Count number of distinct sum subsets within given range
Last Updated :
11 Jul, 2025
Given a set S of N numbers and a range specified by two numbers L (Lower Bound) and R (Upper Bound). Find the number of distinct values of all possible sums of some subset of S that lie between the given range. Examples :
Input : S = { 1, 2, 2, 3, 5 }, L = 1 and R = 5
Output : 5
Explanation : Every number between
1 and 5 can be made out using some subset of S.
{1 as 1, 2 as 2, 3 as 3, 4 as 2 + 2 and 5 as 5}
Input : S = { 2, 3, 5 }, L = 1 and R = 7
Output : 4
Explanation : Only 4 numbers between
1 and 7 can be made out, i.e. {2, 3, 5, 7}.
3 numbers which are {1, 4, 6} can't be made out in any way.
Prerequisites : Bitset | Bit Manipulation
Method 1(Simple) : A naive approach is to generate all possible subsets of given set, calculate their sum subset wise and push them into a hashmap. Iterate over the complete given range and count the numbers which exists in the hashmap. Method 2(Efficient) : An efficient way to solve this problem is by using bitset of size 105. Update the bitset for every element X by left shifting the bitset and doing bitwise OR with previous bitset so that the bitset at the new possible sums become 1. Then by using the concept of Prefix Sums, precompute the required count of numbers between 1 and i for prefix[1..i] to answer each query in O(1) if there are more than query being asked simultaneously. For a query L and R, answer would be simply prefix[R] - prefix[L - 1]
For e.g. S = { 2, 3, 5 }, L = 1 and R = 7 Considering a bitset of size 32 for simplicity. Initially 1 is at 0th position of bitset 00000000000000000000000000000001 For incoming 2, left shifting the bitset by 2 and doing OR with previous bitset 00000000000000000000000000000101 Similarly for 3, 00000000000000000000000000101101 for 5, 00000000000000000000010110101101 This final bitset contains 1 at those positions(possible sums) which can be made out using some subset of S. Hence between position 1 and 7, there are 4 set bits, thus the required answer.
Steps to solve the problem:
- Initialize a bitset BS of size SZ, where SZ is a large enough constant. Initialize the 0th position of the bitset to 1.
- Iterate over the elements S[i] of S from 0 to N-1.
- Left shift the bitset BS by S[i] positions and take the bitwise OR with the previous bitset BS.
- Initialize an array prefix of size SZ, and initialize all elements to 0.
- Iterate over the elements of prefix from 1 to SZ-1.
- Set prefix[i] = prefix[i-1] + BS[i].
- Calculate the answer as prefix[R] - prefix[L-1].
- Return the answer.
Below is the implementation of above approach in C++ :
CPP
// CPP Program to count the number
// distinct values of sum of some
// subset in a range
#include <bits/stdc++.h>
using namespace std;
// Constant size for bitset
#define SZ 100001
int countOfpossibleNumbers(int S[], int N,
int L, int R)
{
// Creating a bitset of size SZ
bitset <SZ> BS;
// Set 0th position to 1
BS[0] = 1;
// Build the bitset
for (int i = 0; i < N; i++) {
// Left shift the bitset for each
// element and taking bitwise OR
// with previous bitset
BS = BS | (BS << S[i]);
}
int prefix[SZ];
// Initializing the prefix array to zero
memset(prefix, 0, sizeof(prefix));
// Build the prefix array
for (int i = 1; i < SZ; i++) {
prefix[i] = prefix[i - 1] + BS[i];
}
// Answer the given query
int ans = prefix[R] - prefix[L - 1];
return ans;
}
// Driver Code to test above functions
int main()
{
int S[] = { 1, 2, 3, 5, 7 };
int N = sizeof(S) / sizeof(S[0]);
int L = 1, R = 18;
cout << countOfpossibleNumbers(S, N, L, R);
return 0;
}
// Java Program to count the number
// distinct values of sum of some
// subset in a range
import java.util.*;
import java.lang.*;
import java.io.*;
class GFG
{
// Constant size for bitset
static int SZ = 100001;
static int countOfpossibleNumbers(int[] S, int N, int L,
