What is the time complexity of Apply Operations to Maximize Frequency Score?

The optimal solution has a time complexity of O(n*log(n)*log(n)).

What data structures are used to solve Apply Operations to Maximize Frequency Score?

This problem uses Array, Binary Search, Sliding Window, Sorting, Prefix Sum. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Apply Operations to Maximize Frequency Score?

Insight 1: Sorting the array allows us to efficiently use a sliding window approach to find the maximum frequency for a given target value. Because the input array `nums` is not sorted, it needs to be sorted first. The elements need not be changed back to their original index. Insight 2: Binary search can be used to find the optimal frequency length. A binary search on the possible frequencies (from 1 to the length of the array) can determine the maximum achievable frequency.

Which companies ask LeetCode 2968. Apply Operations to Maximize Frequency Score in interviews?

This problem is frequently asked at Deutsche Bank, PhonePe.

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Apply Operations to Maximize Frequency Score - LeetCode 2968 Solution

Apply Operations to Maximize Frequency Score - Complete Solution Guide

Apply Operations to Maximize Frequency Score is LeetCode problem 2968, 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 an integer k . You can perform the following operation on the array at most k times: Choose any index i from the array and increase or decrease nums[i] by 1 . The score of the final array is the frequency of the most frequent element in the array. Return the maximum score you can achieve . The frequency of an element is the number of occurences of that element in the array. Example 1: Input: nums = [1,2,6,4], k = 3 Output: 3 Explanation: We can do

Detailed Explanation

The problem asks us to maximize the frequency of a single element in an array by performing at most `k` operations. An operation involves incrementing or decrementing any element in the array by 1. The score is defined as the frequency of the most frequent element after applying the operations. The goal is to find the maximum possible score (frequency) achievable.

Solution Approach

The solution employs a binary search approach to determine the maximum achievable frequency. For a given frequency (length), it uses a sliding window to iterate through the sorted array. For each window, it calculates the minimum cost to make all elements within the window equal to the median value of the window. If the cost is less than or equal to k, it means that the frequency is achievable, and the search space can be narrowed to higher frequencies. If the cost is greater than k, the search space is narrowed to lower frequencies.

Step-by-Step Algorithm

Step 1: Sort the input array `nums` in ascending order.
Step 2: Calculate the prefix sum of the sorted array to efficiently compute the sum of elements within a range.
Step 3: Perform a binary search on the range of possible frequencies (1 to n, where n is the length of `nums`).
Step 4: Within the binary search, for each `mid` value (potential frequency), implement a `check` function that utilizes a sliding window of size `mid` to determine if that frequency can be achieved with at most `k` operations.
Step 5: Inside the `check` function, for each window, find the median element. Calculate the cost to make all elements in the window equal to the median. This cost is the sum of absolute differences between each element in the window and the median.
Step 6: Check if the calculated cost for the window is less than or equal to `k`. If it is, the potential frequency `mid` is achievable; return `true` from the `check` function. Otherwise, continue sliding the window.
Step 7: If the `check` function returns `true`, update the `ans` with the achieved frequency and search for a higher frequency. Otherwise, search for a lower frequency.
Step 8: After the binary search is complete, return the `ans`, which holds the maximum achievable frequency.

Key Insights

Insight 1: Sorting the array allows us to efficiently use a sliding window approach to find the maximum frequency for a given target value. Because the input array `nums` is not sorted, it needs to be sorted first. The elements need not be changed back to their original index.
Insight 2: Binary search can be used to find the optimal frequency length. A binary search on the possible frequencies (from 1 to the length of the array) can determine the maximum achievable frequency.
Insight 3: Prefix sums can be used to efficiently calculate the cost of making all elements within a window equal to a specific value (median). Prefix sum helps in calculating the cost of operations quickly.

Complexity Analysis

Time Complexity: O(n*log(n)*log(n))

Space Complexity: O(n)

Topics

This problem involves: Array, Binary Search, Sliding Window, Sorting, Prefix Sum.

Companies

Asked at: Deutsche Bank, PhonePe.

Apply Operations to Maximize Frequency Score | LeetCode 2968

Hard

Array Binary Search Sliding Window Sorting Prefix Sum

Time: O(n*log(n)*log(n))Space: O(n)

Solution Code

Available in:

import java.util.Arrays;

class Solution {
    public int maxFrequencyScore(int[] nums, long k) {
        int n = nums.length;
        Arrays.sort(nums);

        long[] prefix = new long[n + 1];
        for (int i = 0; i < n; i++) {
            prefix[i + 1] = prefix[i] + nums[i];
        }

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Problem Info

Description

Problem 2968: Apply Operations to Maximize Frequency Score - Detailed explanation and solution approaches.

