What is the time complexity of Find a Value of a Mysterious Function Closest to Target?

The optimal solution has a time complexity of O(n^2).

What data structures are used to solve Find a Value of a Mysterious Function Closest to Target?

This problem uses Array, Binary Search, Bit Manipulation, Segment Tree. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Find a Value of a Mysterious Function Closest to Target?

Insight 1: The bitwise AND operation is monotonic decreasing as the subarray expands. This means func(arr, l, r+1) <= func(arr, l, r). This property is crucial because it allows us to track possible values from the AND operation efficiently. Insight 2: Instead of computing the AND for all possible subarrays [l, r], we can iteratively build up a set of possible values at each index. For each number in the array, we can update the set of possible values by performing a bitwise AND with all the previous possible values and adding the current number to the set.

Which companies ask LeetCode 1521. Find a Value of a Mysterious Function Closest to Target in interviews?

This problem is frequently asked at American Express.

JavaScript is required for the full interactive experience. However, all content is accessible without JavaScript.

Find a Value of a Mysterious Function Closest to Target - LeetCode 1521 Solution

Q: How to solve Find a Value of a Mysterious Function Closest to Target?

The solution uses a dynamic programming approach to iteratively build a set of possible values that can be obtained by performing bitwise AND operations on subarrays ending at the current index. We maintain a set `possible` of the AND values encountered so far. For each number `num` in the array, we create a new set `new_possible` containing `num` itself. We then iterate through the `possible` set, performing a bitwise AND operation between each element `p` in `possible` and `num`, and adding the result to `new_possible`. After calculating the `new_possible` set, we iterate through it and update the minimum absolute difference `ans` with the target. The `possible` set is then updated to `new_possible` for the next iteration. This avoids recalculating overlapping subarrays.

Q: Which companies ask LeetCode 1521. Find a Value of a Mysterious Function Closest to Target in interviews?

This problem is frequently asked at American Express.

Find a Value of a Mysterious Function Closest to Target - Complete Solution Guide

Find a Value of a Mysterious Function Closest to Target is LeetCode problem 1521, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Winston was given the above mysterious function func . He has an integer array arr and an integer target and he wants to find the values l and r that make the value |func(arr, l, r) - target| minimum possible. Return the minimum possible value of |func(arr, l, r) - target| . Notice that func should be called with the values l and r where 0 <= l, r < arr.length . Example 1: Input: arr = [9,12,3,7,15], target = 5 Output: 2 Explanation: Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],

Detailed Explanation

The problem asks us to find the minimum absolute difference between a target value and the result of a mysterious function `func(arr, l, r)`. The function `func` calculates the bitwise AND of a subarray of `arr` from index `l` to `r` (inclusive). We need to iterate through all possible subarrays of `arr`, calculate `func` for each subarray, and find the minimum absolute difference between the result and the target value. The input is an integer array `arr` and an integer `target`, and the output is the minimum possible absolute difference.

Solution Approach

The solution uses a dynamic programming approach to iteratively build a set of possible values that can be obtained by performing bitwise AND operations on subarrays ending at the current index. We maintain a set `possible` of the AND values encountered so far. For each number `num` in the array, we create a new set `new_possible` containing `num` itself. We then iterate through the `possible` set, performing a bitwise AND operation between each element `p` in `possible` and `num`, and adding the result to `new_possible`. After calculating the `new_possible` set, we iterate through it and update the minimum absolute difference `ans` with the target. The `possible` set is then updated to `new_possible` for the next iteration. This avoids recalculating overlapping subarrays.

Step-by-Step Algorithm

Step 1: Initialize `ans` to infinity (or a sufficiently large value) to store the minimum absolute difference and an empty set `possible` to store the possible AND values.
Step 2: Iterate through each number `num` in the input array `arr`.
Step 3: Create a new set `new_possible` and add the current number `num` to it.
Step 4: Iterate through the `possible` set. For each value `p` in `possible`, perform a bitwise AND operation between `p` and `num`, and add the result to `new_possible`.
Step 5: Iterate through the `new_possible` set. For each value `p` in `new_possible`, calculate the absolute difference between `p` and the target, and update `ans` with the minimum value found so far.
Step 6: If `ans` becomes 0, return it immediately, as it's the minimum possible difference.
Step 7: Update the `possible` set to `new_possible`.
Step 8: After iterating through all the numbers in `arr`, return the final value of `ans`.

Key Insights

Insight 1: The bitwise AND operation is monotonic decreasing as the subarray expands. This means func(arr, l, r+1) <= func(arr, l, r). This property is crucial because it allows us to track possible values from the AND operation efficiently.
Insight 2: Instead of computing the AND for all possible subarrays [l, r], we can iteratively build up a set of possible values at each index. For each number in the array, we can update the set of possible values by performing a bitwise AND with all the previous possible values and adding the current number to the set.
Insight 3: The problem's constraints on the input array's elements (1 <= arr[i] <= 10^6) is relatively small, allowing us to keep track of all possible AND values without exceeding memory limits.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(n)

Topics

This problem involves: Array, Binary Search, Bit Manipulation, Segment Tree.

Companies

Asked at: American Express.

