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To maximize the like-time coefficient, we can sort the satisfaction levels in non-descending order and calculate the possible like-time coefficients by considering prefixes from the end of the sorted array. This allows us to selectively choose higher satisfaction dishes to maximize the total like-time coefficient.
Time Complexity: O(n log n) due to sorting.
Space Complexity: O(1) for using a constant amount of extra space.
1#include <iostream>
2#include <vector>
3#include <algorithm>
4
5int maxSatisfaction(std::vector<int>& satisfaction) {
6 std::sort(satisfaction.begin(), satisfaction.end(), std::greater<int>());
7 int total = 0, sum = 0;
8 for (int s : satisfaction) {
9 sum += s;
10 if (sum > 0) {
11 total += sum;
12 } else {
13 break;
14 }
15 }
16 return total;
17}
18
19int main() {
20 std::vector<int> satisfaction = {-1, -8, 0, 5, -9};
21 std::cout << maxSatisfaction(satisfaction);
22 return 0;
23}This C++ solution similarly sorts the satisfaction levels in decreasing order. It then iteratively accumulates the like-time coefficient, stopping when the running sum becomes non-positive.
An alternative approach involves using dynamic programming to maximize the like-time coefficient. We maintain a DP table where `dp[i]` represents the maximum like-time coefficient attainable using the first `i` dishes. This approach computes overlap coefficients dynamically using state transitions calculated for each dish considered.
Time Complexity: O(n^2)
Space Complexity: O(n)
1
class Program {
static int MaxSatisfaction(int[] satisfaction) {
Array.Sort(satisfaction);
int n = satisfaction.Length;
int[] dp = new int[n + 1];
for (int i = n - 1; i >= 0; i--) {
for (int t = 0; t <= i; t++) {
dp[t] = Math.Max(dp[t], dp[t + 1] + satisfaction[i] * (t + 1));
}
}
return dp[0];
}
static void Main() {
int[] satisfaction = {4, 3, 2};
Console.WriteLine(MaxSatisfaction(satisfaction));
}
}The C# program adopts dynamic programming to make state transitions represent time indices for prepared dishes, updating choices dynamically to achieve the maximum like-time coefficient.