What is the time complexity of Maximum Sum Obtained of Any Permutation?

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

What data structures are used to solve Maximum Sum Obtained of Any Permutation?

This problem uses Array, Greedy, Sorting, Prefix Sum. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum Sum Obtained of Any Permutation?

Insight 1: The problem can be solved without explicitly generating all permutations. Instead, focus on how many times each element of `nums` should appear in the total sum across all requests. Insight 2: Use a frequency array (or difference array technique) to keep track of how many times each index of `nums` is included in the requests.

Which companies ask LeetCode 1589. Maximum Sum Obtained of Any Permutation in interviews?

This problem is frequently asked at PayPal.

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Maximum Sum Obtained of Any Permutation - LeetCode 1589 Solution

Maximum Sum Obtained of Any Permutation - Complete Solution Guide

Maximum Sum Obtained of Any Permutation is LeetCode problem 1589, 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

We have an array of integers, nums , and an array of requests where requests[i] = [start i , end i ] . The i th request asks for the sum of nums[start i ] + nums[start i + 1] + ... + nums[end i - 1] + nums[end i ] . Both start i and end i are 0-indexed . Return the maximum total sum of all requests among all permutations of nums . Since the answer may be too large, return it modulo 10 9 + 7 . Example 1: Input: nums = [1,2,3,4,5], requests = [[1,3],[0,1]] Output: 19 Explanation: One permutation o

Detailed Explanation

The problem asks us to find the maximum possible sum of request ranges in an array `nums`, considering all possible permutations of `nums`. Each request `[start_i, end_i]` represents a range of indices within `nums`. We need to find a permutation of `nums` such that the sum of all request range sums is maximized. Since the result can be very large, we need to return the sum modulo 10^9 + 7.

Solution Approach

The solution efficiently computes the frequencies of each index appearing in the requests. Then, it sorts both the original array `nums` and the calculated frequencies. By multiplying the largest numbers from `nums` with the highest frequencies and summing them up, we ensure the maximized sum of requests. The modulo operation prevents integer overflow.

Step-by-Step Algorithm

Step 1: Initialize a frequency array `freqs` of size n+1 (where n is the length of `nums`) with all elements set to 0. The extra element facilitates the difference array technique.
Step 2: Iterate through the `requests` array. For each request [start, end], increment `freqs[start]` by 1 and decrement `freqs[end + 1]` by 1. This effectively marks the range [start, end] in `freqs`.
Step 3: Calculate the prefix sum of the `freqs` array. This step determines how many times each index `i` in `nums` is covered by requests. For example, `freqs[i] += freqs[i-1]` iterates from index 1 to n-1.
Step 4: Remove the last element of the `freqs` array. Since we used the difference array technique and allocated one extra element, we must remove the last element after calculating the prefix sum because it is not relevant.
Step 5: Sort both the `nums` and `freqs` arrays in ascending order.
Step 6: Calculate the total sum by multiplying corresponding elements of the sorted `nums` and `freqs` arrays. Take the modulo at each step to prevent integer overflow.
Step 7: Return the final total sum.

Key Insights

Insight 1: The problem can be solved without explicitly generating all permutations. Instead, focus on how many times each element of `nums` should appear in the total sum across all requests.
Insight 2: Use a frequency array (or difference array technique) to keep track of how many times each index of `nums` is included in the requests.
Insight 3: To maximize the total sum, sort both `nums` and the frequency array. Pair the largest numbers in `nums` with the most frequent indices to maximize the overall sum.

Complexity Analysis

Time Complexity: O(n log n)

Space Complexity: O(n)

Topics

This problem involves: Array, Greedy, Sorting, Prefix Sum.

Companies

Asked at: PayPal.

Maximum Sum Obtained of Any Permutation | LeetCode 1589

Medium

Array Greedy Sorting Prefix Sum

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

Solution Code

Available in:

import java.util.Arrays;

class Solution {
    public int maxSumRangeQuery(int[] nums, int[][] requests) {
        int n = nums.length;
        int MOD = (int)1e9 + 7;

        int[] freqs = new int[n + 1];
        for (int[] request : requests) {
            int start = request[0];
            int end = request[1];
            freqs[start] += 1;
            if (end + 1 < n+1)
            {
              freqs[end + 1] -= 1;
            }
           
        }

        for (int i = 1; i < n; i++) {
            freqs[i] += freqs[i-1];
        }
        
        freqs = Arrays.copyOf(freqs, n);


        Arrays.sort(nums);
        Arrays.sort(freqs);

        long total_sum = 0;
        for (int i = 0; i < n; i++) {
            total_sum = (total_sum + (long)nums[i] * freqs[i]) % MOD;
        }
        
        return (int)total_sum;
    }
}

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Problem Info

Description

Problem 1589: Maximum Sum Obtained of Any Permutation - Detailed explanation and solution approaches.

