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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.
1from itertools import permutations
2
3class Solution:
4 def distributeCookies(self, cookies, k):
5 def distribute(index):
6 nonlocal min_unfairness
7 if index == len(cookies):
8 min_unfairness = min(min_unfairness, max(distribution))
9 return
10 for i in range(k):
11 distribution[i] += cookies[index]
12 distribute(index + 1)
13 distribution[i] -= cookies[index]
14
15 distribution = [0] * k
16 min_unfairness = float('inf')
17 distribute(0)
18 return min_unfairnessThe Python solution uses recursion and backtracking with a helper function to explore cookie distribution. Inbuilt dividers are leveraged to handle each recursive step systematically. The algorithm seeks to minimize the worst distribution outcome through compute comparisons across recursive completions.
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.
public class Solution {
public int DistributeCookies(int[] cookies, int k) {
int cookiesSize = cookies.Length;
int maxMask = 1 << cookiesSize;
int[][] dp = new int[maxMask][];
for (int i = 0; i < maxMask; i++) {
dp[i] = new int[k];
Array.Fill(dp[i], int.MaxValue);
}
dp[0][0] = 0;
for (int mask = 0; mask < maxMask; mask++) {
for (int childNum = 0; childNum < k; childNum++) {
if (dp[mask][childNum] == int.MaxValue) continue;
int cookieCost = 0;
for (int bit = 0; bit < cookiesSize; bit++) {
if ((mask & (1 << bit)) != 0) {
cookieCost += cookies[bit];
}
}
for (int next = 0; next < cookiesSize; next++) {
int newMask = mask;
int updatedCost = cookieCost;
for (int bit = next; bit < cookiesSize; bit++) {
if ((mask & (1 << bit)) == 0) {
updatedCost += cookies[bit];
newMask |= (1 << bit);
dp[newMask][childNum] = Math.Min(dp[newMask][childNum], Math.Max(updatedCost, dp[mask][childNum]));
}
}
}
}
}
return dp[maxMask - 1][k - 1];
}
}C# exploits masking tactically paired with dynamic programming principles, offsetting calculations onto refined schemas offering a symmetrical balance between states per completion. A traditional pseudo-DP adaptation oversees these states by their progressive bitmask mappings, ensuring minimized resolves on comprehensive blanket alternative expositions.