#1402

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Reducing Dishes

Maximize the sum of like-time coefficients by optimally choosing dishes to prepare in this dynamic programming problem.

array dynamic programming greedy sorting

Problem Statement

A chef has recorded the satisfaction levels of n dishes. Each dish takes 1 unit of time to prepare, and its like-time coefficient depends on both the time taken to cook it and its satisfaction level. The like-time coefficient for a dish is the product of the satisfaction level and the time taken to cook it, including prior dishes. Your task is to return the maximum sum of the like-time coefficients the chef can achieve by preparing some of the dishes.

To solve this, the chef may choose any number of dishes, but the order of preparation can affect the result. The challenge lies in optimizing the selection and order of dishes to maximize the sum of like-time coefficients. Use dynamic programming to find the optimal solution, keeping track of previous best results.

Examples

Example 1

Input: satisfaction = [-1,-8,0,5,-9]

Output: 14

After Removing the second and last dish, the maximum total like-time coefficient will be equal to (-1 1 + 0 2 + 5*3 = 14). Each dish is prepared in one unit of time.

Example 2

Input: satisfaction = [4,3,2]

Output: 20

Dishes can be prepared in any order, (2 1 + 3 2 + 4*3 = 20)

Example 3

Input: satisfaction = [-1,-4,-5]

Output: 0

People do not like the dishes. No dish is prepared.

Constraints

n == satisfaction.length
1 <= n <= 500
-1000 <= satisfaction[i] <= 1000

Solution Approach

State Transition Dynamic Programming

Use dynamic programming to track the maximum possible sum of like-time coefficients at each stage, considering subsets of dishes. The state transitions depend on the decision to either include or exclude a dish, factoring in its satisfaction and the time at which it is prepared.

Greedy Ordering of Dishes

To maximize the like-time coefficient, dishes should be considered in order of decreasing satisfaction values. By preparing dishes with higher satisfaction first, you maximize the impact of the time multiplier, which increases as more dishes are prepared.

Space and Time Optimization

Optimize the solution by storing only the necessary results for previous stages of the dynamic programming calculation, thus reducing both time and space complexity. Avoid recalculating values that are already computed.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the dynamic programming approach. The solution typically runs in O(n log n) due to sorting and the dynamic programming step, where n is the number of dishes. The space complexity is O(n), as we need to store intermediate results for each subset of dishes.

What Interviewers Usually Probe

Can the candidate efficiently use dynamic programming with state transitions to reduce computation time?
Does the candidate recognize the importance of sorting dishes in decreasing order of satisfaction for optimizing results?
Is the candidate able to optimize the space complexity while maintaining correctness in the dynamic programming approach?

Common Pitfalls or Variants

Common pitfalls

Not considering the importance of sorting dishes, which can lead to suboptimal solutions.
Confusing the order of operations in dynamic programming, such as forgetting to update previous states or incorrectly tracking the maximum sum.
Overcomplicating the space complexity, when it can be reduced by storing only necessary results.

Follow-up variants

What if the satisfaction values are all negative? Can the algorithm still function effectively?
How would this problem change if there were a maximum limit on how many dishes could be prepared?
How would the solution scale if the number of dishes is significantly larger than 500?

FAQ

How do I approach the 'Reducing Dishes' problem in dynamic programming?

Start by sorting the dishes by satisfaction, and then use dynamic programming to track the best possible sum of like-time coefficients while considering subsets of dishes.

What is the time complexity of the 'Reducing Dishes' problem?

The typical time complexity is O(n log n), which comes from sorting the dishes and then calculating the dynamic programming values.

How does the greedy approach help in solving this problem?

The greedy approach ensures that higher satisfaction dishes are cooked earlier, maximizing their contribution to the like-time coefficient.

What should I focus on to optimize space complexity for this problem?

You should store only the necessary intermediate results, rather than recalculating them at each stage, to reduce space usage.

Can I solve the 'Reducing Dishes' problem without sorting the satisfaction array?

No, sorting the dishes by satisfaction is crucial to maximizing the like-time coefficient and achieving an optimal solution.

terminal

Solution

Solution 1: Greedy + Sorting

Suppose we only choose one dish, then we should choose the dish with the highest satisfaction $s_0$, and check whether $s_0$ is greater than 0. If $s_0 \leq 0$, then we don't cook any dishes, otherwise, we cook this dish, and the total satisfaction is $s_0$.

1

2

3

4

5

6

7

8

9

10

class Solution:
    def maxSatisfaction(self, satisfaction: List[int]) -> int:
        satisfaction.sort(reverse=True)
        ans = s = 0
        for x in satisfaction:
            s += x
            if s <= 0:
                break
            ans += s
        return ans

Continue Practicing

#1363 largest multiple of three #1262 greatest sum divisible by three #646 maximum length of pair chain