1. ICS 353 Design and Analysis of Algorithms Section 01 – 09:20-10:20am – 24:174 Summer Semester 2005 - 2006 (053) King Fahd University of Petroleum & Minerals Information & Computer Science Department

2. Instructor: Dr. Wasfi Al-Khatibوصفي الخطيب Office: (22) 133-1 Office hours: –SMW 08:10 – 09:10 am. –UT 10:20 – 11:20 am. Phone: 1715 email: wasfi@ccse.kfupm.edu.sawasfi@ccse.kfupm.edu.sa Important Preliminaries

3. Important Preliminaries (Cont.) Prerequisite: ICS 202: Data Structures Prerequisites by Topic: –Data structures including linked lists, arrays, stacks, queues, trees, and graphs, and –Knowledge of a high level structured language.

4. Text Book Introduction to Algorithms: Design Techniques and analysis By M. Alsuwaiyel.

5. Course Goals To provide the students with the following: –The fundamentals of algorithms and algorithmic techniques, –The ability to decide on the suitability of a specific technique for a given problem, –The ability to analyze the complexity of a given algorithm, –The ability to design efficient algorithms for new situations, using as building blocks the techniques learned, and –Introducing the concept of NP-complete problems.

6. Course Contents Basic Concepts in Algorithmic Analysis, Chapter1 Mathematical preliminaries, Chapter 2 + Review of Data Structures, Chapter 3 Advanced Data Structures, Chapter 4 Induction, Chapter 5 Divide and Conquer, Chapter 6 Dynamic Programming, Chapter 7 Greedy Algorithms, Chapter 8 Graph Traversal, Chapter 9 NP- Complete Problems, Chapter 10

7. Grading Policy Homeworks10% Quizzes15% Pop Quizzes10% Two Major Exams30% Final Exam35% Absences[-4,2]%

8. Important Dates TaskDate [and Time]LocationWeight Quiz 1Saturday July 1, 2006In class3% Quiz 2Saturday July 8, 2006In class3% Help SessionFriday July 14, 2006 4:00-5:00pmTBAN/A Major Exam ISaturday July 15, 2006 7:00-8:30pmIn class15% Quiz 3Saturday July 22, 2006In class3% Quiz 4Saturday July 29, 2006In class3% Help SessionFriday August 4, 2006 7:00-8:30pmTBAN/A Major Exam IISaturday August 5, 2006 7:00-8:30pmIn class15% Quiz 5Saturday August 12, 2006In class3% Help SessionTuesday August 15, 2006 TBATBAN/A Final ExamWednesday August 16, 2006 7:00-9:00pmTBA35%

9. Final Important Remarks Homeworks: –Due at the beginning of class. –Late homeworks are not accepted. –Discussing homeworks with others (especially on WebCT) is highly encouraged. However, copying homeworks is NOT permitted and will be considered CHEATING. –Homeworks are crucial for learning and obtaining good grades. Quizzes:  15 minute. Each covers material given since the last quiz or major exam. Pop Quizzes: 5-10 minute. Each covers material given during the same lecture. Exams HWs, & quizzes are generally CHALLENGING. MAKE USE of my office hours.

10. Attendance Attendance will be checked each class. A bonus of 0.5 point (out of the maximum 100) will be given for each “additional” hour attended above the 38 hours –This shall come from the help sessions conducted for the two major exams and the final exam. Unexcused Absences Policies: –The first three absences are FREE of charge. –The fourth absence is worth – 2 of your maximum 100 total. –Each subsequent absence, up to the seventh absence, is worth – 0.5. –The eighth absence will result in an automatic DN grade. –An unexcused absence can become an excused absence ONLY by an official letter from the Dean of Student’s office.

11. Your 24-Hour Right One has 24 hours to object to the grade of a homework, [pop] quiz or a major exam starting from the end of the class time in which the graded exam papers have been distributed. –i.e. if you were not present in class during the distribution of exam papers, you loose this right. If for some reason you cannot see me in person, within this period, send me an email requesting an appointment. The email, though, should be sent within the 24-hour time period.

