Subset Sum

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There are traditionally two problems associated with Subset Sum. One is counting the number of ways a list of n numbers make up a given integer. This is also referred to as the Coin Counting problem. (We will call this Problem C for this article.) The other is figuring out if a subset of a given list of integers can sum to a given integer (usually 0). This is, in a way, a special case of the knapsack problem. (We will call this Problem K for this article.)

Problem C (Coin Counting)[edit]

The problem can be defined as: Given a set (or list) of n positive integers a_{1},a_{2},\ldots ,a_{n}, how many solutions does k_{1}a_{1}+k_{2}a_{2}+\ldots +k_{n}a_{n}=T have (where k_{i}\geq 0 for all i)?

Let c(i,j) be the number of ways to sum to j using the subsequence a_{1},a_{2},\ldots ,a_{i}, then the recurrence is simply:

c(i,j)=c(i-1,j)+c(i,j-a_{i})

where

c(0,0)=1

This problem is often asked as a variation of such: Given 1c, 2c, 5c, and 10c pieces, how many ways can you make a dollar?

Problem K (Simplified Knapsack)[edit]

This problem can be thought of as a more specific version of Problem C. It can be defined as: Given a set (or list) of n positive integers a_{1},a_{2},\ldots ,a_{n}, is there a solution such that k_{1}a_{1}+k_{2}a_{2}+\ldots +k_{n}a_{n}=T (where k_{i}>=0)? This is basically a decision variation on Problem C, and can be solved similarly.

The recurrence is essentially:

c(i,j)=c(i-1,j)\lor c(i,j-a_{i})

where

c(0,0)=1

Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.

Examples: set[] = {3, 34, 4, 12, 5, 2}, sum = 9 Output: True //There is a subset (4, 5) with sum 9. Let isSubSetSum(int set[], int n, int sum) be the function to find whether there is a subset of set[] with sum equal to sum. n is the number of elements in set[].

The isSubsetSum problem can be divided into two subproblems …a) Include the last element, recur for n = n-1, sum = sum – set[n-1] …b) Exclude the last element, recur for n = n-1. If any of the above the above subproblems return true, then return true.

Following is the recursive formula for isSubsetSum() problem.

isSubsetSum(set, n, sum) = isSubsetSum(set, n-1, sum) ||

                          isSubsetSum(arr, n-1, sum-set[n-1])

Base Cases: isSubsetSum(set, n, sum) = false, if sum > 0 and n == 0 isSubsetSum(set, n, sum) = true, if sum == 0

Other variations[edit]

There are still other variations and constraints that can be solved similarly. A constraint where \forall i\,\,k_{i}\in \{0,1\} is one such example.