Find K-th Smallest Pair Distance - Practice Coding Problems

Find K-th Smallest Pair Distance - Problem

The distance of a pair of integers a and b is defined as the absolute difference between a and b.

Given an integer array nums and an integer k, return the k^th smallest distance among all the pairs nums[i] and nums[j] where 0 <= i < j < nums.length.

Input & Output

Example 1 — Basic Case

$ Input: nums = [1,3,1], k = 1

› Output: 0

💡 Note: All pairs and their distances: (1,3)→2, (1,1)→0, (3,1)→2. Sorted distances: [0,2,2]. The 1st smallest is 0.

Example 2 — Second Smallest

$ Input: nums = [1,1,1], k = 2

› Output: 0

💡 Note: All pairs have the same values, so all distances are 0. The 2nd smallest distance is still 0.

Example 3 — Larger Array

$ Input: nums = [1,6,1], k = 3

› Output: 5

💡 Note: Pairs: (1,6)→5, (1,1)→0, (6,1)→5. Sorted: [0,5,5]. The 3rd smallest is 5.

Constraints

n == nums.length
2 ≤ n ≤ 10⁴
0 ≤ nums[i] ≤ 10⁶
1 ≤ k ≤ n(n-1)/2

Visualization

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Understanding the Visualization

Input

Array nums and integer k

Process

Calculate all pair distances or use binary search

Output

Return the kth smallest distance

Key Takeaway

🎯 Key Insight: Binary search on answer space with two-pointer counting is much more efficient than generating all O(n²) pairs

Asked in

G Google 15 f Facebook 12 a Amazon 8

The key insight is to use binary search on the answer space rather than generating all pairs. Instead of creating O(n²) pairs, we binary search on possible distance values and use two pointers to efficiently count pairs with distance ≤ candidate. Best approach is binary search with two pointers: Time O(n log n + n log W), Space O(1).

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force - Generate All Pairs	O(n² log n)	O(n²)	Generate all pair distances, sort them, and return the kth element
Binary Search on Answer Space	O(n log n + n log W)	O(1)	Binary search on possible distance values and count pairs with distance ≤ mid

Brute Force - Generate All Pairs — Algorithm Steps

Step 1: Generate all possible pairs and calculate their distances
Step 2: Sort all distances in ascending order
Step 3: Return the kth smallest distance (index k-1)

Visualization

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Step-by-Step Walkthrough

Generate Pairs

Calculate distance for every pair (i,j) where i < j

Sort Distances

Sort all distances in ascending order

Return Kth

Return the kth smallest distance

Code -

solution.c — C

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

int compare(const void *a, const void *b) {
    return (*(int*)a - *(int*)b);
}

int solution(int* nums, int numsSize, int k) {
    int* distances = malloc(numsSize * numsSize * sizeof(int));
    int count = 0;
    
    // Generate all pair distances
    for (int i = 0; i < numsSize; i++) {
        for (int j = i + 1; j < numsSize; j++) {
            distances[count++] = abs(nums[i] - nums[j]);
        }
    }
    
    // Sort and return kth smallest
    qsort(distances, count, sizeof(int), compare);
    int result = distances[k - 1];
    free(distances);
    return result;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    
    // Parse array
    int nums[1000];
    int numsSize = 0;
    char *token = strtok(line, "[],\n");
    while (token != NULL) {
        if (strlen(token) > 0 && token[0] != '[' && token[0] != ']') {
            nums[numsSize++] = atoi(token);
        }
        token = strtok(NULL, "[],\n");
    }
    
    int k;
    scanf("%d", &k);
    
    printf("%d\n", solution(nums, numsSize, k));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n² log n)

O(n²) to generate all pairs plus O(n² log n) to sort them

⚠ Quadratic Growth

Space Complexity

O(n²)

Array to store all O(n²) pair distances

⚠ Quadratic Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force - Generate All Pairs — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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