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CSC 222: Object-Oriented ProgrammingSpring 2013

Searching and sorting sequential search vs. binary search algorithm analysis: big-Oh, rate-of-growth O(N2) sorts: insertion sort, selection sort O(N log N) sorts: merge sort, quick sort recursion

base case, recursive case

recursion vs. iteration

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Searching a list

suppose you have a list, and want to find a particular item, e.g., lookup a word in a dictionary find a number in the phone book locate a student's exam from a pile

searching is a common task in computing searching a database checking a login password lookup the value assigned to a variable in memory

if the items in the list are unordered (e.g., added at random) desired item is equally likely to be at any point in the list need to systematically search through the list, check each entry until found

sequential search

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Sequential search

sequential search traverses the list from beginning to end check each entry in the list if matches the desired entry, then FOUND (return its index) if traverse entire list and no match, then NOT FOUND (return -1)

recall: the ArrayList class has an indexOf method

/** * Performs sequential search on the array field named items * @param desired item to be searched for * @returns index where desired first occurs, -1 if not found */public int indexOf(T desired) { for(int k=0; k < this.items.length; k++) { if (desired.equals(this.items[k])) { return k; } } return -1;}

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How efficient is sequential search?

in the worst case: the item you are looking for is in the last position of the list (or not found) requires traversing and checking every item in the list

if 100 or 1,000 entries NO BIG DEAL if 10,000 or 100,000 entries NOTICEABLE

for this algorithm, the dominant factor in execution time is checking an item the number of checks will determine efficiency

in the average case?

in the best case?

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Big-Oh notation

to represent an algorithm’s performance in relation to the size of the problem, computer scientists use Big-Oh notation

an algorithm is O(N) if the number of operations required to solve a problem is proportional to the size of the problem

sequential search on a list of N items requires roughly N checks (+ other constants) O(N)

for an O(N) algorithm, doubling the size of the problem requires double the amount of work (in the worst case)

if it takes 1 second to search a list of 1,000 items, then it takes 2 seconds to search a list of 2,000 items

it takes 4 seconds to search a list of 4,000 itemsit takes 8 seconds to search a list of 8,000 items. . .

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Searching an ordered list

when the list is unordered, can't do any better than sequential search but, if the list is ordered, a better alternative exists

e.g., when looking up a word in the dictionary or name in the phone book can take ordering knowledge into account pick a spot – if too far in the list, then go backward; if not far enough, go forward

binary search algorithm check midpoint of the list if desired item is found there, then DONE if the item at midpoint comes after the desired item in the ordering scheme, then

repeat the process on the left half if the item at midpoint comes before the desired item in the ordering scheme, then

repeat the process on the right half

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Binary search

/** * Performs binary search on a sorted list. * @param items sorted list of Comparable items * @param desired item to be searched for * @returns index where desired first occurs, -(insertion point)-1 if not found */public static <T extends Comparable<? super T>> int binarySearch(List<T> items, Comparable desired) { int left = 0; // initialize range where desired could

be int right = items.length-1;

while (left <= right) { int mid = (left+right)/2; // get midpoint value and compare int comparison = desired.compareTo(items[mid]);

if (comparison == 0) { // if desired at midpoint, then DONE return mid; } else if (comparison < 0) { // if less than midpoint, focus on left

half right = mid-1; } else { // otherwise, focus on right half left = mid + 1; } } return /* CLASS EXERCISE */ ; // if reduce to empty range, NOT FOUND}

the Collections utility class contains a binarySearch method takes a List of Comparable items and the desired item

List is an interface that specifies basic list operations (ArrayList implements)Comparable is an interface that requires compareTo method (String implements)

MORE ON INTERFACES LATER

this ugly code simply ensures that the list contains Comparable objects

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Visualizing binary search

note: each check reduces the range in which the item can be found by half see http://balance3e.com/Ch8/search.html for demo

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How efficient is binary search?

in the worst case: the item you are looking for is in the first or last position of the list (or not found)

start with N items in listafter 1st check, reduced to N/2 items to searchafter 2nd check, reduced to N/4 items to searchafter 3rd check, reduced to N/8 items to search. . .after log2 N checks, reduced to 1 item to search

again, the dominant factor in execution time is checking an item the number of checks will determine efficiency

in the average case?

in the best case?