12. Cairo Envelopes How much can they increase your letter grade? NONE…………..even if you bring an elder! Examples: –Slept after Fajr –Checkpoints long lines –I got married / I had a divorce –I am graduating and the job is waiting for me

13. So, What Can Increase my Grade? Active participation –In Class –On WebCT –In my office hours The bottom line is: Trying

14. Cheating Policies Cheating in one category will result in a zero for the whole category. For example, if cheating is confirmed in homework 3, this will result in 0 out of 10% [which means that there is no need to submit any additional homework]. An additional 5% will be deducted from the overall total (out of 100).

15. What is an algorithm? An algorithm is defined as a finite set of steps, each of which may require one or more operations and if carried out on a set of inputs, will produce one or more outputs after a finite amount of time. Examples of Algorithms Examples of computations that are not algorithms

16. Properties of Algorithms Definiteness: It must be clear what should be done. Effectiveness: Each step must be such that it can, at least in principle, be carried out by a person using pencil and paper in a finite amount of time. E.g. integer arithmetic. An algorithm produces one or more outputs and may have zero or more externally supplied inputs. Finiteness: Algorithms should terminate after a finite number of operations.

17. Our Objective Find the most efficient algorithm for solving a particular problem. In order to achieve the objective, we need to determine: –How can we find such algorithm? –What does it mean to be an efficient algorithm? –How can one tell that it is more efficient than other algorithms?

18. In the First Chapter We will answer the following two questions –What does it mean to be an efficient algorithm? –How can one tell that it is more efficient than other algorithms? based on some easy-to-understand searching and sorting algorithms that we may have seen earlier.

19. Searching Problem Assume A is an array with n elements A[1], A[2], … A[n]. For a given element x, we must determine whether there is an index j; 1 ≤ j ≤ n, such that x = A[j] Two algorithms, among others, address this problem –Linear Search –Binary Search

20. Linear Search Algorithm Algorithm: LINEARSEARCH Input: array A[1..n] of n elements and an element x. Output: j if x = A[j], 1 ≤ j ≤ n, and 0 otherwise. 1. j  1 2. while (j < n) and (x  A[j]) 3. j  j + 1 4. end while 5. if x = A[j] then return j else return 0

21. Analyzing Linear Search One way to measure efficiency is to count how many statements get executed before the algorithm terminates One should keep an eye, though, on statements that are executed “repeatedly”. What will be the number of “element” comparisons if x –First appears in the first element of A –First appears in the middle element of A –First appears in the last element of A –Doesn’t appear in A.

22. Binary Search We can do “better” than linear search if we knew that the elements of A are sorted, say in non- decreasing order. The idea is that you can compare x to the middle element of A, say A[middle]. –If x < A[middle] then you know that x cannot be an element from A[middle+1], A[middle+2], …A[n]. Why? – If x > A[middle] then you know that x cannot be an element from A[1], A[2], …A[middle-1]. Why?

23. Binary Search Algorithm Algorithm: BINARYSEARCH Input: An array A[1..n] of n elements sorted in nondecreasing order and an element x. Output: j if x = A[j], 1 ≤ j ≤ n, and 0 otherwise. 1. low  1; high  n; j  0 2. while (low ≤ high) and (j = 0) 3. mid   (low + high)/2  4. if x = A[mid] then j  mid 5. else if x < A[mid] then high  mid - 1 6. else low  mid + 1 7. end while 8. return j

24. Worst Case Analysis of Binary Search What to do: Find the maximum number of element comparisons How to do: –The number of “element” comparisons is equal to the number of iterations of the while loop in steps 2-7. HOW? –How many elements of the input do we have in the First iteration Second iteration Thrid iteration … i th iteration –The last iteration occurs when the size of input we have =