Topics

Array Greedy Sorting Prefix Sum

Companies

PayPal

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:1589

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

7 steps

1
Step 1: Initialize a frequency array `freqs` of size n+1 (where n is the length of `nums`) with all elements set to 0. The extra element facilitates the difference array technique.
2
Step 2: Iterate through the `requests` array. For each request [start, end], increment `freqs[start]` by 1 and decrement `freqs[end + 1]` by 1. This effectively marks the range [start, end] in `freqs`.
3
Step 3: Calculate the prefix sum of the `freqs` array. This step determines how many times each index `i` in `nums` is covered by requests. For example, `freqs[i] += freqs[i-1]` iterates from index 1 to n-1.
4
Step 4: Remove the last element of the `freqs` array. Since we used the difference array technique and allocated one extra element, we must remove the last element after calculating the prefix sum because it is not relevant.
5
Step 5: Sort both the `nums` and `freqs` arrays in ascending order.
6
Step 6: Calculate the total sum by multiplying corresponding elements of the sorted `nums` and `freqs` arrays. Take the modulo at each step to prevent integer overflow.
7
Step 7: Return the final total sum.

💡

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-linear - Great for sorting algorithms!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The sorting operations have a time complexity of O(n log n). The creation of the frequency array `freqs` requires O(n) space. Other operations, like prefix sum calculation and summing the products, take O(n) time and constant space, which is dominated by the sorting complexities. Thus the overall time is O(n log n), and overall space is O(n).

📖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 Permutation Generation

Generate all possible permutations of `nums` and calculate the total sum for each permutation. Keep track of the maximum sum found.

Time: O(n! * m * n), where n is the length of `nums` and m is the number of requests. Generating all permutations takes O(n!) time. Calculating the sum for each permutation requires iterating through all requests, and each request requires summing up to n elements.Space: O(n), to store a single permutation.

Trade-offs: This approach is extremely inefficient and not feasible for larger inputs due to the factorial time complexity. It would only be suitable for very small input sizes.

✨

Key Insights

1Insight 1: The problem can be solved without explicitly generating all permutations. Instead, focus on how many times each element of `nums` should appear in the total sum across all requests.
2Insight 2: Use a frequency array (or difference array technique) to keep track of how many times each index of `nums` is included in the requests.
3Insight 3: To maximize the total sum, sort both `nums` and the frequency array. Pair the largest numbers in `nums` with the most frequent indices to maximize the overall sum.

📚

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.

Sorting

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

Edge Cases

Edge case 1: Empty `nums` or `requests` array. The code handles this implicitly because the loops will simply not execute, resulting in a sum of 0.
Edge case 2: `requests` with overlapping ranges. The frequency array correctly handles overlapping ranges by incrementing and decrementing counts, and the prefix sum calculates the correct overall frequency.
Edge case 3: Large values in nums array. Because nums[i] can be up to 10^5 and the length of nums can be up to 10^5 the product of nums[i] * freq could overflow. Therefore, use long long (or long in java) to calculate the product

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Failing to use the modulo operator when calculating the sum, leading to integer overflow.

✗Common mistake 2: Incorrectly calculating the frequencies of indices by not considering overlapping ranges. The difference array method is essential to correctly handle overlaps.

✗Common mistake 3: Not sorting the `nums` and `freqs` arrays, resulting in a suboptimal sum. Sorting ensures that the largest numbers are multiplied by the highest frequencies.

✗Common mistake 4: Not using long long (or long in Java) when doing intermediate calculations because of possible integer overflow

✅

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. If `nums` represents the profit from assigning resources to different tasks, and `requests` represent the priority ranges for certain resources, this algorithm can help determine the optimal assignment of resources to maximize overall profit.

🏦Real-world scenario 2: Network bandwidth allocation. `nums` could represent the bandwidth available at different time slots, and `requests` could represent periods when certain applications require bandwidth. Maximizing the total available bandwidth during those critical periods can improve application performance.

🎮Real-world scenario 3: Data compression. Assigning higher frequency values (like freqs array) to elements that are larger (nums array) allows better space optimization when storing the data.

💡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

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ 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: October 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

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

@TechLeadJul 13

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

@DevExpert92Jul 6

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

@AlgoMasterJun 29

❓

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.

What are common greedy problem patterns?

Activity selection (sort by end time), interval scheduling (sort by start/end), Huffman coding (always combine smallest), fractional knapsack (sort by value/weight ratio), and minimum spanning tree. Look for problems where local decisions don't affect future options.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

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 "Maximum Sum Obtained of Any Permutation"?

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?