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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.
1function distributeCookies(cookies, k) {
2 let minUnfairness = Infinity;
3 let distribution = new Array(k).fill(0);
4
5 function distribute(index) {
6 if (index === cookies.length) {
7 minUnfairness = Math.min(minUnfairness, Math.max(...distribution));
8 return;
9 }
10 for (let i = 0; i < k; i++) {
11 distribution[i] += cookies[index];
12 distribute(index + 1);
13 distribution[i] -= cookies[index];
14 }
15 }
16
17 distribute(0);
18 return minUnfairness;
19}JavaScript employs similar recursive and backtracking measures, utilizing arrays and closures to track state and minimum outcome efficiently. Iterative conditional logic refines possible distributions and perpetuates backtracking to build out possibilities prior to result assessment.
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.
#include <algorithm>
#include <climits>
int distributeCookies(std::vector<int>& cookies, int k) {
int cookiesSize = cookies.size();
int maxMask = 1 << cookiesSize;
std::vector<std::vector<int>> dp(maxMask, std::vector<int>(k, INT_MAX));
dp[0][0] = 0;
for (int mask = 0; mask < maxMask; ++mask) {
for (int children = 0; children < k; ++children) {
if (dp[mask][children] == INT_MAX) continue;
for (int next = 0; next < cookiesSize; ++next) {
int cookieCost = 0;
for (int bit = 0; bit < cookiesSize; ++bit) {
if (mask & (1 << bit)) cookieCost += cookies[bit];
}
int newMask = mask;
for (int bit = next; bit < cookiesSize; ++bit) {
if (!(mask & (1 << bit))) {
cookieCost += cookies[bit];
newMask |= (1 << bit);
dp[newMask][children] = std::min(dp[newMask][children], std::max(cookieCost, dp[mask][children]));
}
}
}
}
}
return dp[maxMask - 1][k - 1];
}The C++ bitmasking and dynamic programming solution encapsulates all distribution states, efficiently cycling through subsets with visibility on child allocations. Here, masks afford subset encoding and assure unique cookie position tracking outbound accelerates search elimination. DP ensures localized caching of maximum states per derived subset.