Minimum Impossible OR

Minimum Impossible OR - Problem

You are given a 0-indexed integer array nums.

We say that an integer x is expressible from nums if there exist some integers 0 <= index1 < index2 < ... < indexk < nums.length for which nums[index1] | nums[index2] | ... | nums[indexk] = x. In other words, an integer is expressible if it can be written as the bitwise OR of some subsequence of nums.

Return the minimum positive non-zero integer that is not expressible from nums.

Input & Output

Example 1 — Basic Case

$ Input: nums = [1,2]

› Output: 4

💡 Note: 1 OR 2 = 3. Expressible: 1, 2, 3. Missing: 4 is the minimum positive integer not expressible.

Example 2 — Single Element

$ Input: nums = [5]

› Output: 1

💡 Note: Only 5 is expressible. 1 is the minimum positive integer not expressible.

Example 3 — Multiple Elements

$ Input: nums = [3,1]

› Output: 2

💡 Note: Expressible: 1, 3, 1|3=3. So expressible are {1,3}. Minimum missing: 2.

Constraints

1 ≤ nums.length ≤ 10⁵
1 ≤ nums[i] ≤ 10⁹

Visualization

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Asked in

G Google 15 f Facebook 12 a Amazon 8

The key insight is that OR operations can only set bits, never unset them. To form any number with bit k set, we need at least one input number with bit k set. Best approach is checking each power of 2 until we find one that cannot be formed. Time: O(n × log(max)), Space: O(1)

Common Approaches

✓ Generate All Possible OR Values

⏱️ Time: O(n × k) Space: O(k)

For each number in the array, we generate all possible OR values by combining it with previously computed OR values. This builds up all expressible numbers.

Power of 2 Insight

⏱️ Time: O(n × log(max)) Space: O(1)

Key insight: OR operations can only set bits, never unset them. To create a number with bit k set, we need at least one number in our array with bit k set. The smallest missing power of 2 is our answer.

Generate All Possible OR Values — Algorithm Steps

Start with empty set of possible OR values
For each number, create new OR values by combining with existing ones
Find the minimum positive integer not in the set

Visualization

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Step-by-Step Walkthrough

Initialize

Start with {0} as base

Process Each Number

OR each existing value with current number

Find Minimum

Find smallest positive integer not in set

Code -

solution.c — C

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

int solution(int* nums, int numsSize) {
    bool possible[1000000] = {false}; // Assume reasonable upper bound
    possible[0] = true;
    
    for (int i = 0; i < numsSize; i++) {
        bool newPossible[1000000] = {false};
        for (int j = 0; j < 1000000; j++) {
            if (possible[j]) {
                int orVal = j | nums[i];
                if (orVal < 1000000) {
                    newPossible[orVal] = true;
                }
            }
        }
        for (int j = 0; j < 1000000; j++) {
            if (newPossible[j]) {
                possible[j] = true;
            }
        }
    }
    
    // Find minimum positive integer not in possible
    int result = 1;
    while (result < 1000000 && possible[result]) {
        result++;
    }
    
    return result;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    
    // Parse JSON array manually
    int nums[1000];
    int numsSize = 0;
    
    char* ptr = line + 1; // Skip '['
    while (*ptr != ']') {
        if (*ptr == ',' || *ptr == ' ') {
            ptr++;
            continue;
        }
        nums[numsSize++] = atoi(ptr);
        while (*ptr != ',' && *ptr != ']') ptr++;
    }
    
    int result = solution(nums, numsSize);
    printf("%d\n", result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n × k)

n numbers, each can create at most k unique OR values (bounded by bit patterns)

⚡ Linearithmic

Space Complexity

O(k)

Set stores all unique OR values, bounded by possible bit combinations

✓ Linear Space

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Smart Actions

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Input & Output

Constraints

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Related Problems

Common Approaches

Generate All Possible OR Values — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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