25. Theorem The number of comparisons performed by Algorithm BINARYSEARCH on a sorted array of size n is at most

26. Merging Two Sorted Lists Problem Description: Given two lists (arrays) that are sorted in non-decreasing order, we need to merge them into one list sorted in non-decreasing order. Example: 37912 1241314 Input Output 123479121314

27. Algorithm MERGE Algorithm: MERGE Input: An array A[1..m] of elements and three indices p, q and r, with 1 ≤ p ≤ q <r ≤ m, such that both the subarrays A[p..q] and A[q + 1..r] are sorted individually in nondecreasing order. Output: A[p..r] contains the result of merging the two subarrays A[p..q] and A[q + 1..r]. Comment: B[p..r] is an auxiliary array.

28. Algorithm MERGE (Cont.) 1. s ← p; t ← q + 1; k ← p 2. while s ≤ q and t ≤ r 3. if A[s] ≤ A[t] then 4. B[k] ← A[s] 5. s ← s + 1 6. else 7. B[k] ←A[t] 8. t ← t + 1 9. end if 10. k ← k + 1 11. end while 12. if (s = q + 1) thenB[k..r] ← A[t..r] 13. else B[k..r] ← A[s..q] 14. end if 15. A[p..r] ← B[p..r]

29. Analyzing MERGE Assuming arrays A[p,q] and A[q+1,r] –The least number of comparisons is which occurs when –The most number of comparisons is which occurs when –The number of element assignments performed is

30. Selection Sort Algorithm: SELECTIONSORT Input: An array A[1..n] of n elements. Output: A[1..n] sorted in nondecreasing order. 1. for i  1 to n - 1 2. k  i 3. for j  i + 1 to n {Find the index of the i th smallest element} 4. if A[j] < A[k] then k  j 5. end for 6. if k  i then interchange A[i] and A[k] 7. end for

31. Selection Sort Example 42859

32. 42859 45829 ik 12 54829 25 94825 35 94825 44

33. Analyzing Selection Sort We need to find the number of comparisons carried out in line #4: –For each iteration of the outer for loop, how many times is line #4 executed? –Therefore, in total, line #4 is executed The number of element Interchanges (swaps): –Minimum: –Maximum:  NOTE: The number of element assignments is 3 times the number of element interchanges

34. Insertion Sort Algorithm: INSERTIONSORT Input: An array A[1..n] of n elements. Output: A[1..n] sorted in nondecreasing order. 1. for i  2 to n 2. x  A[i] 3. j  i - 1 4. while (j > 0) and (A[j] > x) 5. A[j + 1]  A[j] 6. j  j - 1 7. end while 8. A[j + 1]  x 9. end for

35. Insertion Sort Example 42859

36. 42859 x=2 458 29 x=9 458 29 x=8 459 28 x=4 948 25

37. Analyzing Insertion Sort The minimum number of element comparisons is which occurs when The maximum number of element comparisons is which occurs when The number of element assignments is

38. Bottom-Up Merge Sort Informally, the algorithm does the following –1. Divide the array into pairs of elements (with possibly single elements in case the number of elements is ) –2. Merge each pair in non-decreasing order (with possibly a single “pair” left) –3. Repeat step 2 until there is only one “pair” left.

39. Bottom-Up Merge Sort Example 423181075912 6

40. Bottom-Up Merge Sort Example 52984127136 10 52984127136 10 52984127136 10 52984127136 10 52984127136 10 42318 75912 6

41. Algorithm BOTTOMUPSORT Algorithm: BOTTOMUPSORT Input: An array A[1..n] of n elements. Output: A[1..n] sorted in nondecreasing order. 1. t ← 1 2. while t < n 3. s ← t; t ← 2s; i ← 0 4. while i + t ≤ n 5. MERGE(A, i + 1, i + s, i + t) 6. i ← i + t 7. end while 8. if i + s < n then 9. MERGE(A, i + 1, i+ s, n) 10. end while

42. Analyzing Algorithm BOTTOMUPSORT With no loss of generality, assume that the size of the array, n, is a power of 2. –In the first iteration, we have pairs that are merged using element comparisons. –In the second iteration, we have pairs that are merged using –…. –In the j th iteration, we have pairs that are merged using –The outer while loop is executed times. –Therefore,

43. Analyzing Algorithm BOTTOMUPSORT What about the number of element assignments?

44. Time Complexity One way of measuring the performance of an algorithm is how fast it executes. The question is how to measure this “time”? –Is having a digital stop watch suitable?

