node m. and since node X is already at

height+1, it will be appended directly below the root,

so as to balance the height on both left and right sub trees.

Node Y has remained a child of node m. In the scheme of

rotation, we can see that it has been strived to preserve the

parent child relation, in order to simplify the process.


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Advantages of using AVL tree:

1.

Time complexity for search is O(log N) since AVL trees are balanced at

all times.

2.

Operations of insertion and deletion also have a time complexity of

O(log n)

3.

Speed of insertion is not so much impeded by the balancing operation

since it only adds a constant time factor to the operation.

Disadvantages of using AVL tree:1.

AVL tree is difficult to realize in form of code; more space is required for

balancing factor.

2.

AVL tree serves to enable faster operations but rebalancing is a costly

operation in terms of time.

References

Donald Knuth.The Art of Computer Programming, Volume 3: Sorting and

Searching, Second Edition. Addison-Wesley, 1998.

Wikipedia article: Self balancing binary search tree

http://en.wikipedia.org/wiki/Self-balancing_binary_search_tree

Wikipedia article: AVL tree

http://en.wikipedia.org/wiki/AVL_tree

Lectures from University of Washington website
http://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Self-balancing_binary_search_treehttp://en.wikipedia.org/wiki/AVL_treehttp://en.wikipedia.org/wiki/AVL_treehttp://en.wikipedia.org/wiki/Self-balancing_binary_search_treehttp://en.wikipedia.org/wiki/Donald_Knuth


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courses.cs.washington.edu/courses/cse373/04wi/slides/lecture08.ppt

Lectures from University of Wisconsin website

http://pages.cs.wisc.edu/~ealexand/cs367/NOTES/AVL-

Trees/index.html


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Java Collection Framework

What is a collection?Collection is an object that combines/collects multiple elements into a single unit.

What is a collections framework?A collections framework is an architecture which is a combination of different

programming ingredients made for the sole purpose of using collections,

representing them and manipulating them.

A collection framework comprises of these three things:

1. Algorithms:Algorithms are the step by step approach to achieve the

purpose of representing data as we see in a structure theoretically.

2. Implementations:implementations are simply the implementation of the

algorithms which enable the user to use the collection in their own way to

satisfy the needs of a program.

3.

Interfaces:Interfaces are the abstract classes through which we can access

the implementations of the algorithms.

Collections can be used to store, access, manipulate, and communicate aggregate

data or perform different operations on it. Collections represent data items that

belong to a similar group of elements, like:

A dictionary (collection of valid words in a given language)

A deck of cards (collection of valid and playable cards representing a card

system) etc.

Advantages of using JCF (Java Collection Framework)Following some of the advantages that are offered by the used of JCF:


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Reduction in effort required in programming:by providing ready made

and reusable structures to use in the programs, so the programmer does not

have to write those algorithms him/herself.

Increase in performance:by adding in productivity via better

implementations of data structures that are to be used, also that various

interfaces offer different implementations to suit the need of the specific

programs.

Provision of compatibility and interoperability between unrelated APIs:

because using java as a common language these APIs can pass the data

amongst themselves.

Reduces the learning cost:by requiring us to learn different collection

APIs.

Reduces the effort required to design and implement APIs:by not

requiring you to produce ad hoc collections APIs.

Encourages collection reuse:via implementing different algorithms, andinterfaces with which to use and manipulate them to owns benefit.

Architecture and interfaces:

Following are some core collection interfaces:

Collections

Collections are the root structure of collection framework. It represents a

combination of objects known as its members. Some types of collections also

allow similar elements in a set, while others dont. ordered and unordered are

another variety present in the realm of collections. Java platform doesnt have

any implementations as it is for the collections, but set and list are the

subinterfaces that apply the thought behind the collections. Add, clear, clearAll,

retainAll are some of individual and bulk operations that the collection class

exhibits.

Set

Set is a collection of unique elements i.e. it cannot contain duplicate elements.