Find a Value of a Mysterious Function Closest to Target | LeetCode 1521

Hard

Array Binary Search Bit Manipulation Segment Tree

Time: O(n^2)Space: O(n)

Solution Code

Available in:

import java.util.HashSet;
import java.util.Set;

class Solution {
    public int closestToTarget(int[] arr, int target) {
        int ans = Integer.MAX_VALUE;
        Set<Integer> possible = new HashSet<>();

        for (int num : arr) {
            Set<Integer> new_possible = new HashSet<>();
            new_possible.add(num);
            for (int p : possible) {
                new_possible.add(p & num);
            }

            for (int p : new_possible) {
                ans = Math.min(ans, Math.abs(p - target));
            }

            if (ans == 0) {
                return 0;
            }

            possible = new_possible;
        }

        return ans;
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 1521: Find a Value of a Mysterious Function Closest to Target - Detailed explanation and solution approaches.

Topics

Array Binary Search Bit Manipulation Segment Tree

Companies

American Express

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:1521

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Initialize `ans` to infinity (or a sufficiently large value) to store the minimum absolute difference and an empty set `possible` to store the possible AND values.
2
Step 2: Iterate through each number `num` in the input array `arr`.
3
Step 3: Create a new set `new_possible` and add the current number `num` to it.
4
Step 4: Iterate through the `possible` set. For each value `p` in `possible`, perform a bitwise AND operation between `p` and `num`, and add the result to `new_possible`.
5
Step 5: Iterate through the `new_possible` set. For each value `p` in `new_possible`, calculate the absolute difference between `p` and the target, and update `ans` with the minimum value found so far.
6
Step 6: If `ans` becomes 0, return it immediately, as it's the minimum possible difference.
7
Step 7: Update the `possible` set to `new_possible`.
8
Step 8: After iterating through all the numbers in `arr`, return the final value of `ans`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The outer loop iterates 'n' times (once for each element in `arr`). The inner loop iterates through the `possible` set, which has a maximum size of k. The set operations (adding elements, calculating the minimum difference) take O(1) time on average. Therefore, the overall time complexity is O(n*k). The space complexity is dominated by the `possible` set, which stores up to k distinct AND values, hence O(k).

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Brute Force

Calculate the bitwise AND for every possible subarray [l, r] and find the minimum absolute difference. This involves nested loops to iterate through all possible values of l and r.

Time: O(n^2)Space: O(1)

Trade-offs: Simple to implement but inefficient for large arrays due to the quadratic time complexity. The given solution is much better.

✨

Key Insights

1Insight 1: The bitwise AND operation is monotonic decreasing as the subarray expands. This means func(arr, l, r+1) <= func(arr, l, r). This property is crucial because it allows us to track possible values from the AND operation efficiently.
2Insight 2: Instead of computing the AND for all possible subarrays [l, r], we can iteratively build up a set of possible values at each index. For each number in the array, we can update the set of possible values by performing a bitwise AND with all the previous possible values and adding the current number to the set.
3Insight 3: The problem's constraints on the input array's elements (1 <= arr[i] <= 10^6) is relatively small, allowing us to keep track of all possible AND values without exceeding memory limits.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Binary Search

Halving the search space each step. Like finding a word in a dictionary!

Edge Cases

Edge case 1: The array contains only one element. The algorithm correctly calculates the difference between that element and the target.
Edge case 2: The target value is equal to one of the array elements, or an AND result. The algorithm correctly returns 0 as the minimum difference.
Edge case 3: The array contains very large numbers. The algorithm handles these numbers correctly because the bitwise AND operation can be performed on large integers.
Edge case 4: All elements are the same. The set 'possible' will contain only that element, and the algorithm will calculate the difference correctly.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to initialize the `ans` variable to infinity or a sufficiently large value, resulting in incorrect minimum difference calculation.

✗Common mistake 2: Incorrectly calculating the bitwise AND operation or missing the absolute value calculation when determining the difference from the target.

✗Common mistake 3: Using a List instead of a Set for `possible`. This can lead to duplicate values and increase the time complexity, especially if duplicate values are frequently added.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Data compression algorithms can use bitwise operations to identify and exploit patterns in data, similar to how the algorithm finds common bits through the AND operation.

🏦Real-world scenario 2: Network routing protocols can use bitwise AND operations to determine the network address of a device or subnet. Finding the closest target can be applicable to optimizing paths and reducing latency.

🎮Real-world scenario 3: Feature selection in machine learning: Identify the most relevant features (represented as bits) for a model, similar to how the AND operation narrows down possibilities. Finding features closest to a desired outcome or characteristic is relevant.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Hard problems often combine multiple techniques. Break the problem into sub-problems.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: August 21, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 14

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 7

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJun 30

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

Can binary search be applied to non-sorted arrays?

Binary search requires a monotonic property, not necessarily sorted data. It can be applied to any search space where you can determine if the answer is in the left or right half. Examples include searching rotated arrays or finding minimum in optimization problems.

Is it okay to not fully solve hard problems in interviews?

Yes! Interviewers care more about your problem-solving process than the final answer. Demonstrate: clear communication, systematic approach, ability to recognize when to pivot strategies, and graceful handling of hints. Partial progress with good reasoning is valued.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Find a Value of a Mysterious Function Closest to Target"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether binary search techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?