Find a Value of a Mysterious Function Closest to Target

Problem Description

Winston was given a mysterious function func. He has an integer array arr and an integer target, and he wants to find the values l and r that minimize the value |func(arr, l, r) - target|. Return the minimum possible value of |func(arr, l, r) - target| where 0 <= l, r < arr.length.

Key Insights

• The function func operates on subarrays defined by l and r, returning a single integer value.
• The problem requires evaluating all pairs (l, r) and calculating the distance from target.
• Efficient evaluation of the function across all pairs is necessary due to potential large input sizes (up to 100,000).
• The function may exhibit properties that can be leveraged for optimization, like associativity or monotonicity.

Space and Time Complexity

Time Complexity: O(n^2) in the naive approach, but can be optimized to O(n log n) or better depending on properties of func.
Space Complexity: O(1) if the function does not require additional data structures, or O(n) depending on the storage of intermediate results.

Solution

To solve this problem, the approach involves iterating over all possible pairs of indices (l, r) to evaluate func(arr, l, r). The result of func needs to be compared against target to find the minimum absolute difference.

1. Iterate through all pairs (l, r): For each pair, compute the result of func(arr, l, r).
2. Calculate the absolute difference with target: For each result, compute |func(arr, l, r) - target|.
3. Track the minimum difference: Maintain a variable to track the smallest difference encountered during the iterations.

By efficiently managing the range of l and r, and possibly leveraging properties of the function, we can reduce unnecessary calculations.

Code Solutions