This interface is modeled after the representation of sets as in mathematics.

ContainsAll, removeAll, retainAll, addAll are some of the bulk operations that can

be performed on sets.


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List

List is an ordered set of elements. It can have similar elements more than once,

i.e. it can have or can not have unique elements. Every element has an index

number and members can be manipulated via these index numbers. Sort, Shuffle,rotate, binarySearch, indexOfSublist are the examples of the operations that can

be performed on the lists.

Queue

This collection is very common. Most common use for this data structure is to

hold elements before they are to be entered in for processing. A queue comprises

of additional methods which are insertion, extraction etc. queues implement FIFO

(first in first out) sequence of the element entry and removal of elements, but this

not a hard and fast rule. Priority queues are the example of such queues which

defy FIFO structure. Insert, remove and examine are some of the examples of

operations that can be performed on the queues.

Dequeues

Dequeues are double ended queues, they can be operated on from both ends of

the pipeline, they implement both FIFO (Fist in first out) and LIFO (Last in first

out). addFirst, addLast, peekFirst, peekLast are some of the examples of the

operations that can be performed on Dequeues.

Map

Map is basically an object which points to values through keys. Duplicate keys

cannot be entered because that would lead to confusing pointing and retrieval of

information. Java implements maps in three basic forms i.e. Treemaps,

linkedHashMap, HashMaps.

Sorted Set

Sorted sets are different from sets in manner that its elements are arranged in

sorted way. Advantage of sorting is taken using different new operations that are

not present in the Set interface. These find application in dictionaries, roll

numbers of students and other different areas of applications.


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Sorted Map

Sorted maps are different from the Maps in the way that their mappings keys are

maintained in ascending order. Otherwise they are same.

Legacy classes:

Legacy classes have been retrofitted into the collections framework, they can be

used but they are not as efficient as newer data structures. however, the list of

present classes have been mentioned below.

Hashtable:newer alternative would be Hashmap.

Enumeration:collection and iterators should be used instead, butCollection.Enumeration library can be used if needed.

Vectors:Arraylist can be used to replace this data structure.

Stack:linkedlists can be used in place of stacks.

Bitset:Arraylist of Boolean type variables can be used to replace the BitSet Data

structure, but obvious disadvantage would be wastage of space, because one bit

would take 8 bits to represent an array.

ReferencesObject Oriented Software Construction (Lecture notes) University of Auckland

https://www.cs.auckland.ac.nz/~hongyul/tut/3_Collections

Chapter no. 22 Java collections framework

Liang, Introduction to Java programming, Seventh edition, Pearson

education
https://www.cs.auckland.ac.nz/~hongyul/tut/3_Collectionshttps://www.cs.auckland.ac.nz/~hongyul/tut/3_Collections


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Java collections, software engineering notes, University of Washington

http://www.cs.washington.edu/education/courses/403/03wi

Wikipedia article: Java collections framework

http://en.wikipedia.org/wiki/Java_collections_framework

Tutorial: Introduction to Java collections (Sun corporation)

https://docs.oracle.com/javase/tutorial/collections/index.html

Java collections main page

https://docs.oracle.com/javase/7/docs/technotes/guides/collections

/index.html

Java collections framework: overview

https://docs.oracle.com/javase/7/docs/technotes/guides/collections

/overview.html

JCF Lesson: Interfaces (The Java collections > Collections)

https://docs.oracle.com/javase/tutorial/collections/interfaces/
http://www.cs.washington.edu/education/courses/403/03wihttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/index.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/index.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/index.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/overview.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/overview.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/overview.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/overview.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/overview.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/index.htmlhttps://docs.oracle.com/javase/7/docs/technotes/guides/collections/index.htmlhttp://www.cs.washington.edu/education/courses/403/03wi


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Shortest path algorithm

Graph theory and directed graphs:In order to understand the shortest path algorithms we will have to understand what

digraphs are. Graph theory is a branch of mathematics, more specifically of discrete

mathematics. Basically the shortest paths using different algorithms are applied to

digraphs which are a part of the graph theory.

