Partition Array into Two Equal Product Subsets

Partition Array into Two Equal Product Subsets - Problem

You are given an integer array nums containing distinct positive integers and an integer target.

Determine if you can partition nums into two non-empty disjoint subsets, with each element belonging to exactly one subset, such that the product of the elements in each subset is equal to target.

Return true if such a partition exists and false otherwise.

A subset of an array is a selection of elements of the array.

Input & Output

Example 1 — Valid Partition

$ Input: nums = [2,3,6], target = 6

› Output: true

💡 Note: We can partition into {2,3} and {6}. Product of {2,3} = 2×3 = 6, Product of {6} = 6. Both equal target.

Example 2 — No Valid Partition

$ Input: nums = [1,2,3], target = 4

› Output: false

💡 Note: Total product is 1×2×3 = 6, but we need both subsets to have product 4, requiring total product = 16. Impossible.

Example 3 — Single Element Match

$ Input: nums = [4,1], target = 4

› Output: false

💡 Note: Partition into {4} and {1}. Product of {4} = 4 = target, but product of {1} = 1 ≠ target. Both subsets must have product equal to target.

Constraints

2 ≤ nums.length ≤ 10
1 ≤ nums[i] ≤ 100
nums contains distinct integers
1 ≤ target ≤ 10⁶

Visualization

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Asked in

G Google 15 a Amazon 12

The key insight is that if both subsets have product equal to target, then the total product must equal target². The optimal approach uses bit manipulation to enumerate all possible subsets systematically. Time: O(n × 2^n), Space: O(1).

Common Approaches

✓ Bit Manipulation Enumeration

⏱️ Time: O(n × 2^n) Space: O(1)

Generate all possible non-empty subsets using bit manipulation. For each subset, calculate its product and check if the remaining elements also have product equal to target.

Recursive Backtracking

⏱️ Time: O(2^n) Space: O(n)

For each element, recursively try adding it to subset1 or subset2, then check if both subsets have product equal to target. This explores all 2^n possible partitions.

Bit Manipulation Enumeration — Algorithm Steps

Generate all possible subsets using bitmasks
For each subset, calculate product
Check if remaining elements also have product = target

Visualization

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Step-by-Step Walkthrough

Bitmask

Each bit represents element inclusion

Calculate

Compute products for both subsets

Check

Verify both products equal target

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

bool solution(int* nums, int numsSize, int target) {
    long long totalProduct = 1;
    for (int i = 0; i < numsSize; i++) {
        totalProduct *= nums[i];
    }
    
    // If total product is not target^2, impossible
    if (totalProduct != (long long)target * target) {
        return false;
    }
    
    // Try all possible non-empty subsets except the full set
    for (int mask = 1; mask < (1 << numsSize) - 1; mask++) {
        long long subset1Product = 1;
        long long subset2Product = 1;
        
        for (int i = 0; i < numsSize; i++) {
            if (mask & (1 << i)) {
                subset1Product *= nums[i];
            } else {
                subset2Product *= nums[i];
            }
        }
        
        if (subset1Product == target && subset2Product == target) {
            return true;
        }
    }
    
    return false;
}

int main() {
    char line[1000];
    fgets(line, sizeof(line), stdin);
    
    int nums[20];
    int numsSize = 0;
    char* token = strtok(line, "[],\n");
    while (token != NULL) {
        if (strlen(token) > 0) {
            nums[numsSize++] = atoi(token);
        }
        token = strtok(NULL, "[],\n");
    }
    
    int target;
    scanf("%d", &target);
    
    bool result = solution(nums, numsSize, target);
    printf(result ? "true\n" : "false\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n × 2^n)

2^n possible subsets, each requiring O(n) time to calculate product

⚠ Quadratic Growth

Space Complexity

O(1)

Only using constant extra variables for calculations

⚡ Linearithmic Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Bit Manipulation Enumeration — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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