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Big-Oh notation

an algorithm is O(log N) if the number of operations required to solve a problem is proportional to the logarithm of the size of the problem

binary search on a list of N items requires roughly log2 N checks (+ other constants) O(log N)

for an O(log N) algorithm, doubling the size of the problem adds only a constant amount of work

if it takes 1 second to search a list of 1,000 items, then searching a list of 2,000 items will take time to check midpoint + 1 second searching a list of 4,000 items will take time for 2 checks + 1 second searching a list of 8,000 items will take time for 3 checks + 1 second

. . .

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Comparison: searching a phone book

Number of entries in

phone book

Number of checks performed by

sequential search

Number of checks performed by binary search

100 100 7

200 200 8

400 400 9

800 800 10

1,600 1,600 11

… … …

10,000 10,000 14

20,000 20,000 15

40,000 40,000 16

… … …

1,000,000 1,000,000 20

to search a phone book of the United States (~310 million) using binary search?

to search a phone book of the world (7 billion) using binary search?

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Dictionary revisited

public class Dictionary { private ArrayList<String> words;

. . .

public boolean addWord(String newWord) { int index = Collections.binarySearch(this.words, newWord.toLowerCase())

this.words.add(Math.abs(index)-1, newWord.toLowerCase());

return true; }

public boolean addWordNoDupes(String newWord) { int index = Collections.binarySearch(this.words, newWord.toLowerCase()); if (index < 0) {

this.words.add(Math.abs(index)-1, newWord.toLowerCase()); return true;}return false;

}

public boolean findWord(String desiredWord) { return (Collections.binarySearch(this.words, desiredWord.toLowerCase()) >= 0); }}

binary search works as long as the list of words is sorted• dictionary.txt is sorted, so can load the dictionary and do searches• to ensure correct behavior, must also make sure that add methods maintain sorting

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In the worst case…

suppose words are added in reverse order: "zoo", "moo", "foo", "boo"

zoo

to add "moo", must first shift "zoo" one spot to the right

moo zoo

to add "foo", must first shift "moo" and "zoo" each one spot to the right

foo moo zoo

to add "boo", must first shift "foo", "moo" and "zoo" each one spot to the right

boo foo moo zoo

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Worst case (in general)

if inserting N items in reverse order 1st item inserted directly 2nd item requires 1 shift, 1 insertion 3rd item requires 2 shifts, 1 insertion . . . Nth item requires N-1 shifts, 1 insertion---------------------------------------------------------(1 + 2 + 3 + … + N-1) = N(N-1)/2 = (N2 – N)/2 shifts + N insertions

this approach is called "insertion sort" insertion sort builds a sorted list by repeatedly inserting items in correct order

since an insertion sort of N items can take roughly N2 steps, it is an O(N2) algorithm

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Timing the worst case

# items (N) time in msec 5,000 1510,000 4920,000 16240,000 65180,000 2270

160,000 9168 320,000 36463

System.currentTimeMillis method accesses the system clock and returns the time (in milliseconds)

• we can use it to time repeated adds to a dictionary

public class TimeDictionary { public static int timeAdds(int numValues) { Dictionary dict = new Dictionary();

long startTime = System.currentTimeMillis(); for (int i = numValues; i > 0; i--) { String word = "0000000000" + i; dict.addWord(word.substring(word.length()-10)); } long endTime = System.currentTimeMillis();

return (int)(endTime-startTime); }}

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O(N2) performance

as the problem size doubles, the time can quadruple

makes sense for an O(N2) algorithm if X items, then X2 steps required if 2X items, then (2X)2 = 4X2 steps

QUESTION: why is the factor of 4 not realized immediately?

