Problem Description

In this problem, you are given an array prices that represents the cost of chocolates and a 2D array queries, each containing two integers: k_i and m_i. Alice and Bob are purchasing chocolates together, and they have a special arrangement for sharing the cost. For any given chocolate:

• If the chocolate costs less than or equal to k_i, then Bob will pay for it entirely.
• If the chocolate costs more than k_i, Bob will pay k_i, and Alice will cover the remaining cost.

Bob can choose exactly m_i chocolates, and he wants to minimize his relative loss. The relative loss is defined as the difference between the total amount Bob paid and the total amount Alice paid, which he wants to minimize.

Your task is to determine the minimum relative loss Bob can achieve for each query queries[i] and return an array of integers representing these minimum losses.

Intuition

Given that Bob wants to minimize his relative loss (Bob's payment minus Alice's payment), we have to approach this problem by trying to minimize the expenses that go beyond Bob's contribution limit (k_i). To achieve this, we can choose more chocolates with prices within or equal to Bob's limit and try to minimize picking chocolates that require an additional payment from Alice.

To find the optimum distribution of chocolates that Bob should pay for completely and those that Alice contributes to, we can:

1. Sort the prices array in ascending order. This allows us to quickly identify which chocolates can be paid for by Bob and which will require a contribution from Alice.
2. We can use binary search to find the breakpoint where Bob should stop paying for chocolates entirely and start sharing the cost with Alice.

The function f(k, m) performs this binary search by finding the right number of chocolates Bob can fully pay for without incurring too much additional cost from Alice. Given that m chocolates in total need to be purchased, the function calculates the split (using a binary search approach) between the chocolates paid for entirely by Bob and those partially paid for by him (with the rest paid by Alice).

This split helps minimize Bob's relative loss, as it finds the best point where paying an equal or lesser price fully (for as many chocolates as possible) and sharing the cost of the more expensive chocolates leads to the smallest difference between Bob's payment and Alice's payment.

Finally, the minimumRelativeLosses method calculates the minimum loss for each query by considering the split point l, computing the total cost paid by Bob and Alice, and then determining the relative loss (Bob's payment minus Alice's payment). The result for each query is then appended to the ans list, which is returned as the solution to the problem.

Solution Approach

The solution uses a binary search algorithm combined with a prefix sum technique to efficiently compute the required values for minimizing Alice's contribution and thus Bob's relative loss.

Here's a step-by-step breakdown:

1. First, we sort the prices array. This is crucial as it allows us to apply binary search later.

2. We compute the prefix sums of the array using accumulate(prices, initial=0). Prefix sums are efficient for calculating the sum of a subarray in constant time O(1). Having a prefix sum array s means s[i] represents the sum of the prices array from prices[0] to prices[i-1].

3. The code then defines a function f(k: int, m: int) -> int that uses binary search to find the optimal split point l:

   • Initialize two pointers, l and r. l starts at 0, and r is the minimum of m and the index of the rightmost chocolate whose price does not exceed k. This index is found using bisect_right(prices, k) which returns the insertion point to the right of k in the sorted prices list.

   • While l < r, we find the mid-point and calculate how many chocolates to the right of mid Bob would need to share the cost with Alice (right = m - mid).

   • If the price of the chocolate at midpoint is less than 2 * k - prices[n - right], we move the l pointer to mid + 1. This condition checks if the price at the mid is preferred over the price at the symmetric position in the second half (towards the array's end).

   • If the condition is not met, we move the r pointer to mid. This helps us to narrow down the split point where Bob would switch from paying in full to sharing the cost.

4. For each k, m pair in queries, we get the split point l using f(k, m). Calculate Bob's total payment as the sum of the chocolate prices up to l (s[l]) plus doubled k for each of the chocolates from l to m (2 * k * r). Calculate Alice's total payment as the sum of the chocolate prices not covered by Bob (s[n] - s[n - r]).

5. Compute the loss as loss = s[l] + 2 * k * r - (s[n] - s[n - r]), which represents Bob's total payment minus Alice's total payment.

6. Append each computed loss to the ans array, which stores Bob's minimum relative loss for each query.

By combining binary search to efficiently find the optimal split of chocolate prices Bob should pay fully, and prefix sums to quickly compute the sums for Bob's and Alice's payments, the entire algorithm remains efficient and effective at minimizing Bob's relative loss for all the queries.

