What is the time complexity of Maximal Score After Applying K Operations?

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

What data structures are used to solve Maximal Score After Applying K Operations?

This problem uses Array, Greedy, Heap (Priority Queue). Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximal Score After Applying K Operations?

Insight 1: The greedy approach of always picking the largest number available at each step is optimal because we want to maximize our score as quickly as possible. Insight 2: A max-heap data structure is perfectly suited for efficiently tracking the largest number at each step and updating it after each operation.

Which companies ask LeetCode 2530. Maximal Score After Applying K Operations in interviews?

This problem is frequently asked at McKinsey.

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

Maximal Score After Applying K Operations - LeetCode 2530 Solution

Q: How to solve Maximal Score After Applying K Operations?

The solution uses a greedy approach combined with a max-heap data structure. We initialize a max-heap with the elements of the input array `nums`. Then, we iterate `k` times, each time extracting the maximum element from the heap, adding it to the score, calculating the new value (ceiling of val/3), and inserting the new value back into the heap. This ensures that we always pick the largest available number in each operation, maximizing our score.

Maximal Score After Applying K Operations - Complete Solution Guide

Maximal Score After Applying K Operations is LeetCode problem 2530, a Medium 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 have a starting score of 0 . In one operation : choose an index i such that 0 <= i < nums.length , increase your score by nums[i] , and replace nums[i] with ceil(nums[i] / 3) . Return the maximum possible score you can attain after applying exactly k operations . The ceiling function ceil(val) is the least integer greater than or equal to val . Example 1: Input: nums = [10,10,10,10,10], k = 5 Output: 50 Explanation: Apply the op

Detailed Explanation

The problem asks us to find the maximum possible score achievable after applying exactly `k` operations on a given array `nums`. In each operation, we select an index `i` from `nums`, increase our score by the value at that index (`nums[i]`), and then replace `nums[i]` with the ceiling of `nums[i] / 3`. The goal is to maximize the total score after `k` operations.

Solution Approach

The solution uses a greedy approach combined with a max-heap data structure. We initialize a max-heap with the elements of the input array `nums`. Then, we iterate `k` times, each time extracting the maximum element from the heap, adding it to the score, calculating the new value (ceiling of val/3), and inserting the new value back into the heap. This ensures that we always pick the largest available number in each operation, maximizing our score.

Step-by-Step Algorithm

Step 1: Initialize a max-heap with all elements from the `nums` array.
Step 2: Initialize a score variable to 0.
Step 3: Iterate `k` times:
Step 4: In each iteration, extract the maximum element (`val`) from the max-heap.
Step 5: Add `val` to the `score`.
Step 6: Calculate the new value `new_val` as the ceiling of `val / 3`. This can be done as `(val + 2) / 3` in integer arithmetic.
Step 7: Insert `new_val` into the max-heap.
Step 8: After `k` iterations, return the final `score`.

Key Insights

Insight 1: The greedy approach of always picking the largest number available at each step is optimal because we want to maximize our score as quickly as possible.
Insight 2: A max-heap data structure is perfectly suited for efficiently tracking the largest number at each step and updating it after each operation.
Insight 3: The `ceil(nums[i] / 3)` operation needs careful handling to ensure correctness across different programming languages. For example, integer division can truncate, so adding 2 before dividing by 3 can correctly emulate the `ceil` function for positive integers.

Complexity Analysis

Time Complexity: O(n + k log n)

Space Complexity: O(n)

Topics

This problem involves: Array, Greedy, Heap (Priority Queue).

Companies

Asked at: McKinsey.

Maximal Score After Applying K Operations | LeetCode 2530

Medium

Array Greedy Heap (Priority Queue)

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

Solution Code

Available in:

import java.util.PriorityQueue;

class Solution {
    public long maxKelements(int[] nums, int k) {
        PriorityQueue<Integer> max_heap = new PriorityQueue<>((a, b) -> b - a);
        
        for (int num : nums) {
            max_heap.add(num);
        }
        
        long score = 0;
        
        for (int i = 0; i < k; i++) {
            int val = max_heap.poll();
            score += val;
            
            int new_val = (val + 2) / 3;
            
            max_heap.add(new_val);
        }
        
        return score;
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 2530: Maximal Score After Applying K Operations - Detailed explanation and solution approaches.

