Question

Making Change: Given coins of denominations (value) 1 = v1 < v2< … < vn, we wish to make change for an amount A using as few coins as possible. Assume that vi’s and A are integers. Since v1= 1 there will always be a solution.
Formally, an algorithm for this problem should take as input:
 An array V where V[i] is the value of the coin of the ith denomination.
 A value A which is the amount of change we are asked to make
The algorithm should return an array C where C[i] is the number of coins of value V[i] to return as change and m the minimum number of coins it took. You must return exact change so

The objective is to minimize the number of coins returned or:

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a) Describe and give pseudocode for a dynamic programming algorithm to find the minimum number of coins to make change for A.
b) What is the theoretical running time of your algorithm?

Answer 1

Answer:

a)

M(j) = the minimum number of coins required to make change for an amount of money j

                         M(j) = min{M(j − v_i )} + 1

The smallest number of coins required to make j , is the smallest number required to make j − v_i , plus one.

Algorithm Minimum-Change(C,v)

   //all of vi and C are positive integers.

   Input: n types of coin denominations of values v1 < v2 < ... < vn

Output: The minimum number of coins required to make change for an amount of money j

       M[0]ß0

        {M[j]← ∞ where i<0}

    for jß1 to C do

          Min← ∞

          for iß1 to |V| do

              if M[j-v[i]] +1< min then

                    minßM[j-v[i]] +1

              end if

           end for

                  return M[C]

   end for

b)

The theoretical running time of the above algorithm is O(n * A)

Please provide your valuable feedback.