What is the time complexity of Apply Operations on Array to Maximize Sum of Squares?

The optimal solution has a time complexity of O(n + k).

What data structures are used to solve Apply Operations on Array to Maximize Sum of Squares?

This problem uses Array, Hash Table, Greedy, Bit Manipulation. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Apply Operations on Array to Maximize Sum of Squares?

Insight 1: The AND and OR operations can effectively redistribute bits between array elements. The total number of '1' bits across all elements remains constant. Insight 2: To maximize the sum of squares, the largest possible values should be placed in the selected `k` elements. This is because the square function grows faster than the linear sum.

Which companies ask LeetCode 2897. Apply Operations on Array to Maximize Sum of Squares in interviews?

This problem is frequently asked at Sprinklr.

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Apply Operations on Array to Maximize Sum of Squares - LeetCode 2897 Solution

Apply Operations on Array to Maximize Sum of Squares - Complete Solution Guide

Apply Operations on Array to Maximize Sum of Squares is LeetCode problem 2897, 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

You are given a 0-indexed integer array nums and a positive integer k . You can do the following operation on the array any number of times: Choose any two distinct indices i and j and simultaneously update the values of nums[i] to (nums[i] AND nums[j]) and nums[j] to (nums[i] OR nums[j]) . Here, OR denotes the bitwise OR operation, and AND denotes the bitwise AND operation. You have to choose k elements from the final array and calculate the sum of their squares . Return the maximum sum of squa

Detailed Explanation

The problem asks us to maximize the sum of squares of `k` elements chosen from an array `nums`. We are allowed to perform bitwise AND and OR operations on pairs of elements in the array any number of times before selecting the `k` elements. The final result should be returned modulo 10^9 + 7. The key here is to realize that the bitwise operations allow us to freely distribute the '1' bits among the elements of the array. Therefore, the optimal strategy is to concentrate as many '1' bits as possible into the first `k` elements, and then calculate the sum of their squares.

Solution Approach

The solution leverages the bit manipulation property of the AND and OR operations and a greedy strategy. It first counts the number of '1's at each bit position across all elements of the array. Then, it greedily assigns those '1's to the first `k` elements, starting from the least significant bit, to maximize their values. Finally, it calculates the sum of squares of these `k` elements, modulo 10^9 + 7.

Step-by-Step Algorithm

Step 1: Initialize a count array `counts` of size 30 (since the max value of nums[i] is 10^9, which can be represented by 30 bits) to store the number of 1s at each bit position.
Step 2: Iterate through the `nums` array. For each number, iterate through its bits (0 to 29). If the bit is 1, increment the corresponding count in the `counts` array.
Step 3: Initialize a `targets` array of size `k` to store the values of the `k` elements we are going to select. Initially, all elements are 0.
Step 4: Iterate through the bit positions (0 to 29). For each bit position `i`, iterate `min(k, counts[i])` times (the number of 1s we can assign to the first `k` elements). For each of these iterations, add `(1 << i)` to the corresponding element in the `targets` array.
Step 5: Calculate the sum of squares of the elements in the `targets` array, modulo 10^9 + 7. This is the result.
Step 6: Return the result.

Key Insights

Insight 1: The AND and OR operations can effectively redistribute bits between array elements. The total number of '1' bits across all elements remains constant.
Insight 2: To maximize the sum of squares, the largest possible values should be placed in the selected `k` elements. This is because the square function grows faster than the linear sum.
Insight 3: The optimal arrangement involves placing the maximum possible number of '1' bits in the first `k` elements. We should iterate through the bit positions and assign '1's to the initial `k` elements as much as possible.

Complexity Analysis

Time Complexity: O(n + k)

Space Complexity: O(k)

Topics

This problem involves: Array, Hash Table, Greedy, Bit Manipulation.

Companies

Asked at: Sprinklr.

Apply Operations on Array to Maximize Sum of Squares | LeetCode 2897

Hard

Array Hash Table Greedy Bit Manipulation

Time: O(n + k)Space: O(k)

Solution Code

Available in:

import java.util.List;

class Solution {
    public int maxSum(List<Integer> nums, int k) {
        int MOD = (int)1e9 + 7;
        
        int[] counts = new int[30];
        
        for (int num : nums) {
            for (int i = 0; i < 30; i++) {
                if (((num >> i) & 1) == 1) {
                    counts[i]++;
                }
            }
        }
        
        int[] targets = new int[k];
        
        for (int i = 0; i < 30; i++) {
            for (int j = 0; j < Math.min(k, counts[i]); j++) {
                targets[j] += (1 << i);
            }
        }

        long total_sum_sq = 0;
        for (int x : targets) {
            total_sum_sq = (total_sum_sq + (long)x * x) % MOD;
        }
        
        return (int)total_sum_sq;
    }
}

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Problem Info

Description

Problem 2897: Apply Operations on Array to Maximize Sum of Squares - Detailed explanation and solution approaches.

