Algorithm Design Techniques. Divide and conquer Problem

Transcription

1 Algorithm Desig Techiques Divide ad coquer Problem

2 Divide ad Coquer Algorithms Divide ad Coquer algorithm desig works o the priciple of dividig the give problem ito smaller sub problems which are similar to the origial problem. The sub problems are ideally of the same size. These sub problems are solved idepedetly usig recursio The solutios for the sub problems are combied to get the solutio for the origial problem 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 2

3 Divide ad Coquer Algorithms The Divide ad Coquer strategy ca be viewed as oe which has three steps. The first step is called Divide which is othig but dividig the give problems ito smaller sub problems which are idetical to the origial problem ad also these sub problems are of the same size. The secod step is called Coquer where i we solve these sub problems recursively. The third step is called Combie where i we combie the solutios of the sub problems to get the solutio for the origial problem. 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 3

4 Quick Sort Quick sort is oe of the most powerful sortig algorithm. It works o the Divide ad Coquer desig priciple. Quick sort works by fidig a elemet, called the pivot, i the give iput array ad partitios the array ito three sub arrays such that The left sub array cotais all elemets which are less tha or equal to the pivot The middle sub array cotais pivot The right sub array cotais all elemets which are greater tha or equal to the pivot Now the two sub arrays, amely the left sub array ad the right sub array are sorted recursively 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 4

5 Quick Sort 1. Begi 2. Set left = 1, right = 3. If (left < right) the 3.1 Partitio a[left right] such that a[left p-1] are all less tha a[p] ad a[p+1 right] are all greater tha a[p] 3.2 Quick Sort a[left p-1] 3.3 Quick Sort a[p+1 right] 4. Ed 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 5

6 Quick Sort To partitio the give array a[1 ] such that every elemet to the left of the pivot is less tha the pivot ad every elemet to the right of the pivot is greater tha the pivot. 1. Begi 2. Set left = 1, right =, pivot = a[left], p = left 3. For r = left+1 to right do 3.1 If a[r] < pivot the a[p] = a[r], a[r] = a[p+1], a[p+1] = pivot Icremet p 4. Ed with output as p 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 6

7 Master theorem T( ) at ( / c b) f ( ) if if d d a= # of sub problems b size of sub problem f() time to combie sub problems 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 7

8 Master Method May divide-ad-coquer recurrece equatios have the form: T( ) c at ( / b) f ( ) if if d d The Master Theorem: logb a 1. if f ( ) is O( ), the T ( ) is 2. if f 3. if f ( ) is ( ) is ( ( log log b b a a log ( ), the T( ) is ( ), the T( ) is ( f ( )), provided af ( / b) f ( ) for some 1. k log b a ) log b a log k 1 ) 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 8

9 Master Theorem Case 1: Whe f() is polyomially smaller the the special fuctio log b a Whe f() is close to the special fuctio Whe f() is polyomially larger the the special fuctio 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 9

10 Master Theorem : Examples T( ) 4T ( / 2) Solutio: log b a=2, so case 1 says T() is O( 2 ). T( ) 2T ( / 2) log Solutio: log b a=1, so case 2 says T() is O( log 2 ). T( ) T( /3) log Solutio: log b a=0, so case 3 says T() is O( log ). 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 10

11 Biary Search Biary Search takes a sorted array as the iput It works by comparig the target (search key) with the middle elemet of the array ad termiates if it is the same, else it divides the array ito two sub arrays ad cotiues the search i left (right) sub array if the target is less (greater) tha the middle elemet of the array. 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 11

12 Biary Search Begi 2. Set left = 1, right = 3. While (left < right) do 3.1 m = floor(( left + right )/2) 3.2 If (target = a[m]) the Ed with output as m 3.3 If (target < a[m]) the right = m If (target > a[m]) the left = m+1 3. Ed with output as oe 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 12

13 Complexity-Biary Search Worst Case Aalysis: Step 3.2 to Step 3.4 each performs 1 compariso ad these steps executes log times. log is the umber of times the while loop executes. Each time we are reducig the problem size by half. i.e /2, /4, /8, The loop termiates whe we hit o a sigle elemet array ad this would occur whe the while loop has executed log times. The maximum umber of comparisos i Biary Search would be 3 * log. Hece the worst case complexity of Biary search is O(log). 5/1/2006 Algorithm Aalysis & Desig CS 007 BE CS 5th Semester 13