45. Order of Growth As measuring time is subjective to many factors, we look for a more “objective” measure, i.e. the number of operations Since counting the exact number of operations is cumbersome, sometimes impossible, we can always focus our attention to asymptotic analysis, where constants and lower-order terms are ignored. –E.g. n 3, 1000n 3, and 10n 3 +10000n 2 +5n-1 are all “the same” –The reason we can do this is that we are always interested in comparing different algorithms for arbitrary large number of inputs.

46. Example Growth rate for some function

47. Example Growth rate for same previous functions showing larger input sizes

48. Running Times for Different Sizes of Inputs of Different Functions

49. Asymptotic Analysis: Big-oh (O()) Definition: For T(n) a non-negatively valued function, T(n) is in the set O(f(n)) if there exist two positive constants c and n 0 such that T(n)  cf(n) for all n > n 0. Usage: The algorithm is in O(n 2 ) in [best, average, worst] case. Meaning: For all data sets big enough (i.e., n>n 0 ), the algorithm always executes in less than or equal to cf(n) steps in [best, average, worst] case.

50. Big O() O() notation indicates an upper bound. Usually, we look for the tightest upper bound: – while T(n) = 3n 2 is in O(n 3 ), we prefer O(n 2 ).

51. Big O() Examples Example 1: Find c and n 0 to show that T(n) = (n+2)/2 is in O(n) Example 2: Find c and n 0 to show that T(n)=c 1 n 2 +c 2 n is in O(n 2 ) Example 3: T(n) = c. We say this is in O(1).

52. Asymptotic Analysis: Big-Omega (  ()) Definition: For T(n) a non-negatively valued function, T(n) is in the set  (g(n)) if there exist two positive constants c and n 0 such that T(n) >= cg(n) for all n > n 0. Meaning: For all data sets big enough (i.e., n > n 0 ), the algorithm always executes in more than or equal to cg(n) steps.  () notation indicates a lower bound.

53.  () Example Find c and n 0 to show that T(n) = c 1 n 2 + c 2 n is in  (n 2 ).

54. Asymptotic Analysis: Big Theta (  ()) When O() and  () meet, we indicate this by using  () (big-Theta) notation. Definition: An algorithm is said to be  (h(n)) if it is in O(h(n)) and it is in  (h(n)).

55. Example Show that log(n!) is in  (n log n).

56. Complexity Classes and small-oh (o()) Using  () notation, one can divide the functions into different equivalence classes, where f(n) and g(n) belong to the same equivalence class if f(n) =  (g(n)) To show that two functions belong to different equivalence classes, the small-oh notation has been introduced Definition: Let f(n) and g(n) be two functions from the set of natural numbers to the set of non-negative real numbers. f(n) is said to be in o(g(n)) if for every constant c > 0, there is a positive integer n 0 such that f(n) < cg(n) for all n  n 0.

57. Simplifying Rules If f(n) is in O(g(n)) and g(n) is in O(h(n)), then f(n) is in O(h(n)) If f(n) is in O(kg(n)) for any constant k > 0, then f(n) is in ……… If f 1 (n) is in O(g 1 (n)) and f 2 (n) is in O(g 2 (n)), then (f 1 + f 2 )(n) is in ……… If f 1 (n) is in O(g 1 (n)) and f 2 (n) is in O(g 2 (n)) then f 1 (n)f 2 (n) is in ……… You can safely “globally” replace O with  or  in the above, where the above rules will still hold.

