CSE 373: Analysis of Algorithms Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Fall 2014

Transcription

1 CSE : Analysis of Algorithms Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Fall 0 Theory is when you know everything but nothing works. Practice is when everything works but no one knows why. In our lab, theory and practice are combined: nothing works and no one knows why. A practical theoretician ( no one knows who )

2 Some Mostly Useless Information Lecture Time:TuTh:00 pm -:0 pm Location: Light Engineering Lab 0, West Campus Instructor: Rezaul A. Chowdhury Office Hours:TuTh:00 pm -:0 pm, Computer Science TA: Ibrahim Hammoud Office Hours:MoWe0:00 am -:0 am, 0 Computer Science firstname.lastname@stonybrook.edu Class Webpage:

3 Topics to be Covered The following topics will be covered ( hopefully ) elementary data structures sorting and searching greedy algorithms divide-and-conquer algorithms dynamic programming graph algorithms randomized algorithms parallel algorithms and multithreaded computations NP-completeness and approximation algorithms

4 Grading Policy Problem solving ( homework problem sets ): 0% ( highest score %, lowest score %, and others 0% each ) Problem design ( themes, one per homework problem set ): 0% ( each worth.% ) In-class midterm ( Thursday, Oct, :00pm :0pm ): % Final exam ( Monday, Dec, :0pm :00pm, location: TBD ): % Each homework problem set and exam will include additional problems for graduate students taking the course as CSE 8. Graduate and undergraduate students will be graded separately.

5 Groups and Supergroups Groups for Problem Solving: Each group will consist of a pair of students each group will submit only one copy of hand-written solutions for each homework problem set each group member must write down solutions for two problem sets Supergroups for Problem Design: Each supergroupwill consist of a pair of groups ( students ) each supergroupwill submit only one copy of hand-written problem for each theme each supergroupmember must write down problem for one theme

6 Textbooks Required Thomas Cormen, Charles Leiserson, Ronald Rivest, and Clifford Stein. Introduction to Algorithms(rd Edition), MIT Press, 009. Recommended SanjoyDasgupta, Christos Papadimitriou, and UmeshVazirani. Algorithms(st Edition), McGraw-Hill, 00. Jon Kleinberg and ÉvaTardos. Algorithm Design(st Edition), Addison Wesley, 00. Steven Skiena. The Algorithm Design Manual(nd Edition), Springer, 008.

7 What is an Algorithm? An algorithm is a well-defined computational procedurethat solves a well-specified computational problem. It accepts a value or set of values as input, and produces a value or set of values as output

8 What is an Algorithm? An algorithm is a well-defined computational procedurethat solves a well-specified computational problem. It accepts a value or set of values as input, and produces a value or set of values as output Example:mergesortsolves the sorting problem specified as a relationship between the input and the output as follows. Input:A sequence of numbers,,,. Output:A permutation,,, of the input sequence such that.

9 Desirable Properties of an Algorithm Correctness Designing an incorrect algorithm is straight-forward Efficiency Efficiency is easily achievable if we give up on correctness Surprisingly, sometimes incorrect algorithms can also be useful! If you can control the error rate Tradeoff between correctness and efficiency: Randomized algorithms ( Monte Carlo: always efficient but sometimes incorrect, Las Vegas: always correct but sometimes inefficient ) Approximation algorithms ( always incorrect! )

10 Algorithmic Puzzles

11 Tromino Cover A right trominois an L-shaped tile formed by three adjacent squares. Puzzle:You are given a board with one missing square. you must cover all squares except the missing one exactly using right trominoes the trominoes must not overlap board

12 Steps Tromino Cover board

13 Steps Tromino Cover Divide the board into disjoint subboards. board

14 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. board

15 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! board

16 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

17 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

18 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

19 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

20 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

21 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

22 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

23 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

24 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

25 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

26 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

27 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

28 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

29 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

30 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

31 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

32 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

33 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

34 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

35 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

36 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

37 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

38 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

39 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

40 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board

41 Steps Tromino Cover Divide the board into disjoint subboards. Place a trominoat the center so that it fully covers one square from each of the three ( ) subboards with no missing square, and misses the fourth subboard completely. This reduces the original problem into smaller instances of the same problem! Solve each smaller subproblem recursively using the same technique. board This algorithm design technique is called recursive divide & conquer.

42 Collecting Coins Puzzle:A robot moves from the top-left corner to the bottom-right corner of an grid. At each step it either moves one cell to the right or one cell down from its current location. Each cell of the grid contains zero or more coins. The robot collects the coins from every cell it visits. What path the robot should take to collect the maximum number of coins? 8 grid

43 Collecting Coins Let us first count the number of coins in each cell

44 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

45 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

46 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

47 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

48 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

49 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

50 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

51 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

52 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

53 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

54 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

55 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

56 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

57 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

58 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

59 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

60 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

61 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

62 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

63 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

64 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

65 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

66 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

67 0 0 0 Collecting Coins 8 8 Let, number of coins in cell, Let, max number of coins the robot can collect if it goes from cell, to cell,, 0, 0 0 max,,,,,

68 Collecting Coins This algorithm design technique is called dynamic programming. We computed solutions to larger and larger instances of the given problem using saved solutions to smaller overlapping instances of the same problem until we found a solution to the given instance.

69 Rooster Chase Puzzle:A framer (F) is trying to catch a rooster (R) on a grid. F s initial position: bottom-left corner of the grid R s initial position: top-right corner of the grid Fand Rmove alternately until Ris captured each move is to a neighboring cell horizontally or vertically Ris captured when Fmoves to a cell occupied by R 8 R 8 What algorithm should Fuse to catch Rwhen F moves first? R moves first? F 8 8board

70 Rooster Chase Initially, both F and Rare on cells of the same color. If Fis to catch Rin the next move, Rmust be in a cell horizontally or vertically adjacent to F s current cell. So, right before Fcatches R, they must be in cells of opposite color. 8 R 8 But that will never happen if Fmoves first! F 8 8board So, if Fmoves first, Fwill never be able to catch R!

71 Rooster Chase If Rmoves first, then F will be able to catch Rin at most moves of its own ( or total moves )! Let, F s current location, and, R s current location. F s algorithm is as follows: 8 R 8 : Fmoves up : Fmoves right : cannot happen F 8 8board This is a greedy algorithmas Falways chooses the option that looks the best ( i.e., reduces the Manhattan distance to R) at the time of choice.

72 Three Jugs Puzzle: You are given three jugs one 8-pint jug full of water one empty -pint jug one empty -pint jug Your task is to get exactly pints of water in one of the jugs by completely filling up or emptying jugs into others. Minimize the number of times you fill up / empty jugs. -pint -pint 8-pint initial state

73 Three Jugs 0

74 Three Jugs 0

75 0 Three Jugs

76 Three Jugs 0

77 Three Jugs 0

78 Three Jugs 0

79 Three Jugs 0

80 Three Jugs 0 We have just used a state-space search algorithmcalled the breadth-first search.