Non-decreasing Subsequences

Non-decreasing Subsequences - Problem

Given an integer array nums, return all the different possible non-decreasing subsequences of the given array with at least two elements. You may return the answer in any order.

A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

Note that the solution set must not contain duplicate subsequences.

Input & Output

Example 1 — Basic Case

$ Input: nums = [4,6,7,7]

› Output: [[4,6],[4,6,7],[4,6,7,7],[4,7],[4,7,7],[6,7],[6,7,7],[7,7]]

💡 Note: All possible non-decreasing subsequences with at least 2 elements. Note that [4,7,7] and [6,7,7] are different subsequences even though they contain the same values because they use different indices.

Example 2 — Minimum Size

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

› Output: [[4,4]]

💡 Note: Only [4,4] forms a valid non-decreasing subsequence with at least 2 elements. All other pairs are decreasing.

Example 3 — All Same Values

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

› Output: [[1,1],[1,1,1]]

💡 Note: We can form [1,1] (using any two 1s) and [1,1,1] (using all three 1s). Note that duplicate subsequences are eliminated.

Constraints

1 ≤ nums.length ≤ 15
-100 ≤ nums[i] ≤ 100

Visualization

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Asked in

G Google 15 a Amazon 12 f Facebook 8 M Microsoft 6

The optimal approach uses backtracking to build non-decreasing subsequences while preventing duplicates by tracking used values at each recursion level. Key insight: Use a set to avoid processing the same value multiple times at the same recursion depth. Best approach: Backtracking. Time: O(2^n), Space: O(n)

Common Approaches

✓ Backtracking with Duplicate Prevention

⏱️ Time: O(2^n) Space: O(n)

Use backtracking to build subsequences by choosing to include or exclude each element, ensuring non-decreasing order and preventing duplicates by tracking used values at each recursion level.

Brute Force - Generate All Combinations

⏱️ Time: O(2^n × n) Space: O(2^n × n)

Generate all 2^n possible subsequences using bit manipulation, then filter those that are non-decreasing with at least 2 elements, and remove duplicates.

Backtracking with Duplicate Prevention — Algorithm Steps

Start backtracking from each position in array
At each step, try adding elements that are ≥ current last element
Use set to prevent duplicate values at same recursion level
Add valid subsequences with length ≥ 2 to result

Visualization

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Step-by-Step Walkthrough

Choose Element

Pick an element that maintains non-decreasing order

Recurse

Build subsequence by adding more valid elements

Backtrack

Try other possibilities by removing last element

Code -

solution.c — C

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

int** globalResult;
int* globalColumnSizes;
int globalReturnSize;
int globalCapacity = 1000;

void backtrack(int* nums, int numsSize, int start, int* path, int pathSize) {
    if (pathSize >= 2) {
        if (globalReturnSize >= globalCapacity) {
            globalCapacity *= 2;
            globalResult = realloc(globalResult, globalCapacity * sizeof(int*));
            globalColumnSizes = realloc(globalColumnSizes, globalCapacity * sizeof(int));
        }
        
        globalResult[globalReturnSize] = malloc(pathSize * sizeof(int));
        for (int i = 0; i < pathSize; i++) {
            globalResult[globalReturnSize][i] = path[i];
        }
        globalColumnSizes[globalReturnSize] = pathSize;
        globalReturnSize++;
    }
    
    int used[1000] = {0}; // Assuming values fit in reasonable range
    for (int i = start; i < numsSize; i++) {
        // Simple duplicate check (assuming reasonable range)
        int val = nums[i] + 500; // Offset for negative numbers
        if (val >= 0 && val < 1000 && used[val]) {
            continue;
        }
        
        // Only add if it maintains non-decreasing order
        if (pathSize == 0 || nums[i] >= path[pathSize - 1]) {
            if (val >= 0 && val < 1000) used[val] = 1;
            path[pathSize] = nums[i];
            backtrack(nums, numsSize, i + 1, path, pathSize + 1);
        }
    }
}

int** solution(int* nums, int numsSize, int* returnSize, int** returnColumnSizes) {
    globalResult = malloc(globalCapacity * sizeof(int*));
    globalColumnSizes = malloc(globalCapacity * sizeof(int));
    globalReturnSize = 0;
    
    int path[20];
    backtrack(nums, numsSize, 0, path, 0);
    
    *returnSize = globalReturnSize;
    *returnColumnSizes = globalColumnSizes;
    return globalResult;
}

void parseArray(const char* str, int* arr, int* size) {
    *size = 0;
    const char* p = str;
    while (*p && *p != '[') p++;
    if (*p == '[') p++;
    while (*p && *p != ']') {
        while (*p == ' ' || *p == ',') p++;
        if (*p == ']' || *p == '\0') break;
        arr[(*size)++] = (int)strtol(p, (char**)&p, 10);
    }
}

int main() {
    char line[1000];
    fgets(line, sizeof(line), stdin);
    
    // Simple array parsing
    int nums[20];
    int numsSize = 0;
    char* token = strtok(line, "[,]");
    while (token != NULL) {
        if (strlen(token) > 0 && token[0] != '\n') {
            nums[numsSize++] = atoi(token);
        }
        token = strtok(NULL, "[,]");
    }
    
    int returnSize;
    int* returnColumnSizes;
    int** result = solution(nums, numsSize, &returnSize, &returnColumnSizes);
    
    printf("[");
    for (int i = 0; i < returnSize; i++) {
        printf("[");
        for (int j = 0; j < returnColumnSizes[i]; j++) {
            printf("%d", result[i][j]);
            if (j < returnColumnSizes[i] - 1) printf(",");
        }
        printf("]");
        if (i < returnSize - 1) printf(",");
    }
    printf("]\n");
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(2^n)

Each element can be included or excluded, with pruning for duplicates

⚠ Quadratic Growth

Space Complexity

O(n)

Recursion depth and temporary subsequence storage

⚠ Quadratic Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Backtracking with Duplicate Prevention — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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