Question

1. [10 pts.] Consider the following coin changing problem. You are given a value N and an infinite supply of coins with values di, d2,.. , dk. Give an O(Nk) algorithm for finding the smallest number of coins that add up to the value N.

Answer 1

Let OPT(n) be the minimum number of coins required to get value n using the coins d1,d2,...,dk. Then we can think of dynamic programming recurrence equation to be

OPT(n)= min (1+OPT(n-di) for all i from 1 to k

This means that if I use one coin of value di for getting value n, then remaining coins size require to add value n-di should be minimum.

This we will create dynamic programming table name OPT whose value will grow linearly from 1 to N and it will check the minimum number of coins needed to compute OPT(n) using past value.

Base case:- OPT(di) = 1 for all i in range 1 to k

OPT(n) = INFINITE for n< d1 because there is no way to get value less than d1

Below is the algorithm which run in time O(Nk) in which OPT(N) will be minimum numbers of coin needed to have value N using the coins of value d1,d2,...,dk

$Minimum_Coins(N) 1. for i =1 to N:- 2 3. for i1 to k:- 4 5. for id1+1 to N: we already have estimate for value d1 OPT[1] = INFINITE \\initialization step OPT[di]1 only 1 coin is best solution for j=1 to k and djCzi:- 7 8 9. Return OPT(N) OPT[i] 1+0PT[i-dj] Was per recurrence equation =$

Please comment for any clarification