Algorithm Analysis Divide and Conquer. Chung-Ang University, Jaesung Lee

Transcription

1 Algorithm Analysis Divide and Conquer Chung-Ang University, Jaesung Lee

2 Introduction 2

3 Divide and Conquer Paradigm 3

4 Advantages of Divide and Conquer Solving Difficult Problems Algorithm Efficiency Parallelism Memory Access Roundoff Control 4

5 Implementation Issues Recursion Explicit Stack Stack Size Choosing the Base Cases Sharing Repeated Subproblems 5

6 Divide and Conquer Overview General method Design strategy Large improvement in Time Complexity 6

7 Divide and Conquer Overview Divide and Conquer Strategy Divide the problem Solve smaller instances recursively Assemble the solutions Recursion stops when the division becomes impossible 7

8 Divide and Conquer Overview Divide and Conquer Steps Divide Conquer 8

9 Control Abstraction of D/C 9

10 Control Abstraction of D/C 10

11 Binary Search records, instances, datum, Keys: < < < < < Objective of Binary Search Given a key, find corresponding element from the list Successful search: = Unsuccessful search: = 0 where 0 = 1 11

12 Binary Search Jump into the middle of the array: [ ] Compare and [ ] If = then has been found If < then must be in the former portion of If > then must be in the latter portion of 12

13 Binary Search Algorithm 13

14 Binary Search Algorithm 14

15 Example for Binary Search Index Elements

16 Example for Binary Search Index Elements

17 Example for Binary Search low mid high Index Elements

18 Example for Binary Search low mid high Index Elements

19 Example for Binary Search Index low 8 high 9 Elements mid 19

20 Example for Binary Search high Index Elements mid low 20

21 Example for Binary Search Index Elements

22 Example for Binary Search high Index Elements mid low 22

23 Example for Binary Search Index high 1 low Elements mid 23

24 Example for Binary Search Index Elements Comp

25 Example for Binary Search If is not presented, for example < 1 1 < < [2] 2 < < [3] 3 < < [4] 4 < < [5] 5 < < [6] 6 < < [7] 7 < < [8] 8 < < [9] 9 < Requires maximum 4 comparisons 25

26 Example for Binary Search Average comparisons for an unsuccessful search is = = 3.4 The time complexity of Binary Search is log Let s prove this! 26

27 Analysis for the worst case ( ): the time complexity of Binary search 0 = 0 ( ) = 1 = [ ] = < [ ] = 1 + > [ ] 27

28 Analysis for the worst case Assume = 2 1 where is non-negative integer, then Array Algebraically, = = 2 for 1. 28

29 Analysis for the worst case Thus, we can simplify as 0 = = 1 = [ ] = < [ ] = > [ ] 29

30 Analysis for the worst case In the worst case, the comparison always fails. 0 = = is time complexity for worst case! 30

31 Analysis for the worst case 2 1 = = = = 2 1 =

32 Analysis for the worst case 2 1 = For, 2 1 = + 0 = Because 2 1 =, = log + 1 = log + 1 = log 32

33 External and Internal Path Length An illustration of Binary tree ( ): external (square) nodes of tree ( ): internal (round) nodes 33

34 External and Internal Path Length h( ): the height of node 34

35 External and Internal Path Length ( ): the depth of node 35

36 External and Internal Path Length Internal path length: External path length: = = = 4 and = 4 = 12 and = 4 36

37 External and Internal Path Length Internal path length: External path length: = = Theorem: = + 2 where is a binary tree. 37