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This method uses a backtracking technique, where we recursively build each combination by adding numbers incrementally and backtrack whenever we've constructed combinations of size k or cannot extend further.
Time Complexity: O(C(n, k) * k), where C(n, k) is the binomial coefficient. Each combination is built in O(k) time.
Space Complexity: O(k) due to the recursion stack storing the current combination.
1function combine(n, k) {
2 const result = [];
3 const backtrack = (start, current) => {
4 if (current.length === k) {
5 result.push([...current]);
6 return;
7 }
8 for (let i = start; i <= n; i++) {
9 current.push(i);
10 backtrack(i + 1, current);
11 current.pop();
12 }
13 };
14
15 backtrack(1, []);
16 return result;
17}
18
19console.log(combine(4, 2));The JavaScript implementation uses closures to achieve backtracking. The backtrack function leverages an inner function to track the current path and recursion start, exploring every possible k-sized combination.
Bit masking provides a novel way to generate combinations by representing choice decisions (e.g., pick or skip) in binary. A bit mask is applied to the set of numbers [1...n], and for each bit position, if the bit is set, that number is included in the combination.
Time Complexity: O(2n * n), as we iterate over all possible subsets of [1...n] and check the bit count.
Space Complexity: O(C(n, k) * k) for storing all combinations.
1#include <vector>
std::vector<std::vector<int>> combine(int n, int k) {
std::vector<std::vector<int>> result;
int total = 1 << n;
for (int mask = 0; mask < total; ++mask) {
if (__builtin_popcount(mask) == k) {
std::vector<int> current;
for (int i = 0; i < n; ++i) {
if (mask & (1 << i)) {
current.push_back(i + 1);
}
}
result.push_back(current);
}
}
return result;
}
int main() {
int n = 4, k = 2;
std::vector<std::vector<int>> combinations = combine(n, k);
for (const auto& combo : combinations) {
std::cout << "[";
for (size_t i = 0; i < combo.size(); ++i) {
std::cout << combo[i];
if (i < combo.size() - 1) std::cout << ", ";
}
std::cout << "]\n";
}
return 0;
}In C++, we traverse numbers 0 to 2n-1, representing each as a bitmask. We use the __builtin_popcount function to count 1s in the mask, equating to picking elements for a combination.