Topics

Array Greedy Heap (Priority Queue)

Companies

McKinsey

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:2530

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 a max-heap with all elements from the `nums` array.
2
Step 2: Initialize a score variable to 0.
3
Step 3: Iterate `k` times:
4
Step 4: In each iteration, extract the maximum element (`val`) from the max-heap.
5
Step 5: Add `val` to the `score`.
6
Step 6: Calculate the new value `new_val` as the ceiling of `val / 3`. This can be done as `(val + 2) / 3` in integer arithmetic.
7
Step 7: Insert `new_val` into the max-heap.
8
Step 8: After `k` iterations, return the final `score`.

💡

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 log n)

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity comes from building the heap (O(n)) and performing k heap operations (O(k log n)). The space complexity is dictated by the heap which stores at most n 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

Using a sorted array

Instead of a heap, you could use a sorted array. After each operation, you would need to re-sort the array to maintain the order. You can insert the `new_val` and then sort the array. This can be optimized using binary search to find the correct location to insert `new_val`.

Time: O(k * n log n) for sorting n elements k times or O(k*n) for inserting into a sorted array of size nSpace: O(n)

Trade-offs: The heap-based approach is generally more efficient (O(n + k log n)) than sorting the array in each iteration (O(k * n log n)). However, a sorted array may be simpler to implement for some.

✨

Key Insights

1Insight 1: The greedy approach of always picking the largest number available at each step is optimal because we want to maximize our score as quickly as possible.
2Insight 2: A max-heap data structure is perfectly suited for efficiently tracking the largest number at each step and updating it after each operation.
3Insight 3: The `ceil(nums[i] / 3)` operation needs careful handling to ensure correctness across different programming languages. For example, integer division can truncate, so adding 2 before dividing by 3 can correctly emulate the `ceil` function for positive integers.

📚

Key Terms Explained

For Students

Array

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

Greedy Algorithm

Making the best choice at each step without looking ahead.

Edge Cases

Edge case 1: `nums` array contains only small values (e.g., 1). The algorithm still works correctly because the heap ensures the largest available element is always chosen, even if the score increases slowly.
Edge case 2: `k` is larger than the number of elements in `nums`. The algorithm still works correctly because the heap operations are performed `k` times, modifying the elements as required.
Edge case 3: `nums` array contains only a single element. The algorithm will repeatedly pick that element and divide it by 3 (ceiling) `k` times.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not using a max-heap (or equivalent) and trying to find the maximum element by iterating through the array in each operation. This would lead to a much slower time complexity (O(n*k)).

✗Common mistake 2: Incorrectly implementing the ceiling function. It's important to ensure that `ceil(val / 3)` is calculated correctly, especially when dealing with integer division, for which adding 2 before dividing by 3 is crucial.

✗Common mistake 3: Forgetting to re-heapify or re-sort after modifying an element, leading to incorrect ordering and a suboptimal score.

✅

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, where you want to maximize profit by allocating resources to tasks that yield the highest immediate return, but decrease in value after each allocation (similar to dividing by 3).

🏦Real-world scenario 2: Scheduling tasks with diminishing returns. Prioritize tasks with the highest initial value, but whose value decreases over time. The heap helps to dynamically reschedule tasks based on their updated value.

💡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

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

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

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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 11, 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

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

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

@AlgoMasterJul 6

❓

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 verify if a greedy approach is correct?

Prove the greedy choice property: the locally optimal choice leads to globally optimal solution. Use exchange argument: show that swapping any element with the greedy choice doesn't improve the solution. If you can't prove it, test extensively or consider DP.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

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 "Maximal Score After Applying K Operations"?

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

🤖 Need help with this problem?