#2944

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Number of Coins for Fruits

Calculate the minimum coins to buy fruits using state transition dynamic programming while handling rewards and purchase choices optimally.

array dynamic programming queue heap priority queue monotonic queue

Problem Statement

You are given an integer array prices where prices[i] represents the number of coins required to buy the (i + 1)th fruit. Each fruit may also grant a reward that allows taking other fruits for free.

Even if a fruit can be taken for free as a reward from a previously purchased fruit, you may choose to buy it to collect its reward. Determine the minimum coins needed to acquire all fruits while considering these reward interactions.

Examples

Example 1

Input: prices = [3,1,2]

Output: 4

Note that even though you could take the 2 nd fruit for free as a reward of buying 1 st fruit, you purchase it to receive its reward, which is more optimal.

Example 2

Input: prices = [1,10,1,1]

Output: 2

Example 3

Input: prices = [26,18,6,12,49,7,45,45]

Output: 39

Note that even though you could take the 6 th fruit for free as a reward of buying 3 rd fruit, you purchase it to receive its reward, which is more optimal.

Constraints

1 <= prices.length <= 1000
1 <= prices[i] <= 105

Solution Approach

Dynamic Programming State Definition

Define dp[i] as the minimum coins needed to purchase the first i fruits considering all possible reward interactions. This state captures the cumulative minimal cost and allows structured transitions.

Transition with Reward Consideration

For each fruit, compute dp[i] by evaluating the cost of buying it versus taking it as a reward. Update the dp array by checking if purchasing now leads to a smaller total coins spent later, handling the free fruit scenario efficiently.

Optimize with Queue or Heap

Use a monotonic queue or priority queue to keep track of minimal dp values for segments influenced by rewards. This ensures that the state transition remains efficient and scales to the input constraints.

Complexity Analysis

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

Time complexity depends on how transitions are implemented; using a DP array with monotonic queue can reduce redundant calculations. Space complexity is primarily the DP array plus auxiliary structures for tracking rewards.

What Interviewers Usually Probe

Expect candidates to recognize state transition dynamic programming is required.
Check if the candidate correctly models free fruits as reward states in DP.
Look for efficient handling of DP transitions with queue or heap to meet constraints.

Common Pitfalls or Variants

Common pitfalls

Ignoring the benefit of purchasing a fruit even if it could be received for free.
Incorrectly updating DP states, leading to overcounting or missing reward interactions.
Not optimizing transitions, resulting in timeouts on large input arrays.

Follow-up variants

Varying reward rules where a fruit gives multiple free fruits instead of one.
Adding a limit on the number of fruits that can be purchased or taken for free.
Modifying the price array to include zero-cost fruits and verifying DP correctness.

FAQ

What is the main pattern used in Minimum Number of Coins for Fruits?

The primary pattern is state transition dynamic programming, tracking minimal coins while considering reward effects from purchased fruits.

Can a fruit be purchased even if it is available for free?

Yes, purchasing a fruit can sometimes lead to a better overall solution because it may unlock rewards that reduce total coins spent.

What data structures can optimize the DP transitions?

Using a monotonic queue or priority queue helps efficiently track the minimum cost states influenced by rewards during DP updates.

What are the constraints on the prices array?

The array length is 1 to 1000, and each price is between 1 and 105 coins.

How do I handle reward interactions in the solution?

Update DP states carefully for each fruit by considering both buying it and taking it as a free reward, choosing the option that minimizes the total coins.

terminal

Solution

Solution 1: Memoization Search

We define a function $\textit{dfs}(i)$ to represent the minimum number of coins needed to buy all the fruits starting from the $i$-th fruit. The answer is $\textit{dfs}(1)$.

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class Solution:
    def minimumCoins(self, prices: List[int]) -> int:
        @cache
        def dfs(i: int) -> int:
            if i * 2 >= len(prices):
                return prices[i - 1]
            return prices[i - 1] + min(dfs(j) for j in range(i + 1, i * 2 + 2))

        return dfs(1)

Solution 2: Dynamic Programming

We can rewrite the memoization search in Solution 1 into a dynamic programming form.

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class Solution:
    def minimumCoins(self, prices: List[int]) -> int:
        @cache
        def dfs(i: int) -> int:
            if i * 2 >= len(prices):
                return prices[i - 1]
            return prices[i - 1] + min(dfs(j) for j in range(i + 1, i * 2 + 2))

        return dfs(1)

Solution 3: Dynamic Programming + Monotonic Queue Optimization

Observing the state transition equation in Solution 2, we can see that for each $i$, we need to find the minimum value of $f[i + 1], f[i + 2], \cdots, f[2i + 1]$. As $i$ decreases, the range of these values also decreases. This is essentially finding the minimum value in a sliding window with a narrowing range, which can be optimized using a monotonic queue.

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class Solution:
    def minimumCoins(self, prices: List[int]) -> int:
        @cache
        def dfs(i: int) -> int:
            if i * 2 >= len(prices):
                return prices[i - 1]
            return prices[i - 1] + min(dfs(j) for j in range(i + 1, i * 2 + 2))

        return dfs(1)

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