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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.
1class 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 = Integer.MAX_VALUE;
8 for (int piv = start; piv <= end; piv++) {
9 int cost = piv + Math.max(piv - 1 >= start ? dp[start][piv - 1] : 0,
10 piv + 1 <= 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 System.out.println("Minimum money needed: " + sol.getMoneyAmount(10));
22 }
23}In the Java solution, the function getMoneyAmount computes the minimum cost using a bottom-up dynamic programming approach. It considers all possible guesses for each range, selects the guess with the minimum of the maximum costs, and updates the dp table accordingly. This solution yields the optimal result stored at dp[1][n].
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
In this Python approach, memoization is achieved using an instance class variable dp initialized to zeros. The recursive function ensures subproblems are evaluated only once per unique input values, avoiding overlapping calculations and thus improving efficiency.