A directed graph / digraph is a set of nodes connected by edges / paths which either have

or dont have a direction associated with them.

Graphs are expressed in the notation G=(V,A) where:

V denotes the nodes or in other words, vertices. A is a set of ordered pairs of vertices which are also called edges in a graph, they can

be directed or non-directed.

Analogy of digraphs to road maps

An analogy can be given from the map of a city where the shortest path between various

destinations can be found out by traversing the path between the destinations. Nodes of the

graphs correspond to the intersections on the road map, and edges are road segments,

weights can be assigned to the segments according to the length of the road segments.

Shortest path problem:

Shortest path problems are defined in Wikipedia as:

Ingraph theory,the shortest path problem is the problem of finding apath

between twovertices (or nodes) in agraph such that the sum of theweights of its

constituent edges is minimized.

Categorization of Shortest path problems

Shortest path algorithms can be categorized as follows:

1.

Single pair shortest path problem

2.

single source shortest path problem

3.

single destination shortest path problem

4. all pairs shortest path problem
http://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Path_(graph_theory)http://en.wikipedia.org/wiki/Vertex_(graph_theory)http://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Weighted_graphs_and_networkshttp://en.wikipedia.org/wiki/Glossary_of_graph_theory#Weighted_graphs_and_networkshttp://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Vertex_(graph_theory)http://en.wikipedia.org/wiki/Path_(graph_theory)http://en.wikipedia.org/wiki/Graph_theory


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Shortest path algorithms:

Following are different shortest path algorithms which can be used in different scenarios

and settings, their efficiency can be different based on the type of problem given:

Dijkstra's algorithm

BellmanFord algorithm

A* search algorithm

FloydWarshall algorithm

Johnson's algorithm

Viterbi algorithm

Some of the algorithms will be explained here briefly:

Floyd-Warshallsshortest path algorithm:The FloydWarshall algorithm was published byRobert Floyd in 1962. However, it is

similar to the algorithms previously published byBernard Roy in 1959.

FloydWarshall algorithm is also known as Floyd's algorithm, RoyWarshall algorithm,

RoyFloyd algorithm. it is agraph analysisalgorithm for finding theshortest path in a

weighted digraph with positive or negative edge weights, but comprising of negative cycles.

A single execution will find the lengths of the shortest paths between all paired nodes. Time

complexity in the worst case scenario is O (|V|3)

Algorithm:

1.

it compares all possible paths through the graph between each paired node.

2. It does this with the help of (|V |3) comparisons in a graph.

3. there may be up to (|V |2) edges in the graph, and every combination of edges is

tested.

4.

It does so by step by step improving the estimate on the shortest path between two

vertices.

5. until the estimate is optimum the loop is continued.

Dijkstras shortest path algorithm:

Dijkstra's algorithm was published byEdsger Dijkstra in 1959.

It is agraph search algorithm that solves the single-sourceshortest path problem.It is used

in graphs that cannot use negative edge cost. It produces ashortest path tree.This

algorithm is often used inrouting along with other graph algorithms.
http://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/A*_search_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Johnson%27s_algorithmhttp://en.wikipedia.org/wiki/Viterbi_algorithmhttp://en.wikipedia.org/wiki/Robert_Floydhttp://en.wikipedia.org/wiki/Bernard_Royhttp://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Weighted_graphhttp://en.wikipedia.org/wiki/Edsger_W._Dijkstrahttp://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Edsger_W._Dijkstrahttp://en.wikipedia.org/wiki/Weighted_graphhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Graph_(mathematics)http://en.wikipedia.org/wiki/Bernard_Royhttp://en.wikipedia.org/wiki/Robert_Floydhttp://en.wikipedia.org/wiki/Viterbi_algorithmhttp://en.wikipedia.org/wiki/Johnson%27s_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/A*_search_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm


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A source node is given in the graph in which the shortest path is to be found, the algorithm

finds the path with lowest edge cost which obviously is the shortest path from that node to

every other node in the graph (single source - multiple destination problem). It can also be

used to find the shortest path in single source single destination problems by stopping

the algorithm once the algorithm has reached the destination and shortest path is achieved.

