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In this approach, we use dynamic programming (DP) to solve the problem efficiently. The idea is to set up a DP table where dp[k][n] represents the minimum number of moves required to find the critical floor with k eggs and n floors.
We can use binary search to determine which floor to drop the egg from in each step, which helps to optimize the number of trials needed. The recurrence relation can be formulated as:
dp[k][n] = 1 + min(max(dp[k-1][x-1], dp[k][n-x])) for all 1 <= x <= n.
This relation means that for each floor x, you either continue with the remaining floors if the egg doesn't break, or with one fewer egg if it does break. The goal is to find the minimal worst-case scenario.
Time Complexity: O(k * n * log n) because each fill operation of the DP table involves a binary search over the number of floors.
Space Complexity: O(k * n) because of the DP table holding computed results.
1using System;
2
3public class Solution {
4 public int SuperEggDrop(int k, int n) {
5 int[,] dp = new int[k + 1, n + 1];
6 for (int i = 0; i <= k; i++) {
7 dp[i, 0] = 0;
8 if (i > 0) {
9 dp[i, 1] = 1;
10 }
11 }
12 for (int j = 0; j <= n; j++) {
13 dp[1, j] = j;
14 }
for (int i = 2; i <= k; i++) {
for (int j = 2; j <= n; j++) {
int low = 1, high = j, mid, ans = int.MaxValue;
while (low <= high) {
mid = low + (high - low) / 2;
int broken = dp[i - 1, mid - 1];
int notBroken = dp[i, j - mid];
ans = Math.Min(ans, 1 + Math.Max(broken, notBroken));
if (broken > notBroken) {
high = mid - 1;
} else {
low = mid + 1;
}
}
dp[i, j] = ans;
}
}
return dp[k, n];
}
public static void Main(string[] args) {
Solution sol = new Solution();
Console.WriteLine(sol.SuperEggDrop(3, 14)); // Output: 4
}
}This C# solution also creates a 2D array dp where dp[i,j] calculates the minimum moves required for a given i eggs and j floors utilizing binary search. Base case results are directly assigned, subsequently filling the table iteratively as it explores every possible drop scenario.
Here, we use a mathematical approach combined with dynamic programming. Instead of trying to calculate the minimum number of moves directly, we use a theoretical approach that considers the problem of breaking down moves into binary forms.
We change the problem into one of maximizing the number of floors we can test with a given number of drops. We can calculate m moves where the result is T[k][m] = T[k-1][m-1] + T[k][m-1] + 1. If T[k][m] ≥ n, we found the minimum m.
Time Complexity: O(k * n), because we directly shift through combinations.
Space Complexity: O(n) since we're using array-based exploratory floor encoding.
#include <vector>
using namespace std;
int superEggDrop(int k, int n) {
vector<int> dp(n + 1, 0);
for (int i = 0; i <= n; ++i) dp[i] = i;
for (int i = 2; i <= k; ++i) {
vector<int> dp_new(n + 1, 0);
int x = 1;
for (int j = 1; j <= n; ++j) {
while (x < j && max(dp[x - 1], dp_new[j - x]) > max(dp[x], dp_new[j - x - 1])) {
++x;
}
dp_new[j] = 1 + max(dp[x - 1], dp_new[j - x]);
}
dp = dp_new;
}
return dp[n];
}
int main() {
cout << superEggDrop(2, 6) << endl; // Output: 3
return 0;
}This C++ implementation uses a 1D dynamic programming array to track the maximum floors that can be tested with a given number of drops.
The two arrays dp and dp_new represent the current and the next state of testing, respectively. The variable x helps dynamically find the balance point in egg drops.