This approach uses recursion and backtracking to generate all possible combinations by exploring each candidate. If a candidate is chosen, we explore further with the remaining target reduced by the candidate's value. We use backtracking to remove a candidate and try the next one. This way, we explore all possible combinations using depth-first search.
Time Complexity: O(2^T), where T is the target, as it potentially explores every combination of the candidates.
Space Complexity: O(T) for the recursion call stack and the path being explored.
1#include <stdio.h>
2#include <stdlib.h>
3#include <string.h>
4
5void findCombination(int* candidates, int candidatesSize, int target, int** result, int* returnColSize, int* returnSize, int* currentCombo, int currentComboSize, int startIndex) {
6 if (target < 0) return;
7 if (target == 0) {
8 result[*returnSize] = (int*)malloc(sizeof(int) * currentComboSize);
9 memcpy(result[*returnSize], currentCombo, sizeof(int) * currentComboSize);
10 returnColSize[*returnSize] = currentComboSize;
11 (*returnSize)++;
12 return;
13 }
14
15 for (int i = startIndex; i < candidatesSize; i++) {
16 currentCombo[currentComboSize] = candidates[i];
17 findCombination(candidates, candidatesSize, target - candidates[i], result, returnColSize, returnSize, currentCombo, currentComboSize + 1, i);
18 }
19}
20
21int** combinationSum(int* candidates, int candidatesSize, int target, int** columnSizes, int* returnSize) {
22 int** result = (int**)malloc(150 * sizeof(int*));
23 *columnSizes = (int*)malloc(150 * sizeof(int));
24 *returnSize = 0;
25 int* currentCombo = (int*)malloc(target * sizeof(int));
26 findCombination(candidates, candidatesSize, target, result, *columnSizes, returnSize, currentCombo, 0, 0);
27 free(currentCombo);
28 return result;
29}
30
31int main() {
32 int candidates[] = {2, 3, 6, 7};
33 int candidatesSize = 4;
34 int target = 7;
35 int* columnSizes;
36 int returnSize;
37 int** result = combinationSum(candidates, candidatesSize, target, &columnSizes, &returnSize);
38 for (int i = 0; i < returnSize; i++) {
39 printf("[");
40 for (int j = 0; j < columnSizes[i]; j++) {
41 printf("%d%s", result[i][j], j == columnSizes[i] - 1 ? "" : ", ");
42 }
43 printf("]\n");
44 free(result[i]);
45 }
46 free(result);
47 free(columnSizes);
48 return 0;
49}
50This C solution uses recursion to explore combinations and backtracks when a target less than zero is reached. If the target is zero, a valid combination is found and added to the result. The helper function findCombination iterates through candidates allowing reuse of the current candidate.
We can use a dynamic programming (DP) approach to solve the Combination Sum problem. We maintain a DP array where each index represents the number of ways to form that particular sum using the candidates. The approach involves iterating through candidates and updating the DP array for each possible sum that can be formed with that candidate.
Time Complexity: Roughly O(T * N), for target T and candidates N.
Space Complexity: O(T) for the dp array itself in context of storage.
1using System;
2using System.Collections.Generic;
3
4class CombinationSumDPSolution {
5 public IList<IList<int>> CombinationSum(int[] candidates, int target) {
6 List<IList<int>>[] dp = new List<IList<int>>[target + 1];
7 dp[0] = new List<IList<int>> { new List<int>() };
8
9 for (int i = 1; i <= target; i++) {
10 dp[i] = new List<IList<int>>();
11 foreach (int candidate in candidates) {
12 if (i >= candidate) {
13 foreach (var combination in dp[i - candidate]) {
14 var newComb = new List<int>(combination);
15 newComb.Add(candidate);
16 dp[i].Add(newComb);
17 }
18 }
19 }
20 }
21 return dp[target];
22 }
23
24 static void Main(string[] args) {
25 int[] candidates = {2, 3, 6, 7};
26 int target = 7;
27 CombinationSumDPSolution solution = new CombinationSumDPSolution();
28 var result = solution.CombinationSum(candidates, target);
29 foreach (var combination in result) {
30 Console.WriteLine("[" + string.Join(", ", combination) + "]");
31 }
32 }
33}
34In C#, the DP approach initializes arrays at indices corresponding to all sums up to target using candidates progressively, storing all valid combinations for each sum index after iterating through possible sums.