Big-Oh captures rate-of-growth behavior in the long run• when determining Big-Oh, only the dominant factor is significant (in the long run)

cost = N(N-1)/2 shifts (+ N inserts + additional operations) O(N2)N=1,000: 499,500 shifts + 1,000 inserts + … overhead cost is significantN=100,000: 4,999,950,000 shifts + 100,000 inserts + … only N2 factor is significant

# items (N) time in msec 5,000 1510,000 4920,000 16240,000 65180,000 2270

160,000 9168 320,000 36463

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Best case for insertion sort

while insertion sort can require ~N2 steps in worst case, it can do much better sometimes BEST CASE: if items are added in order,

then no shifting is required only requires N insertion steps, so O(N)

if double size, roughly double time

on average, might expect to shift only half the time (1 + 2 + … + N-1)/2 = N(N-1)/4 = (N2 – N)/4 shifts, so still O(N2)

would expect faster timings than worst case, but still quadratic growth

list size (N) time in msec40,000 3280,000 79

160,000 194 320,000 400

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Timing insertion sort (average case)

import java.util.Random;

public class TimeDictionary { public static long timeAdds(int numValues) { Dictionary1 dict = new Dictionary1();

Random randomizer = new Random();

long startTime = System.currentTimeMillis(); for (int i = 0; i < numValues; i++) { String word = "0000000000" + randomizer.nextInt(); dict.addWord(word.substring(word.length()-10)); } long endTime = System.currentTimeMillis();

return (endTime - startTime); }}

can use a Random object to pick random numbers and add to a String

list size (N) time in msec10,000 8720,000 11940,000 39780,000 1420

160,000 5306 320,000 20442

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A more generic insertion sortwe can code insertion sort independent of the Dictionary class

could use a temporary list for storing the sorted numbers, but not needed

don't stress about <T extends Comparable<? super T>> specifies that the parameter must be an ArrayList of items that either implements or extends a class that

implements the Comparable interface (???) more later, for now, it ensures the class has a compareTo method

public static <T extends Comparable<? super T>> void insertionSort(ArrayList<T> items) { for (int i = 1; i < items.size(); i++) { // for each index i, T itemToPlace = items.get(i); // save the value at index i int j = i; // starting at index i, while (j > 0 && itemToPlace.compareTo(items.get(j-1)) < 0) { items.set(j, items.get(j-1)); // shift values to the right j--; // until find spot for the value } items.set(j, itemToPlace); // store the value in its spot } }

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Other O(N2) sorts

alternative algorithms exist for sorting a list of items

e.g., selection sort: find smallest item, swap into the 1st index find next smallest item, swap into the 2nd index find next smallest item, swap into the 3rd index . . .

public static <T extends Comparable<? super T>> void selectionSort(ArrayList<T> items) { for (int i = 0; i < items.size()-1; i++) { // for each index i, int indexOfMin = i; // find the ith smallest item for (int j = i+1; j < items.size(); j++) { if (items.get(j).compareTo(items.get(indexOfMin)) < 0) { indexOfMin = j; } } T temp = items.get(i); // swap the ith smallest items.set(i, items.get(indexOfMin)); // item into position i items.set(indexOfMin, temp); } }

HW5: Hunt the Wumpus

you are to implement a text-based adventure game from the 70's

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Cave class

you must implement a class that models a single cave each cave has a name & number, and is connected to three other caves via tunnels by default, caves are empty & unvisited (although these can be updated)

how do we represent the cave contents? we could store the contents as a string: "EMPTY", "WUMPUS", "BATS", "PIT"

Cave c = new Cave("Cavern of Doom", 0, 1, 2, 3);c.setContents("WUMPUS");

potential problems?

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there are only 4 possible values for cave contents the trouble with using a String to represent these is no error checking

c.setContents("WUMPAS"); // perfectly legal, but ???

