#1521

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Find a Value of a Mysterious Function Closest to Target

In this problem, you'll use binary search to find the closest value of a mysterious function to a given target.

array binary search bit manipulation segment tree

Problem Statement

Given an integer array arr and an integer target, you're asked to find two indices l and r such that the value of a mysterious function func(arr, l, r) is as close as possible to the target. The function func takes arr, l, and r, and computes an unknown value based on the subarray arr[l...r]. Your task is to find the minimum possible difference between func(arr, l, r) and the target.

You need to return the minimum value of |func(arr, l, r) - target| for valid indices l and r where 0 <= l, r < arr.length. The problem focuses on applying binary search over the valid answer space to minimize this difference efficiently.

Examples

Example 1

Input: arr = [9,12,3,7,15], target = 5

Output: 2

Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],[3,3],[4,4],[0,1],[1,2],[2,3],[3,4],[0,2],[1,3],[2,4],[0,3],[1,4],[0,4]], Winston got the following results [9,12,3,7,15,8,0,3,7,0,0,3,0,0,0]. The value closest to 5 is 7 and 3, thus the minimum difference is 2.

Example 2

Input: arr = [1000000,1000000,1000000], target = 1

Output: 999999

Winston called the func with all possible values of [l,r] and he always got 1000000, thus the min difference is 999999.

Example 3

Input: arr = [1,2,4,8,16], target = 0

Output: 0

Example details omitted.

Constraints

1 <= arr.length <= 105
1 <= arr[i] <= 106
0 <= target <= 107

Solution Approach

Binary Search Over the Answer Space

The key to solving this problem efficiently is binary search. The idea is to binary search for the closest result by checking pairs of indices l and r and evaluating the function. By systematically narrowing the possible answer space, you can efficiently find the minimum difference from the target.

Optimization with Array Manipulation

Manipulate subarrays in such a way that their behavior over different values of l and r is easily calculable. Using properties like the 'and' value of subarrays helps ensure that the space of possible answers is reduced in a structured way, allowing binary search to focus on the most promising regions.

Efficient Calculation of `func`

Instead of recalculating the function func(arr, l, r) for every pair of indices, optimize it by considering cumulative properties of the array. By recognizing patterns in the array, you can speed up the calculation and reduce unnecessary work.

Complexity Analysis

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

The time complexity depends on the final approach used to implement the binary search and function evaluations. Generally, it will involve O(log n) for binary search combined with O(n) for array evaluations, making the total complexity O(n log n) in most cases. Space complexity will depend on the auxiliary data structures used during the process but should be manageable within the problem's constraints.

What Interviewers Usually Probe

Understanding of binary search and how to adapt it to a range of answers is crucial.
The ability to optimize through recognizing patterns in array manipulation.
Efficiently managing the evaluation of subarrays and ensuring that the approach scales well with larger inputs.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the binary search pattern for this problem could lead to inefficient solutions.
Failing to optimize the calculation of func(arr, l, r) could result in excessive computation time.
Not taking full advantage of array properties like the 'and' value or cumulative calculations might hinder the solution's efficiency.

Follow-up variants

Variation with different types of functions to minimize (e.g., sum, product).
Introducing constraints on how the indices l and r are chosen (e.g., must satisfy additional conditions).
Modifying the target to be dynamic or based on a range instead of a fixed value.

FAQ

How does binary search apply to this problem?

Binary search is used to efficiently explore the valid space of potential solutions for the indices l and r. By narrowing down the search space, we can find the closest possible function result to the target value.

What role does the mysterious function `func` play?

The function func(arr, l, r) calculates a value based on the subarray arr[l...r], and the goal is to minimize the difference between this result and the given target.

Why is array manipulation important in this problem?

Array manipulation helps optimize the process of evaluating func(arr, l, r) by exploiting properties such as cumulative values or patterns that can be reused, reducing unnecessary recalculations.

What are the common mistakes when solving this problem?

Misunderstanding how binary search fits into the problem's structure or not optimizing the function calculations can lead to inefficient solutions that fail to handle larger inputs effectively.

What is the best approach for solving this problem efficiently?

The best approach combines binary search over the valid answer space with efficient manipulation of subarrays and function optimizations, ensuring the solution works within the given constraints.

terminal

Solution

Solution 1: Hash Table + Enumeration

According to the problem description, we know that the function $func(arr, l, r)$ is actually the bitwise AND result of the elements in the array $arr$ from index $l$ to $r$, i.e., $arr[l] \& arr[l + 1] \& \cdots \& arr[r]$.

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class Solution:
    def closestToTarget(self, arr: List[int], target: int) -> int:
        ans = abs(arr[0] - target)
        s = {arr[0]}
        for x in arr:
            s = {x & y for y in s} | {x}
            ans = min(ans, min(abs(y - target) for y in s))
        return ans

Continue Practicing

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