Topics

Array Binary Search Sliding Window Sorting Prefix Sum

Companies

Deutsche BankPhonePe

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:2968

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: Sort the input array `nums` in ascending order.
2
Step 2: Calculate the prefix sum of the sorted array to efficiently compute the sum of elements within a range.
3
Step 3: Perform a binary search on the range of possible frequencies (1 to n, where n is the length of `nums`).
4
Step 4: Within the binary search, for each `mid` value (potential frequency), implement a `check` function that utilizes a sliding window of size `mid` to determine if that frequency can be achieved with at most `k` operations.
5
Step 5: Inside the `check` function, for each window, find the median element. Calculate the cost to make all elements in the window equal to the median. This cost is the sum of absolute differences between each element in the window and the median.
6
Step 6: Check if the calculated cost for the window is less than or equal to `k`. If it is, the potential frequency `mid` is achievable; return `true` from the `check` function. Otherwise, continue sliding the window.
7
Step 7: If the `check` function returns `true`, update the `ans` with the achieved frequency and search for a higher frequency. Otherwise, search for a lower frequency.
8
Step 8: After the binary search is complete, return the `ans`, which holds the maximum achievable frequency.

💡

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*log(n)*log(n))

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The dominant factor is the O(n log n) factor coming from sorting and the product n*log n derived from binary search combined with the sliding window check.

📖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

Without Binary Search (Potentially slower for large n)

Iterate through all possible frequencies from 1 to n. For each frequency, use a sliding window to check if it's achievable. This eliminates the binary search, but in the worst case, it degrades the search to an O(n^2) within the sorting routine.

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

Trade-offs: Easier to implement but less efficient for large arrays. Use only if binary search proves too complex to implement during time constrained situations.

✨

Key Insights

1Insight 1: Sorting the array allows us to efficiently use a sliding window approach to find the maximum frequency for a given target value. Because the input array `nums` is not sorted, it needs to be sorted first. The elements need not be changed back to their original index.
2Insight 2: Binary search can be used to find the optimal frequency length. A binary search on the possible frequencies (from 1 to the length of the array) can determine the maximum achievable frequency.
3Insight 3: Prefix sums can be used to efficiently calculate the cost of making all elements within a window equal to a specific value (median). Prefix sum helps in calculating the cost of operations quickly.

📚

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!

Sliding Window

A moving "window" over data to process subsets efficiently.

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

Edge case 1: When k is 0, we return the maximum frequency in the array without performing any operations. The current solution handles this case because the binary search will eventually check the initial frequency.
Edge case 2: When all numbers in array are same, the answer is the size of nums, regardless of the value of k. The current solution correctly handles this case.
Edge case 3: When k is large enough to make all numbers same. The binary search will find the numsSize, since we can make all elements equal.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not sorting the array first. Sorting is crucial for the sliding window approach to work efficiently.

✗Common mistake 2: Incorrectly calculating the cost of making all elements in the window equal to the median. Need to consider the absolute differences and efficiently calculate them (using prefix sums).

✗Common mistake 3: Integer overflow when calculating costs. Since k can be up to 10^14, using `int` may cause overflows. Using `long` data types is crucial when manipulating large numbers.

✅

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: Optimizing resource allocation. Imagine resources represented by numbers. 'k' can represent the budget or tolerance for adjustments. Goal is to maximize how many resources are at the same level after adjustments.

🏦Real-world scenario 2: Data normalization. The given algorithm can be adapted to normalize a dataset by bringing elements closer to a central value while adhering to a constraint representing the maximum allowed adjustment ('k').

💡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

It's okay to not finish hard problems in interviews. Show your problem-solving process.

⚠️ 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 28, 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

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

@CodeNinjaJul 14

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

@TechLeadJul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I avoid off-by-one errors in binary search?

Use the pattern: left = 0, right = n-1, while (left <= right), mid = left + (right - left) / 2. Decide if you need to find exact match, first occurrence, or last occurrence, and adjust your conditions accordingly. Test with arrays of size 0, 1, and 2.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Apply Operations to Maximize Frequency Score"?

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.

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