Enumerated types

there is a better alternative for when there is a small, fixed number of values an enumerated type is a new type (class) whose value are explicitly enumerated

public enum CaveContents { EMPTY, WUMPUS, PIT, BATS}

note that these values are NOT Strings – they do not have quotes

you specify a enumerated type value by ENUMTYPE.VALUE

c.setContents(CaveContents.WUMPUS);

since an enumerated type has a fixed number of values, any invalid input would be caught by the compiler

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CaveMaze

the CaveMaze class reads in & stores a maze of cavessince the # of caves is set, simpler to use an array

provided version only allows limited movementyou must add functionality

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public class CaveMaze { private Cave[] caves; private Cave currentCave; private boolean alive; public CaveMaze(String filename) throws java.io.FileNotFoundException { Scanner infile = new Scanner(new File(filename)); int numCaves = infile.nextInt(); this.caves = new Cave[numCaves]; for (int i = 0; i < numCaves; i++) { int num1 = infile.nextInt(); int num2 = infile.nextInt(); int num3 = infile.nextInt(); int num4 = infile.nextInt(); String name = infile.nextLine().trim(); this.caves[num1] = new Cave(name, num1, num2, num3, num4); } this.alive = true; this.currentCave = this.caves[0]; this.currentCave.markAsVisited(); }

. . .}

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O(N log N) sorts

there are sorting algorithms that do better than insertion & selection sorts

merge sort & quick sort are commonly used O(N log N) sorts recall from sequential vs. binary search examples:

when N is large, log N is much smaller than N thus, when N is large, N log N is much smaller than N2

N N log N N2

1,000 10,000 1,000,000 2,000 22,000 4,000,000 4,000 48,000 16,000,000 8,000 104,000 64,000,00016,000 224,000 256,000,00032,000 480,000 1,024,000,000

they are both recursive algorithmsi.e., each breaks the list into pieces, calls itself to sort the smaller pieces, and combines the results

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Recursion

a recursive algorithm is one that refers to itself when solving a problem to solve a problem, break into smaller instances of problem, solve & combine recursion can be a powerful design & problem-solving technique

examples: binary search, merge sort, hierarchical data structures, …

classic (but silly) examples:

Fibonacci numbers:1st Fibonacci number = 12nd Fibonacci number = 1Nth Fibonacci number = (N-1)th Fibonacci number + (N-2)th Fibonacci number

Euclid's algorithm to find the Greatest Common Divisor (GCD) of a and b (a ≥ b)• if a % b == 0, the GCD(a, b) = b• otherwise, GCD(a, b) = GCD(b, a % b)

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Recursive methods

these are classic examples, but pretty STUPID both can be easily implemented using iteration (i.e., loops) recursive approach to Fibonacci has huge redundancy

we will look at better examples later, but first analyze these simple ones

/** * Computes Nth Fibonacci number. * @param N sequence index * @returns Nth Fibonacci number */public int fibonacci(int N){ if (N <= 2) { return 1; } else { return fibonacci(N-1) + fibonacci(N-2); }}

/** * Computes Greatest Common Denominator. * @param a a positive integer * @param b positive integer (a >= b) * @returns GCD of a and b */ public int GCD(int a, int b) { if (a % b == 0) { return b; } else { return GCD(b, a%b); }}

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Understanding recursion

every recursive definition has 2 parts:BASE CASE(S): case(s) so simple that they can be solved directlyRECURSIVE CASE(S): more complex – make use of recursion to solve smaller

subproblems & combine into a solution to the larger problem

int fibonacci(int N){ if (N <= 2) { // BASE CASE return 1; } else { // RECURSIVE CASE return fibonacci(N-1) + fibonacci(N-2); }}

int GCD(int a, int b) { if (a % b == 0) { // BASE CASE return b; } else { // RECURSIVE return GCD(b, a%b); }}

to verify that a recursive definition works: convince yourself that the base case(s) are handled correctly ASSUME RECURSIVE CALLS WORK ON SMALLER PROBLEMS, then convince

yourself that the results from the recursive calls are combined to solve the whole

