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This approach leverages backtracking to explore all possible ways of distributing the bags of cookies among the children. We utilize a recursive function to attempt to give each bag of cookies to any of the available children. At each step, we update the total cookies received by the child to whom the current bag is assigned. The goal is to minimize the maximum number of cookies any child receives, known as unfairness.
Time Complexity: O(k^n), where n is the number of cookie bags and k is the number of children. The solution explores every distribution possibility.
Space Complexity: O(k + n), due to the recursion stack and the distribution array.
1#include <vector>
2#include <algorithm>
3#include <climits>
4
5void distribute(std::vector<int>& cookies, std::vector<int>& distribution, int index, int& minUnfairness) {
6 if (index == cookies.size()) {
7 minUnfairness = std::min(minUnfairness, *std::max_element(distribution.begin(), distribution.end()));
8 return;
9 }
10 for (int& d : distribution) {
11 d += cookies[index];
12 distribute(cookies, distribution, index + 1, minUnfairness);
13 d -= cookies[index];
14 }
15}
16
17int distributeCookies(std::vector<int>& cookies, int k) {
18 std::vector<int> distribution(k, 0);
19 int minUnfairness = INT_MAX;
20 distribute(cookies, distribution, 0, minUnfairness);
21 return minUnfairness;
22}The C++ solution uses a recursive function similar to the C solution. It takes advantage of C++ STL utilities such as vectors and algorithms. The recursion matches cookie bags to children, checking the resultant unfairness after all distributions, seeking to update the minimum.
This approach uses bitmasking in conjunction with dynamic programming to keep track of all combinations of cookie distributions among k children. We aim to optimize processing by encoding choices using bitmasks to represent different subsets of cookies. This method alleviates computational overhead by avoiding permutation/combination evaluation via iterative states and sub-state record tracking. We leverage a DP technique where dp[mask][children] represents the minimum unfairness achievable assigning the selected cookie combination (mask) across given children.
Time Complexity: O(n * 2^n * k), dealing with both subset and child assignment constraints.
Space Complexity: O(2^n * k), necessary for complete subset and branch caching.
The C solution here employs a 2D array to track the minimal unfairness through a combination of subsets and children allocation states. Utilizes numerous possible sub-states via masks representing cookie allocation realization across valid compartments. Varied subsets are computed iteratively ensuring that the unnecessary re-evaluation of states is avoided through DP caching.