# Subset Sum Queries in a Range using Bitset

Given an array[] of **N** positive integers and **M** queries. Each query consists of two integers **L** and **R** represented by a range. For each query, find the count of numbers that lie in the given range which can be expressed as the sum of any subset of given array.

**Prerequisite :** Subset Sum Queries using Bitset**Examples:**

Input :arr[] = { 1, 2, 2, 3, 5 }, M = 4

L = 1, R = 2

L = 1, R = 5

L = 3, R = 6

L = 9, R = 30Output :

2

5

4

5Explanation :For the first query, in range [1, 2] all numbers i.e. 1 and 2 can be expressed as a subset sum, 1 as 1, 2 as 2. For the second query, in range [1, 5] all numbers i.e. 1, 2, 3, 4 and 5 can be expressed as subset sum, 1 as 1, 2 as 2, 3 as 3, 4 as 2 + 2 or 1 + 3, 5 as 5. For the third query, in range [3, 6], all numbers i.e. 3, 4, 5 and 6 can be expressed as subset sum. For the last query, only numbers 9, 10, 11, 12, 13 can be expressed as subset sum, 9 as 5 + 2 + 2, 10 as 5 + 2 + 2 + 1, 11 as 5 + 3 + 2 + 1, 12 as 5 + 3 + 2 + 2 and 13 as 5 + 3 + 2 + 2 + 1.

**Approach :** The idea is to use a bitset and iterate over the array to represent all possible subset sums. The current state of bitset is defined by ORing it with the previous state of bitset left shifted X times where X is the current element processed in the array. To answer the queries in O(1) time, we can precompute the count of numbers upto every number and for a range [L, R], answer would be **pre[R] – pre[L – 1]**, where pre[] is the precomputed array.

Below is the implementation of above approach.

`// CPP Program to answer subset` `// sum queries in a given range` `#include <bits/stdc++.h>` `using` `namespace` `std;` ` ` `const` `int` `MAX = 1001;` `bitset<MAX> bit;` ` ` `// precomputation array` `int` `pre[MAX];` ` ` `// structure to represent query` `struct` `que {` ` ` `int` `L, R;` `};` ` ` `void` `answerQueries(` `int` `Q, que Queries[], ` `int` `N,` ` ` `int` `arr[])` `{` ` ` `// Setting bit at 0th position as 1` ` ` `bit[0] = 1;` ` ` `for` `(` `int` `i = 0; i < N; i++)` ` ` `bit |= (bit << arr[i]);` ` ` ` ` `// Precompute the array` ` ` `for` `(` `int` `i = 1; i < MAX; i++)` ` ` `pre[i] = pre[i - 1] + bit[i];` ` ` ` ` `// Answer Queries` ` ` `for` `(` `int` `i = 0; i < Q; i++) {` ` ` `int` `l = Queries[i].L;` ` ` `int` `r = Queries[i].R;` ` ` `cout << pre[r] - pre[l - 1] << endl;` ` ` `}` `}` ` ` `// Driver Code to test above function` `int` `main()` `{` ` ` `int` `arr[] = { 1, 2, 2, 3, 5 };` ` ` `int` `N = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `int` `M = 4;` ` ` `que Queries[M];` ` ` `Queries[0].L = 1, Queries[0].R = 2;` ` ` `Queries[1].L = 1, Queries[1].R = 5;` ` ` `Queries[2].L = 3, Queries[2].R = 6;` ` ` `Queries[3].L = 9, Queries[3].R = 30;` ` ` `answerQueries(M, Queries, N, arr);` ` ` `return` `0;` `}` |

**Output:**

2 5 4 5

**Time Complexity :** Each query can be answered in O(1) time and precomputation requires O(MAX) time.