Same road map analogy can be applied here. For example, if we want to find the shortest

distance from our current location to all other destinations or road intersections in the city,

then Dijkstras algorithm can be used, or in the second case as we have discussed, we will

have to stop midway in order to find the shortest path from current location to a specific

location. But the algorithm will also find the shortest paths for all the destinations in

between of that specific path.

Algorithm:

Source node is called initial node. Let the distance of node Y be the distance from the initial

node to Y. Following steps will be followed to find the shortest path from initial node tonode Y.

1. An initial value is set to all nodes. This is 0 for initial node and infinity for all other

nodes.

2.

All nodes are marked unvisited. Initial node is set as current. A set called unvisited

set is created comprising of all the unvisited nodes.

3. Consider all of its unvisited neighbors of the current node and calculate their

tentative distances. Compare the newly calculated distance to the current assigned

value and assign the smaller one, if the previous value was greater.

4.

When the distance is measured for all the neighbors from the current node, set the

current node as visited and eliminate it from the unvisited set.5.

If the destination node is set to visited (in single source single destination

problem) or if the smallest tentative distance from current node to the nodes in

unvisited set is infinity (in single source multiple destination proble, when the

unvisited nodes are disjointed from the current node) then the algorithm has

finished.

6. Select the unvisited node that is marked with the smallest tentative distance, and set

it as the new "current node" then go back to step 3.

Comparison between Floyd-Warshall and Dijkstras

algorithm:

Floyd-Warshall Algorithm:

1.

Finds a shortest-path for all node-pairs (x, y).

2.

We can have one or more links of negative cost, c(x, y)


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3.

no cycle of negative cost. (Assume that c(xi,x)=0for each

4. node xi, which is the same as not having the links (x).)

5.

Complexity: O(N3), where N = number of nodes in the digraph.

Dijkstras shortest path Algorithm:

1. Finds shortest path from a given startNode to all other nodes reachable from it in a

digraph.

2. Assumes that each link cost c(x, y) = 0.

3. Complexity: O(N2), N = number of nodes in the digraph

Applications of Shortest path algorithms:

Maps and navigation Shortest-paths is a broadly useful problem-solving model

Robot navigation.

Texture mapping.

Urban traffic planning.

Optimal pipelining of VLSI chip.

Subroutine in advanced algorithms.

Telemarketer operator scheduling.

Routing of telecommunications messages.

Network routing protocols (OSPF, BGP, RIP)


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References:

Schaums outlines: Discrete mathematics.

3rdedition by Seymor Lipschutz & Marc Lipson

Shortest path problem (Wikipedia article).

http://en.wikipedia.org/wiki/Shortest_path_problem

Princeton University, article on the introduction of Dijkstras algorithm.

http://www .cs.princeton.edu/introalgsds/55dijkstra

Louisiana State University, article on Floyd-Warshalls algorithm.

http://www.csc.lsu.edu/~kundu/dstr/3-floyd.pdf

Floyd-Warshalls shortest path algorithm (Wikipedia Article)

http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm

Dijkstras Algorithm (Wikipedia article)

http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm
http://en.wikipedia.org/wiki/Shortest_path_problemhttp://www.csc.lsu.edu/~kundu/dstr/3-floyd.pdfhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://www.csc.lsu.edu/~kundu/dstr/3-floyd.pdfhttp://en.wikipedia.org/wiki/Shortest_path_problem

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Shortest path algorithms, Java collections framework and AVL tree