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Avoiding infinite(?) recursion

to avoid infinite recursion: must have at least 1 base case (to terminate the recursive sequence) each recursive call must get closer to a base case

int fibonacci(int N){ if (N <= 2) { // BASE CASE return 1; } else { // RECURSIVE CASE return fibonacci(N-1) + fibonacci(N-2); }}

int GCD(int a, int b) { if (a % b == 0) { // BASE CASE return b; } else { // RECURSIVE return GCD(b, a%b); }}

with each recursive call, the number is getting smaller closer to base case (≤ 2)

with each recursive call, a & b are getting smaller closer to base case (a % b == 0)

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Merge sort

a better example of recursion is merge sortBASE CASE: to sort a list of 0 or 1 item, DO NOTHING!RECURSIVE CASE:

1. Divide the list in half2. Recursively sort each half using merge sort3. Merge the two sorted halves together

12 9 6 20 3 15

12 9 6 20 3 151.

6 9 12 3 15 202.

3 6 9 12 15 203.

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Merging two sorted lists

merging two lists can be done in a single pass since sorted, need only compare values at front of each, select smallest requires additional list structure to store merged items

public <T extends Comparable<? super T>> void merge(ArrayList<T> items, int low, int high) { ArrayList<T> copy = new ArrayList<T>();

int size = high-low+1; int middle = (low+high+1)/2; int front1 = low; int front2 = middle; for (int i = 0; i < size; i++) { if (front2 > high || (front1 < middle && items.get(front1).compareTo(items.get(front2)) < 0)) { copy.add(items.get(front1)); front1++; } else { copy.add(items.get(front2)); front2++; } }

for (int k = 0; k < size; k++) { items.set(low+k, copy.get(k)); }}

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Merge sort

once merge has been written, merge sort is simple for recursion to work, need to be able to specify range to be sorted initially, want to sort the entire range of the list (index 0 to list size – 1) recursive call sorts left half (start to middle) & right half (middle to end) . . .

private <T extends Comparable<? super T>> void mergeSort(ArrayList<T> items, int low, int high) { if (low < high) { int middle = (low + high)/2; mergeSort(items, low, middle); mergeSort(items, middle+1, high); merge(items, low, high); }}

public <T extends Comparable<? super T>> void mergeSort(ArrayList<T> items) { mergeSort(items, 0, items.size()-1);}

note: private helper method does the recursion; public method calls the helper with appropriate inputs

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Recursive analysis of a recursive algorithm

cost of sorting N items = cost of sorting left half (N/2 items) + cost of sorting right half (N/2 items) +cost of merging (N items)

more succinctly: Cost(N) = 2*Cost(N/2) + C1*N

Cost(N) = 2*Cost(N/2) + C1*N can unwind Cost(N/2)= 2*( 2*Cost(N/4) + C2*N/2 ) + C1*N= 4*Cost(N/4) + (C1 + C2)*N can unwind Cost(N/4)= 4*( 2*Cost(N/8) + C3*N/4 ) + (C1 + C2)*N= 8*Cost(N/8) + (C1 + C2 + C3)*N can continue unwinding= …= N*Cost(1) + (C1 + C2/2 + C3/4 + … + ClogN/N)*N= (C0 + C1 + C2 + C3 + … + ClogN )*N where C0 = Cost(1)≤ (max(C0, C1,…, ClogN)*log N) * N = C * N log N where C = max(C0, C1,…, ClogN) O(N log N)

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Dictionary revisited

recall most recent version of Dictionary inserts each new word in order (i.e., insertion sort) & utilizes binary search

searching is fast (binary search), but adding is slow

N adds + N searches: N*O(N) + N*O(log N) = O(N2) + O(N log N) = O(N2)

if you are going to do lots of adds in between searches: simply add each item at the end O(1) before the first search, must sort – could use merge sort

N adds + sort + N searches: N*O(1) + O(N log N) + N*log(N) = O(N log N)

Collections class contains a sort method that implements quick sort in practice, quick sort is a faster O(N log N) sort than merge sort

1. picks a pivot element from the list (can do this at random or be smarter)2. partitions the list so that all items ≤ pivot are to left, al items > pivot are to right3. recursively (quick) sorts the partitions