Example Walkthrough

Let us consider a small example to illustrate the solution approach using the following prices array and queries.

Suppose the prices array is [1, 4, 5, 7, 8] representing the cost of chocolates in ascending order. And there is one query in the queries array: [4, 3] which means that Bob's payment limit k_i is 4 and he can choose m_i = 3 chocolates.

Here are the steps we would take to find the minimum relative loss for this query:

1. Since our prices array is already sorted, we don't need to sort it again.

2. We calculate the prefix sums of the prices array. The prefix sums array s would be [0, 1, 5, 10, 17, 25].

3. Next, using the f(k, m) function, we perform a binary search to find the optimal number of chocolates that Bob will pay for entirely, without sharing the cost with Alice.

   Given k = 4 and m = 3, we initialize l = 0 and r = min(3, 2) since there are 2 chocolates at indices 0 and 1 that are less than or equal to k.

   So, r starts at 2. We examine the midpoint mid between l and r, which is 1. Calculating the number of chocolates to the right of mid that Bob has to share the cost for: right = m - mid = 3 - 1 = 2.

   The condition in our binary search checks if prices[mid] < 2 * k - prices[n - right], which is prices[1] < 2 * 4 - prices[5 - 2], meaning 4 < 8 - 5. Since this is false, we don't adjust l; instead, we make r equal to mid.

   With l = 0 and r = 1, we find that mid = 0, and repeating the above condition, we get prices[0] < 2 * 4 - prices[5 - 2], which means 1 < 8 - 5. This condition is true, so we update l to mid + 1, which makes l = 1.

   As l cannot be less than r, our binary search concludes that l = 1 is the optimal split point.

4. For the query [4, 3], we have l = 1 and r = m - l = 2. Bob's total payment is the sum of prefix sums up to l plus k times r or s[1] + 2 * k * r, which gives us 1 + 2 * 4 * 2 = 17. Alice's total payment is s[5] - s[5 - r] = 25 - 10 = 15.

5. The relative loss for Bob is thus 17 - 15 = 2. We append 2 to our answer array, which means for the query (4, 3), the minimum relative loss Bob can achieve is 2.

By going through the above steps, we conclude that by choosing the first one chocolate costing 1 (within his limit) and contributing to the costs of two more expensive chocolates (with Alice covering the remaining), Bob can minimize his relative loss to 2.

Solution Implementation

Time and Space Complexity

Time Complexity

The minimumRelativeLosses function consists of a main for-loop that processes each query, a sorted operation on the prices list, and a helper function (f) that uses binary search.

1. Sorting the prices list has a time complexity of O(n log n), where n is the length of the prices.

2. The accumulate function, which calculates the prefix sum array s, has a time complexity of O(n) because it processes each element in the prices array exactly once.

3. The function f uses a binary search to find the optimal split point that minimizes relative losses. The binary search within the f function runs in O(log min(m, n)), where m is the number of elements in one query and n is the length of the prices. This is due to the binary search being potentially limited by m, the number of elements that the query wants to examine.

4. The main for-loop iterates over each query in queries. Assuming q is the number of queries, the for-loop runs q times. Inside this loop, the f function is called, which has the O(log min(m, n)) complexity.

So the overall time complexity of the for-loop including the call to the f function is O(q * log min(m, n)).

Combining these complexities, we get O(n log n) + O(n) + O(q * log min(m, n)). Since q * log min(m, n) could potentially be larger than n log n and n, especially when q is large, we can consider O(q * log min(m, n)) to be the dominating term in typical scenarios.

Final Time Complexity: O(n log n) + O(n) + O(q * log min(m, n)).

Space Complexity

1. The prices list is sorted in-place, so no additional space is required for that.

2. The accumulate function creates a prefix sum array s with n + 1 elements, giving a space complexity of O(n).

3. The ans list will hold q elements (one for each query), which gives a space complexity of O(q).

The only additional space used is the s and ans lists, since binary search does not require additional space beyond the few variables for pointers and indices.

Final Space Complexity: O(n) + O(q) or simply O(n + q) if we assume q can be as significant as n.