Chapter 5: Algorithms

Transcription

1 Chapter 5: Algorithms Computer Science: An Overview Tenth Edition by J. Glenn Brookshear Presentation files modified by Farn Wang Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2 Chapter 5: Algorithms 5.1 The Concept of an Algorithm 5.2 Algorithm Representation 5.3 Algorithm Discovery 5.4 Iterative Structures 5.5 Recursive Structures 5.6 Efficiency and Correctness Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-2

3 Definition of Algorithm An algorithm is an ordered set of unambiguous, executable steps that defines a terminating process. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-3

4 Algorithm Representation Requires well-defined primitives A collection of primitives constitutes a programming language. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-4

5 Folding a bird from a square piece of paper Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-5

6 Origami primitives 折 り 紙 Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-6

7 Problem Solving Steps Designed the method in Hungary teaching the son of a baron. 1. Understand the problem. 2. Devise a plan for solving the problem. 3. Carry out the plan. 4. Evaluate the solution for accuracy and its potential as a tool for solving other problems. George Pólya ( ) Stanford, ETH-Zurich Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-7

8 Getting a Foot in the Door Basic approaches: Try working the problem backwards Solve an easier related problem Relax some of the problem constraints Solve pieces of the problem first (bottom up methodology) Stepwise refinement: Divide the problem into smaller problems (top-down methodology) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-8

9 A trial case study in solving problems - Ages of Children Problem Person A is charged with the task of determining the ages of B s three children. B tells A that the product of the children s ages is 36. A replies that another clue is required. B tells A the sum of the children s ages. A replies that another clue is needed. B tells A that the oldest child plays the piano. A tells B the ages of the three children. How old are the three children? Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-9

10 Understanding the domain instances for a solution You can spend all the time fretting without a clue. Or you can start to play with the numbers. The only pair that needs clue 3 to distinguish. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-10

11 Representation of algorithms - Pseudocode Primitives Assignment name expression Conditional selection if condition then action Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-11

12 Representation of algorithms - Pseudocode Primitives (continued) Repeated execution while condition do activity Procedure procedure name (generic names) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-12

13 Representation of algorithms - The procedure Greetings in pseudocode Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-13

14 Iterative Structures Pretest loop: while (condition) do (loop body) Posttest loop: repeat (loop body) until(condition) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-14

15 The sequential search algorithm in pseudocode Assumption: sorted list. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-15

16 Components of repetitive control Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-16

17 Representation of algorithms - flowchart The while loop structure: Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-17

18 Representation of algorithms - flowchart The repeat-until loop structure Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-18

19 How to design an algorithm? - Intuition behind a sorting algorithm Given a data item b and a sorted list d 1,,d k, we can easily insert b into d 1,,d k. sort d 1 sort d 1, d 2 Insert d 1 to the right position in. Insert d 2 to the right position in d 1. sort d 1, d 2, d 3 Insert d 3 to the right position in d 1,d 2. sort d 1, d 2,., d n Insert d n to the right position in d 1,,d n-1. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-19

20 Test run of the algorithm in mind: Sorting the list Fred, Alex, Diana, Byron, and Carol alphabetically Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-20

21 Pseudo-code for insertion sort - an example of loop algorithms This is the pseudocode in the textbook. I think it is a bad example. Why? Too close to the real code. Too many details. Difficult to understand. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-21

22 A better pseudo-code for insertion sort High-level mathematical notations. Procedure sort(list) { } Loop for N = 2 to List { Insert List[N] to the right position in List[1..N-1] to make List[1..N] sorted; High-level and intuitive operations in natural languages. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-22

23 Recursion The execution of a procedure leads to another execution of the procedure. Multiple activations of the procedure are formed, all but one of which are waiting for other activations to complete. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-23

24 Recursion F(n) = F(n-1) + F(n-2) F(0) = 0 F(1) = 1 F(n) { } if (n <= 0) return(0); else if (n == 1) return 1; else return F(n-1)+F(n-2); How many times F() is called? f(n) = 1+f(n-1)+f(n-2); f(0)=1, f(1)=1; Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-24

25 Recursion - implemented with iteration F(n) = F(n-1) + F(n-2) F(0) = 0 F(1) = 1 F(n) { } F(0) = 0; F(1) = 1; k = 2; while (k <= n) { F(k)=F(k-1)+F(k-2); k=k+1;} return (F(n)); How many steps? f (n) = 1 + n; Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-25

26 2013/04/17 stopped here. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-26

27 Recursion - with improved performance F(n) = F(n-1) + F(n-2) ; F(0) = 1; F(1) = 1 F(n) { if (n < 0) exit(0); else if (n == 0 n == 1) return n; else if (D[n] == -1) D[n] = F(n-1)+F(n-2); return(d[n]); } Main (n) { } k=0; while (k<=n) {D[k]=-1; k=k+1;} return F(n); Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Initialize the D array. 5-27