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Modified Dictionary classpublic class Dictionary2 {

private ArrayList<String> words;private boolean isSorted;

public Dictionary2() { this.words = new ArrayList<String>(); this.isSorted = true;}

public Dictionary2(String fileName) { this(); try { Scanner infile = new Scanner(new File(fileName));

while (infile.hasNext()) { String nextWord = infile.next(); this.addWord(nextWord); } infile.close(); } catch (java.io.FileNotFoundException e) { System.out.println("No such file: " +

fileName); }

} public boolean addWord(String newWord) { this.words.add(newWord.toLowerCase()); this.isSorted = false; return true;}

public boolean findWord(String desiredWord) { if (!this.isSorted) { Collections.sort(this.words); this.isSorted = true; } return Collections.binarySearch(this.words,

desiredWord.toLowerCase()) >= 0;}

. . .}

the isSorted field keeps track of whether the list is sorted (i.e., no addWords have been performed since last findWord)

we could do a little more work in addWord to avoid unnecessary sorts

note: gives better performance if N adds are followed by N searches

what if the adds & searches alternate?

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Dictionary2 timings

# items (N) Dictionary1 (msec) Dictionary2 (msec)

100,000 2121 176 200,000 8021 386 400,000 31216 864

import java.util.Random;

public class TimeDictionary { public static int timeAdds(int numValues) { Dictionary2 dict = new Dictionary2();

Random randomizer = new Random();

long startTime = System.currentTimeMillis(); for (int i = 0; i < numValues; i++) { String word = "0000000000" + randomizer.nextInt(); dict.addWord(word.substring(word.length()-10)); } for (int i = 0; i < numValues; i++) { dict.findWord("zzz"); } long endTime = System.currentTimeMillis();

return (int)(endTime-startTime); } }

N adds followed by N searches:

• Dictionary1 used insertion sort & binary search

• O(N2) + O(N log N) O(N2)

• Dictionary2 uses add-at-end, quick sort before first search, then binary search

• O(N) + O(N log N) + O(N log N) O(N log N)

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Recursion vs. iteration

it wouldn't be difficult to code fibonacci and GCD without recursion

public int fibonacci(int N) { public int GCD(int a, int b) { int previous = 1; while (a % b != 0) { int current = 1; int temp = b; for (int i = 3; i <= N; i++) { b = a % b; int newCurrent = current + previous; a = temp; previous = current; } current = newCurrent; return b; } }

return current; }

in theory, any recursive algorithm can be rewritten iteratively (using a loop) but sometimes, a recursive definition is MUCH clearer & MUCH easier to write

e.g., merge sort

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Recursion & efficiency

there is some overhead cost associated with recursionpublic int fibonacci(int N) { public int GCD(int a, int b) { int previous = 1; while (a % b != 0) { int current = 1; int temp = b; for (int i = 3; i <= N; i++) { b = a % b; int newCurrent = current + previous; a = temp; previous = current; } current = newCurrent; return b; } }

return current; }

with recursive version: each refinement requires a method callinvolves saving current execution state, allocating memory for the method instance, allocating and initializing parameters, returning value, …

with iterative version: each refinement involves a loop iteration + assignments

the cost of recursion is relatively small, so usually no noticeable difference in practical terms, there is a limit to how deep recursion can go

e.g., can't calculate the 10 millionth fibonacci number in the rare case that recursive depth can be large (> 1,000), consider iteration

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Recursion & redundancy

in the case of GCD, there is only a minor efficiency difference number of recursive calls = number of loop iterations

this is not always the case efficiency can be significantly different(due to different underlying algorithms)

consider the recursive fibonacci method:

fibonacci(5)

fibonacci(4) + fibonacci(3)

fibonacci(3) + fibonacci(2) fibonacci(2) + fibonacci(1)

fibonacci(2) + fibonacci(1)

there is a SIGNIFICANT amount of redundancy in the recursive versionnumber of recursive calls > number of loop iterations (by an exponential amount!)

recursive version is MUCH slower than the iterative onein fact, it bogs down on relatively small values of N