58. Very Useful Simplifying Rule Let f(n) and g(n) be be two functions from the set of natural numbers to the set of non-negative real numbers such that: Then if L <  then f(n) is in if L > 0 then f(n) is in if 0 < L <  then f(n) is in if L = 0 then f(n) is in

59. Space Complexity Space complexity refers to the number of memory cells needed to carry out the computational steps required in an algorithm excluding memory cells needed to hold the input. Compare additional space needed to carry out SELECTIONSORT to that of BOTTOMUPSORT if we have an array with 2 million elements!

60. Examples What is the space complexity for –Linear search –Binary search –Selection sort –Insertion sort –Merge (that merges two sorted lists) –Bottom up merge sort

61. Estimating the Running Time of an Algorithm As mentioned earlier, we need to focus on counting those operations which represent, in general, the behavior of the algorithm This is achieved by –Counting the frequency of basic operations. Basic operation is an operation with highest frequency to within a constant factor among all other elementary operations –Recurrence Relations

62. Counting the Frequency of Basic Operations Sometimes, it is easier to compute the frequency of an operation that is a good representative of the overall time complexity of the algorithm –For example, Algorithm MERGE. Counting the number of iterations –The number of iterations in a while loop and/or a for loop is a good indication of the total number of operations

63. Example 1 sum = 0; for (j=1; j<=n; j++) for (i=1; i<=j; i++) sum++; for (k=0; k<n; k++) A[k] = k;

64. Example 2 for j := 1 to  n do sum[j] := 0; for i := 1 to j 2 do sum[j] := sum[j] + i; end for; return sum[1..  n];

65. Example 3 count := 0; for i := 1 to n do m :=  n/i  for j := 1 to m do count := count + 1 ; end for; return count;

66. Example 4 count := 0; while n >= 1 do for j := 1 to n do execute_algorithm_x; count := count + 1; end for n := n / 2; end while return count;

67. Examples 5 & 6 sum1 = 0; for (k=1; k<=n; k*=2) for (j=1; j<=n; j++) sum1++; sum2 = 0; for (k=1; k<=n; k*=2) for (j=1; j<=k; j++) sum2++;

68. Example 7 count := 0; for i := 1 to n do j := 2; while j <= n do j := j 2 ; count := count + 1; end while end for; return count;

69. Recurrence Relations The number of operations can be represented as a recurrence relation. There are very well known techniques, other than expanding the recurrence relation, which we will study in order to solve these recurrences

70. Example Recursive Merge Sort MergeSort(A,p,r) if p < r then q :=  (p+r)/2  ; MergeSort(A,p,q); MergeSort(A,q+1,r); Merge(A,p,q,r); end if; –What is the call to sort an array with n elements? –Let us assume that the overall cost of sorting n elements is T(n), assuming that n is a power of two. If n = 1, do we know T(n)? What is the cost of MergeSort(A,p,q)? What is the cost of MergeSort(A,q+1,r)? What is the cost of Merge(A,p,q,r)?

71. Average Case Analysis Probabilities of all inputs is an important piece of prior knowledge in order to compute the number of operations on average Usually, average case analysis is lengthy and complicated, even with simplifying assumptions.

72. Computing the Average Running Time The running time in this case is taken to be the average time over all inputs of size n. –Assume we have k inputs, where each input costs C i operations, and each input can occur with probability P i, 1  i  k, the average running time is given by

73. Average Case Analysis of Linear Search Assume that the probability that key x appears in any position in the array (1, 2, …, n) or does not appear in the array is equally likely –This means that we have a total of ……… different inputs, each with probability ……… –What is the number of comparisons for each input? –Therefore, the average running time of linear search = ………

74. Average Case Analysis of Insertion Sort Assume that array A contains the numbers from 1..n ( i.e. elements are distinct) Assume that all n! permutations of the input are equally likely. What is the number of comparisons for inserting A[i] in its proper position in A[1..i]? What about on average? Therefore, the total number of comparisons on average is