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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 <limits.h>
2
3void distribute(int *cookies, int cookiesSize, int k, int *distribution, int index, int *minUnfairness) {
4 if (index == cookiesSize) {
5 int maxCookies = INT_MIN;
6 for (int i = 0; i < k; i++) {
7 if (distribution[i] > maxCookies) {
8 maxCookies = distribution[i];
9 }
10 }
11 if (maxCookies < *minUnfairness) {
12 *minUnfairness = maxCookies;
13 }
14 return;
15 }
16 for (int i = 0; i < k; i++) {
17 distribution[i] += cookies[index];
18 distribute(cookies, cookiesSize, k, distribution, index + 1, minUnfairness);
19 distribution[i] -= cookies[index];
20 }
21}
22
23int distributeCookies(int* cookies, int cookiesSize, int k) {
24 int distribution[k];
25 for (int i = 0; i < k; i++) {
26 distribution[i] = 0;
27 }
28 int minUnfairness = INT_MAX;
29 distribute(cookies, cookiesSize, k, distribution, 0, &minUnfairness);
30 return minUnfairness;
31}This solution uses recursion to assign each cookie bag to one of the k children. Each recursive call explores adding the current cookie bag to any child's current collection. The recursion depth corresponds to processing each cookie bag. Once all bags are distributed, it computes the maximum cookies received by any child in that distribution (unfairness). The recursive solution uses backtracking to explore different paths and aims to minimize the maximum unfairness across all potential distributions.
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.
Utilizing Python’s computational proficiency, the solution employs a dynamic programming framework with bitmasking methodologies. This method entails armed subsets capable of tracking present distributions down to respective resources, allocating these across a root data figure. Consequential DPs facilitate computations along possible distributions, maintaining constant storage alignment on key objective modulations.