Topics

Array Hash Table Greedy Bit Manipulation

Companies

Sprinklr

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:2897

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

6 steps

1
Step 1: Initialize a count array `counts` of size 30 (since the max value of nums[i] is 10^9, which can be represented by 30 bits) to store the number of 1s at each bit position.
2
Step 2: Iterate through the `nums` array. For each number, iterate through its bits (0 to 29). If the bit is 1, increment the corresponding count in the `counts` array.
3
Step 3: Initialize a `targets` array of size `k` to store the values of the `k` elements we are going to select. Initially, all elements are 0.
4
Step 4: Iterate through the bit positions (0 to 29). For each bit position `i`, iterate `min(k, counts[i])` times (the number of 1s we can assign to the first `k` elements). For each of these iterations, add `(1 << i)` to the corresponding element in the `targets` array.
5
Step 5: Calculate the sum of squares of the elements in the `targets` array, modulo 10^9 + 7. This is the result.
6
Step 6: Return the result.

💡

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 + k)

💾

Space Complexity

O(k)

📝

Why?

The time complexity is derived from iterating through the input array `nums` once, and then iterating through the bit positions (constant 30) and assigning them to the `k` target elements. The space complexity comes from storing the `k` target elements.

📖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

Sorting

Another approach could involve generating all possible array states using the allowed bitwise operations, then sorting the array and picking the top k elements to compute the sum of squares. However, generating all states would be computationally very expensive and not feasible with the given constraints.

Time: ExponentialSpace: Exponential

Trade-offs: This is purely a theoretical alternative and totally impractical. Avoid this at all costs.

✨

Key Insights

1Insight 1: The AND and OR operations can effectively redistribute bits between array elements. The total number of '1' bits across all elements remains constant.
2Insight 2: To maximize the sum of squares, the largest possible values should be placed in the selected `k` elements. This is because the square function grows faster than the linear sum.
3Insight 3: The optimal arrangement involves placing the maximum possible number of '1' bits in the first `k` elements. We should iterate through the bit positions and assign '1's to the initial `k` elements as much as possible.

📚

Key Terms Explained

For Students

Array

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

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Greedy Algorithm

Making the best choice at each step without looking ahead.

Edge Cases

Edge case 1: `nums` is empty. Although the problem constraints guarantee `nums.length >= 1`, a robust solution would handle the empty case gracefully (e.g., return 0). The given solutions work correctly for empty list, as it will result in all counts being zero and return 0.
Edge case 2: `k` is equal to `nums.length`. The solution works correctly in this case as it simply redistributes the bits among all the array elements. The result is always optimal.
Edge case 3: `k` is 1. The solution correctly concentrates all '1' bits into the first element and calculates its square.
Edge case 4: Numbers with all bits set to 0, this will not impact the solution as there are no 1 bits to move around.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not taking the modulo at each step of calculating the sum of squares, which can lead to integer overflow issues.

✗Common mistake 2: Incorrectly implementing the bitwise operations, or not understanding how they redistribute bits.

✗Common mistake 3: Forgetting to consider the constraint that we have to choose 'k' elements, thus not greedily assigning bits to the top 'k' elements.

✗Common mistake 4: Assuming that the initial arrangement of numbers is optimal and not performing the bitwise operations. It's crucial to understand that the operations can significantly increase the sum of squares.

✅

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: Resource allocation. Imagine `nums` representing the amount of resource each user has, and `k` represents the number of users to be prioritized. The bitwise operations model a redistribution mechanism. The algorithm maximizes the total 'square' value of resource given to priority users, perhaps to prevent starvation or boost performance in critical systems.

🏦Real-world scenario 2: Data compression. Consider `nums` representing data blocks, and each bit representing a feature. By applying the operations, you can concentrate key features in a smaller set of blocks (`k`), potentially simplifying compression algorithms.

💡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

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ 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: June 18, 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

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

@InterviewProJul 14

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

@CodeNinjaJul 7

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJun 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.

How do I choose between a hash map and a hash set?

Use a hash set when you only need to track presence/absence of elements (like detecting duplicates). Use a hash map when you need to associate values with keys (like counting frequencies or storing indices). Hash sets use less memory since they only store keys.

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 "Apply Operations on Array to Maximize Sum of Squares"?

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 hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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