Question

5. Dynamic Programming (a) Given a set of four matrices for the following dimensions: We need to compute Al* A2 A3 A4 Al=2X3; A2=3X5; A3=5X2: A4=2X4 Find the order in which the matrix pairs should be multiplied to produce the optimum number of operations. Show all your steps (10) (b) For the problems given below, determine whether it is more efficient to use a divide and conquer strategy or a dynamic programming strategy. Give the reasons for your choice (5*3=15) i. In a given set of integers, find the maximum number that is divisible by a given constant m. ii. Find whether there is a subset of integers in a given set that adds upto 10 ii. Finding the most frequently occuring number in a given set iv. Given a set of numbers, can they set be partitioned into two groups such that the sum of each group is equal; ie {1.5,11,5} can be partitioned to {1,5,5} and {11} Identifying whether any part of a given string is a palindrome. For ex- ample "Trinidad", contains "dad" which is a palindrome. V.

Answer 1

Q.5 AL [2x3 AR XS, AB - SX2, 2 Au=2x4 23524 A, A2 A3 Au 58 llo 168 178 A. (A2 A3 A4) ax4 454 Min (100,ry) (A1A2) (A₂A4) sx4 (A1 A2 A3 A4 ax4 Ilo 42 42 $0 A₃ (A3A4) 5x4 TADA₂A3) (A,A2) A35x2 (A2A3) A4 Qxs '3X2 Costa (A2A2) Cest (A3A4) = 5XRX4 40 . Cost AIAZ - 30 30 cost (A,A2) 2 X3 X5 30 (A3 A4) 90 tota | A1 A2 (A3 A4) 5x4 2x6x4 cost (A2A3), 3x5x2 30 (A2A3) Au 3X ex4 24 म 30+84-54 A1 (A₂A2) 2 X3 X2 2 (A,A2 A3 2X5X2 3x5x4 total loo 30+2 242 & 40 total=40+40+30 So Minimum cost 58 Pairs are {(A, CA, 13) W

given set of integers find Maximum No that is 06 W In a divisible by divide and conquer In this greedy approach is better Algorithm: o Enter array $ 3 oln) set Min entido; for(i=0; i<n sitt) 2 iblaci] %m) && a[i]>min) Min = a[i]; 3 9 At the end in Minimum le. Min we have Maximum No that is divisible m. since we con cwrite Non recursive brof by so we can also losite recursive programme also. Complessity (1) Find Whetter there is subset of integer in given set that adds upto 1o. This is better with dynamic bozofsamming bince in divide and Conquer all cases will not be cover. (ii) finding the most frequently occucing no In a given set. Here divide and conquer is better. Sost the no's Now . o Make a count variable and do one scan of array

(IV Given a det of number can they set be bantitioned into two groups such that the sum of each group is equal Here dynamic programming is better because we can't sem que We need to take all the cases (w Identity Whetten a part of given string is balindrome Here dynamic programming is better way