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This approach involves using dynamic programming with recursion to solve the problem. The main idea is to break the problem into subproblems: determining the cost of guessing any number within a given range. For each number in the current range, calculate the worst-case cost of guessing that number, accounting for both possibilities where the picked number is less or greater than the guessed number. Store solutions to subproblems to avoid recalculating them.
Time Complexity: O(n^3) due to the three nested loops through possible ranges and guesses.
Space Complexity: O(n^2) for storing results of subproblems.
1public class Solution {
2 public int GetMoneyAmount(int n) {
3 int[,] dp = new int[n + 1, n + 1];
4 for (int len = 2; len <= n; len++) {
5 for (int start = 1; start <= n - len + 1; start++) {
6 int end = start + len - 1;
7 int minCost = int.MaxValue;
8 for (int piv = start; piv <= end; piv++) {
9 int cost = piv + Math.Max(piv > start ? dp[start, piv - 1] : 0,
10 piv < end ? dp[piv + 1, end] : 0);
11 minCost = Math.Min(minCost, cost);
12 }
13 dp[start, end] = minCost;
14 }
15 }
16 return dp[1, n];
17 }
18
19 public static void Main(string[] args) {
20 Solution sol = new Solution();
21 Console.WriteLine("Minimum money needed: " + sol.GetMoneyAmount(10));
22 }
23}The C# solution applies the same dynamic programming strategy as other languages. Here, an integer matrix dp is used to store the minimum costs for various number ranges. The outer loop ensures increasing range sizes, and the cheapest worst-case cost within each range is stored. Finally, dp[1, n] is returned as the solution.
A more advanced recursive approach with memoization achieves similar results, storing results of previously solved subproblems in a cache for fast retrieval. This is especially useful for large inputs, reducing computational overhead by avoiding repeated calculations of the same conditions and ranges.
Time Complexity: O(n^2), reduced with memoization.
Space Complexity: O(n^2), for memoizing results.
1 private int[,] dp;
private int MinCost(int start, int end) {
if (start >= end) return 0;
if (dp[start, end] != 0) return dp[start, end];
int res = int.MaxValue;
for (int piv = start; piv <= end; piv++) {
int cost = piv + Math.Max(MinCost(start, piv - 1), MinCost(piv + 1, end));
res = Math.Min(res, cost);
}
dp[start, end] = res;
return res;
}
public int GetMoneyAmount(int n) {
dp = new int[n+1, n+1];
return MinCost(1, n);
}
public static void Main(string[] args) {
Solution sol = new Solution();
System.Console.WriteLine("Minimum amount to guarantee a win: " + sol.GetMoneyAmount(10));
}
}This C# implementation utilizes recursive memoization in its code structure. It efficiently calculates required costs for all possible number ranges by storing intermediate results and retrieving them as needed, enhancing the overall speed and reducing redundant processing.