Distribute Elements Into Two Arrays II

Distribute Elements Into Two Arrays II - Problem

You are given a 1-indexed array of integers nums of length n.

We define a function greaterCount such that greaterCount(arr, val) returns the number of elements in arr that are strictly greater than val.

You need to distribute all the elements of nums between two arrays arr1 and arr2 using n operations.

In the first operation, append nums[1] to arr1. In the second operation, append nums[2] to arr2.

Afterwards, in the ith operation:

If greaterCount(arr1, nums[i]) > greaterCount(arr2, nums[i]), append nums[i] to arr1.
If greaterCount(arr1, nums[i]) < greaterCount(arr2, nums[i]), append nums[i] to arr2.
If greaterCount(arr1, nums[i]) == greaterCount(arr2, nums[i]), append nums[i] to the array with a lesser number of elements.
If there is still a tie, append nums[i] to arr1.

The array result is formed by concatenating the arrays arr1 and arr2.

Return the integer array result.

Input & Output

Example 1 — Basic Distribution

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

› Output: [2,3,1,3]

💡 Note: Start: arr1=[2], arr2=[1]. For nums[2]=3: count1=1 (2>3? no), count2=0 (1>3? no), so 0<1, add to arr2. For nums[3]=3: count1=1, count2=1, tie goes to smaller array (arr1). Result: arr1=[2,3], arr2=[1,3] → [2,3,1,3]

Example 2 — Equal Counts

$ Input: nums = [5,4,3,8]

› Output: [5,3,4,8]

💡 Note: Start: arr1=[5], arr2=[4]. For nums[2]=3: count1=1 (5>3), count2=1 (4>3), equal counts and equal sizes, tie goes to arr1. For nums[3]=8: count1=0, count2=0, equal counts, arr1 smaller, add to arr1.

Example 3 — Minimum Size

$ Input: nums = [1,2]

› Output: [1,2]

💡 Note: Only two elements, first goes to arr1, second to arr2. Result is concatenation: [1,2]

Constraints

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

Visualization

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Asked in

G Google 15 M Microsoft 12 a Amazon 8

The key insight is efficiently counting elements greater than a given value in two dynamic arrays. The brute force approach uses linear scanning (O(n²)), while the optimal solution uses Binary Indexed Trees with coordinate compression for O(n log n) time complexity.

Common Approaches

✓ Binary Indexed Tree (Fenwick Tree)

⏱️ Time: O(n log n) Space: O(n)

Coordinate compress the values and use Binary Indexed Trees to maintain counts for each array. This allows O(log n) queries for counting elements greater than a given value.

Brute Force Simulation

⏱️ Time: O(n²) Space: O(n)

For each element from index 3 onwards, count how many elements in arr1 and arr2 are greater than the current element. Make the decision based on these counts and distribute accordingly.

Binary Indexed Tree (Fenwick Tree) — Algorithm Steps

Coordinate compress all values to handle large ranges
Maintain two BITs for arr1 and arr2 counts
For each element, query BIT for greater count and update accordingly

Visualization

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Step-by-Step Walkthrough

Coordinate Compression

Map values to smaller range [1, m]

BIT Query

Query range [val+1, m] for count > val

Update

Add element to chosen array and update BIT

Code -

solution.c — C

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

typedef struct {
    int n;
    int* tree;
} BIT;

BIT* createBIT(int n) {
    BIT* bit = (BIT*)malloc(sizeof(BIT));
    bit->n = n;
    bit->tree = (int*)calloc(n + 1, sizeof(int));
    return bit;
}

void updateBIT(BIT* bit, int i, int delta) {
    while (i <= bit->n) {
        bit->tree[i] += delta;
        i += i & (-i);
    }
}

int queryBIT(BIT* bit, int i) {
    int res = 0;
    while (i > 0) {
        res += bit->tree[i];
        i -= i & (-i);
    }
    return res;
}

int rangeQueryBIT(BIT* bit, int l, int r) {
    if (l > r) return 0;
    return queryBIT(bit, r) - queryBIT(bit, l - 1);
}

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

int* solution(int* nums, int n, int* returnSize) {
    *returnSize = n;
    if (n == 0) return NULL;
    if (n <= 2) {
        int* result = (int*)malloc(n * sizeof(int));
        for (int i = 0; i < n; i++) result[i] = nums[i];
        return result;
    }
    
    // Simple coordinate compression
    int* sortedVals = (int*)malloc(n * sizeof(int));
    for (int i = 0; i < n; i++) sortedVals[i] = nums[i];
    qsort(sortedVals, n, sizeof(int), compare);
    
    int m = 1;
    for (int i = 1; i < n; i++) {
        if (sortedVals[i] != sortedVals[m-1]) {
            sortedVals[m++] = sortedVals[i];
        }
    }
    
    BIT* bit1 = createBIT(m);
    BIT* bit2 = createBIT(m);
    
    int* result = (int*)malloc(n * sizeof(int));
    int* arr1 = (int*)malloc(n * sizeof(int));
    int* arr2 = (int*)malloc(n * sizeof(int));
    int size1 = 0, size2 = 0;
    
    arr1[size1++] = nums[0];
    arr2[size2++] = nums[1];
    
    // Find coordinates and update BITs
    int coord1 = 0, coord2 = 0;
    for (int i = 0; i < m; i++) {
        if (sortedVals[i] == nums[0]) coord1 = i + 1;
        if (sortedVals[i] == nums[1]) coord2 = i + 1;
    }
    updateBIT(bit1, coord1, 1);
    updateBIT(bit2, coord2, 1);
    
    for (int i = 2; i < n; i++) {
        int val = nums[i];
        int valCoord = 0;
        for (int j = 0; j < m; j++) {
            if (sortedVals[j] == val) {
                valCoord = j + 1;
                break;
            }
        }
        
        int count1 = rangeQueryBIT(bit1, valCoord + 1, m);
        int count2 = rangeQueryBIT(bit2, valCoord + 1, m);
        
        if (count1 > count2) {
            arr1[size1++] = val;
            updateBIT(bit1, valCoord, 1);
        } else if (count2 > count1) {
            arr2[size2++] = val;
            updateBIT(bit2, valCoord, 1);
        } else if (size1 <= size2) {
            arr1[size1++] = val;
            updateBIT(bit1, valCoord, 1);
        } else {
            arr2[size2++] = val;
            updateBIT(bit2, valCoord, 1);
        }
    }
    
    for (int i = 0; i < size1; i++) result[i] = arr1[i];
    for (int i = 0; i < size2; i++) result[size1 + i] = arr2[i];
    
    free(sortedVals);
    free(bit1->tree);
    free(bit1);
    free(bit2->tree);
    free(bit2);
    free(arr1);
    free(arr2);
    
    return result;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    
    int nums[1000], n = 0;
    char* token = strtok(line, "[,]");
    while (token != NULL) {
        if (strlen(token) > 0 && token[0] != '\n') {
            nums[n++] = atoi(token);
        }
        token = strtok(NULL, "[,]");
    }
    
    int returnSize;
    int* result = solution(nums, n, &returnSize);
    
    printf("[");
    for (int i = 0; i < returnSize; i++) {
        printf("%d", result[i]);
        if (i < returnSize - 1) printf(",");
    }
    printf("]\n");
    
    free(result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n log n)

Coordinate compression takes O(n log n), then n operations each taking O(log n) for BIT queries and updates

⚠ Quadratic Growth

Space Complexity

O(n)

Space for coordinate mapping and two Binary Indexed Trees

⚡ Linearithmic Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Binary Indexed Tree (Fenwick Tree) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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