28 Recursion for search? - Trying out an example for the entry John Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-28

29 Binary search technique - A first draft Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-29

30 Binary search algorithm - pseudocode with recursion concept Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-30

31 Binary search in recursion bsearch(l[j,k], t) { if (j>k) return search failure. m = (j+k)/2 ; if (t==l[m]) return m; else (t<l[m]) return bsearch(l[j,m-1],t); else return bsearch(l[m+1,k], t); } Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-31

32 Binary search in recursion bsearch(l[j,k], t) { while (j<=k) { m = (j+k)/2 ; if (t==l[m]) return m; else (t<l[m]) k = m-1; else j=m+1; } return search failure. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley } 5-32

33 Conceptualizing the recursion (I/III) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-33

34 Conceptualizing the recursion (II/III) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-34

35 Conceptualizing the recursion (III/III) Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-35

36 Algorithm Efficiency Measured as number of instructions executed Big O notation: Used to represent efficiency classes. Examples: Insertion sort for n elements is in O(n 2 ) binary search for n elements is in O(log n) F(n) iterative: O(n) F(n) purely recurisive: O(f(n)), f(n)=1+f(n-1)+f(n-2). Best, worst, and average case analysis Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-36

37 Applying the insertion sort in a worstcase situation Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-37

38 Graph of the worst-case analysis of the insertion sort algorithm Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-38

39 Graph of the worst-case analysis of the binary search algorithm Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-39

40 Algorithm Efficiency Big O concepts for worst-case analysis Insertion sort for n elements is in O(n 2 ) n n sec with efficient compiler & CPU binary search for n elements is in O(log n) log n sec with lousy compiler & CPU As n becomes bigger and bigger, n n >> log n sec Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-40

41 The lower-bound for worst cases? - An example Chain Separating Problem A traveler has a gold chain of seven links. He must stay at an isolated hotel for seven nights. The rent each night consists of one link from the chain. What is the fewest number of links that must be cut so that the traveler can pay the hotel one link of the chain each morning without paying for lodging in advance? Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-41

42 Separating the chain using only three cuts Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-42

43 Solving the problem with only one cut Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-43

44 Lower-bound of worst-case efficiency How do we know if an algorithm is good enough for a task? What is the assumptions of basic operations? Have we tied up ourselves? Lemma: One cut is the minimum. Proof: At least one chain is given in the first morning. So one cut is the least. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-44

45 Lower-bound of worst-case efficiency other techniques reduction: translate any instance of a problem A with known lower-bound to the given problem B. Then we can show B s lower-bound is no lower than A s. information theory: Is log(n) the best lower-bound for the worst case of searching algorithms? Is n log(n) the best for sorting algorithms? Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-45

46 Software Verification Proof of correctness Assertions Preconditions Loop invariants Testing send testcases to SW check the SW behavior or output Verification analysis of models of SW Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-46

47 The assertions associated with a typical while structure Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-47

48 Assertions - an example on recursive code. F(n) = F(n-1) + F(n-2), F(0) =0, F(1) = 1 F(n) { } F(0) = 0; F(1) = 1; k = 2; while (k <= n) { } F(k)=F(k-1)+F(k-2); k=k+1; return (F(n)); Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley precondition: true precondition: k=2 Loop invariants: k<=n, i<k, F(i)=F(i-1)+F(i-2) can be proved with mathematical induction. precondition: F(n)=F(n-1)+F(n-2) 5-48

49 Various algorithms (I) Numerical factorization, prime number testing, Constraint solving equations, inequalities, differential systems Sequence: Sorting, searching, merging, pattern matching, data mining Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-49

50 Various algorithms (II) Graph algorithms searching, traversal, critical paths, flow analysis,. structure identification, spanning trees Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-50

51 Various algorithms (III) Graph algorithms searching, traversal, critical paths, flow analysis,. structure identification, spanning trees Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-51

52 Various algorithms (III) Logics solutions, Boolean operations, greatest fixpoint, least fixpoint Distributed algorithms leader elections, snapshots, deadlock detection, consensus. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-52

53 Algorithms for nonterminating systems Operating systems deadlock mutual exclusion security Medical systems Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-53

54 When there is no algorithm genetic algorithm swamp intelligence ant intelligence machine learning heuristics solution as proofs.. Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-54

55 Complexities Average case? Worst case? Algorithms: time complexities space complexities Problems: How efficient can we make the program run? Is there an algorithm? Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-55

56 Data structures linear structures arrays, sequence, stacks, queues tree-structures binary trees, 2-3 trees, splay trees general graphs DAG (directed acyclic graphs) directed, undirected, multigraphs Copyright 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5-56