int R)
{
// Initially BS is 1
int BS = 1;
for (int i = 0; i < N; i++) {
// Left shift the bitset for each
// element and taking bitwise OR
// with previous bitset
BS = BS | (BS << S[i]);
}
// Convert BS to bitset array
String BS_ = Integer.toBinaryString(BS);
BS_ = String.format("%" + (-SZ) + "s", BS_).replace(' ', '0');
// Initializing the prefix array to zero
int[] prefix = new int[SZ];
for (int i = 0; i < SZ; i++)
prefix[i] = 0;
// Build the prefix array
for (var i = 1; i < SZ; i++) {
prefix[i]
= prefix[i - 1] + ((BS_.charAt(i) == '1') ? 1 : 0);
}
// Answer the given query
int ans = prefix[R] - prefix[L - 1];
return ans;
}
// Driver Code to test above functions
public static void main(String[] args)
{
int[] S = { 1, 2, 3, 5, 7 };
int N = S.length;
int L = 1;
int R = 18;
// Function call
System.out.println(countOfpossibleNumbers(S, N, L, R));
}
}
// This code is contributed by phasing17
# Python3 Program to count the number
# distinct values of sum of some
# subset in a range
# Constant size for bitset
SZ = 100001
def countOfpossibleNumbers(S, N, L, R):
# Initially BS is 1
BS = 1
for i in range(N):
# Left shift the bitset for each
# element and taking bitwise OR
# with previous bitset
BS = BS | (BS << S[i])
# Convert BS to bitset array
BS = bin(BS)[2::]
BS = [int(i) for i in BS.zfill(SZ)][::-1]
# Initializing the prefix array to zero
prefix = [0 for i in range(SZ)]
# Build the prefix array
for i in range(1, SZ):
prefix[i] = prefix[i - 1] + BS[i]
# Answer the given query
ans = prefix[R] - prefix[L - 1]
return ans
# Driver Code to test above functions
S = [1, 2, 3, 5, 7]
N = len(S)
L, R = 1, 18
print(countOfpossibleNumbers(S, N, L, R))
# This code is contributed by phasing17
// C# Program to count the number
// distinct values of sum of some
// subset in a range
using System;
class GFG
{
// Constant size for bitset
static int SZ = 100001;
static int countOfpossibleNumbers(int[] S, int N, int L,
int R)
{
// Initially BS is 1
int BS = 1;
for (int i = 0; i < N; i++) {
// Left shift the bitset for each
// element and taking bitwise OR
// with previous bitset
BS = BS | (BS << S[i]);
}
// Convert BS to bitset array
string BS_ = Convert.ToString(BS, 2);
BS_ = BS_.PadRight(SZ, '0');
// Initializing the prefix array to zero
int[] prefix = new int[SZ];
for (int i = 0; i < SZ; i++)
prefix[i] = 0;
// Build the prefix array
for (var i = 1; i < SZ; i++) {
prefix[i]
= prefix[i - 1] + ((BS_[i] == '1') ? 1 : 0);
}
// Answer the given query
int ans = prefix[R] - prefix[L - 1];
return ans;
}
// Driver Code to test above functions
public static void Main(string[] args)
{
int[] S = { 1, 2, 3, 5, 7 };
int N = S.Length;
int L = 1;
int R = 18;
// Function call
Console.WriteLine(
countOfpossibleNumbers(S, N, L, R));
}
}
// This code is contributed by phasing17
// JS Program to count the number
// distinct values of sum of some
// subset in a range
// Constant size for bitset
const SZ = 100001;
function countOfpossibleNumbers(S, N, L, R)
{
//Initially BS is 1
var BS = 1;
for (var i = 0; i < N; i++) {
// Left shift the bitset for each
// element and taking bitwise OR
// with previous bitset
BS = BS | (BS << S[i]);
}
//Convert BS to bitset array
BS = BS.toString(2);
BS = BS.padEnd(SZ, "0");
BS = BS.split("");
BS = BS.map(str => {
return Number(str);
});
// Initializing the prefix array to zero
var prefix = new Array(SZ).fill(0);
// Build the prefix array
for (var i = 1; i < SZ; i++) {
prefix[i] = prefix[i - 1] + BS[i];
}
// Answer the given query
var ans = prefix[R] - prefix[L - 1];
return ans;
}
// Driver Code to test above functions
var S = [ 1, 2, 3, 5, 7 ];
var N = S.length;
var L = 1;
var R = 18;
console.log(countOfpossibleNumbers(S, N, L, R));
// This code is contributed by phasing17
Time Complexity: O(S*Z) where S*Z is the maximum sum for given constraints, i.e. 105
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem