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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.
This JavaScript function sorts the array and calculates the total like-time coefficient from the back, selecting positive prefix sums only.
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)
1def max_satisfaction(satisfaction):
2 satisfaction.sort()
3 n = len(satisfaction)
4 dp = [0] * (n + 1)
5 for i in range(n - 1, -1, -1):
6 for t in range(i + 1):
7 dp[t] = max(dp[t], dp[t + 1] + satisfaction[i] * (t + 1))
8 return dp[0]
9
10satisfaction = [4, 3, 2]
11print(max_satisfaction(satisfaction))This Python solution handles the problem using a dynamic programming array that is updated in reverse order while considering whether or not to include each dish for maximum results.