What is the time complexity of Maximum Star Sum of a Graph?

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

What is the space complexity of Maximum Star Sum of a Graph?

The space complexity is O(n + m).

What data structures are used to solve Maximum Star Sum of a Graph?

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

What are the key insights for solving Maximum Star Sum of a Graph?

Insight 1: The core idea is to iterate through each node and treat it as the potential center of a star graph. Insight 2: For each potential center, we need to consider its neighbors and select the 'k' largest positive neighbors to maximize the star sum. Negative neighbors always decrease the sum, so we only include positive neighbors in our adjacency list.

Which companies ask LeetCode 2497. Maximum Star Sum of a Graph in interviews?

This problem is frequently asked at Akuna Capital.

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Maximum Star Sum of a Graph - LeetCode 2497 Solution

Maximum Star Sum of a Graph - Complete Solution Guide

Maximum Star Sum of a Graph is LeetCode problem 2497, 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

There is an undirected graph consisting of n nodes numbered from 0 to n - 1 . You are given a 0-indexed integer array vals of length n where vals[i] denotes the value of the i th node. You are also given a 2D integer array edges where edges[i] = [a i , b i ] denotes that there exists an undirected edge connecting nodes a i and b i. A star graph is a subgraph of the given graph having a center node containing 0 or more neighbors. In other words, it is a subset of edges of the given graph such tha

Detailed Explanation

The problem asks us to find the maximum possible sum of a 'star graph' within a given undirected graph. A star graph is a subgraph centered around a single node, including a subset of its neighbors. The input consists of node values (vals), edge connections (edges), and a limit 'k' on the number of neighbors a star graph can have. The goal is to find the central node and its neighbors (up to 'k'), such that the sum of their values is maximized.

Solution Approach

The solution iterates through each node in the graph, considering it as the center of a potential star graph. For each node, it builds an adjacency list containing only the positive values of its neighbors. It then sorts this list in descending order and selects the top 'k' neighbors to calculate the star sum. Finally, it keeps track of the maximum star sum encountered across all nodes and returns it.

Step-by-Step Algorithm

Step 1: Create an adjacency list to represent the graph. The adjacency list stores only positive neighbor values for each node, as negative neighbors will always reduce the overall sum. The list is indexed by the node number.
Step 2: Iterate through each node from 0 to n-1. For each node, consider it as the center of a potential star graph.
Step 3: For the current center node, retrieve its neighbors from the adjacency list.
Step 4: Sort the neighbors' values in descending order.
Step 5: Select the top 'k' neighbors (or all neighbors if there are fewer than 'k') and sum their values. Add this sum to the center node's value to get the star sum for this center node.
Step 6: Update the maximum star sum encountered so far.
Step 7: After iterating through all nodes, return the maximum star sum.

Key Insights

Insight 1: The core idea is to iterate through each node and treat it as the potential center of a star graph.
Insight 2: For each potential center, we need to consider its neighbors and select the 'k' largest positive neighbors to maximize the star sum. Negative neighbors always decrease the sum, so we only include positive neighbors in our adjacency list.
Insight 3: Sorting the neighbors' values allows us to easily pick the top 'k' largest neighbors.

Complexity Analysis

Time Complexity: O(n log n + m)

Space Complexity: O(n + m)

Topics

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

Companies

Asked at: Akuna Capital.

Maximum Star Sum of a Graph | LeetCode 2497

Medium

Array Greedy Graph Sorting Heap (Priority Queue)

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

Solution Code

Available in:

import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.List;

class Solution {
    public int maxStarSum(int[] vals, int[][] edges, int k) {
        int n = vals.length;
        List<List<Integer>> adj = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            adj.add(new ArrayList<>());
        }

        for (int[] edge : edges) {
            int u = edge[0];
            int v = edge[1];
            if (vals[v] > 0) {
                adj.get(u).add(vals[v]);
            }
            if (vals[u] > 0) {
                adj.get(v).add(vals[u]);
            }
        }

        int max_s = Integer.MIN_VALUE;

        for (int i = 0; i < n; i++) {
            Collections.sort(adj.get(i), Collections.reverseOrder());

            int curr_s = vals[i];
            int num_to_take = Math.min(k, adj.get(i).size());

            for (int j = 0; j < num_to_take; j++) {
                curr_s += adj.get(i).get(j);
            }

            max_s = Math.max(max_s, curr_s);
        }

        return max_s;
    }
}

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Problem Info

Description

Problem 2497: Maximum Star Sum of a Graph - Detailed explanation and solution approaches.

Topics

Array Greedy Graph Sorting Heap (Priority Queue)

Companies

Akuna Capital

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:2497

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: Create an adjacency list to represent the graph. The adjacency list stores only positive neighbor values for each node, as negative neighbors will always reduce the overall sum. The list is indexed by the node number.
2
Step 2: Iterate through each node from 0 to n-1. For each node, consider it as the center of a potential star graph.
3
Step 3: For the current center node, retrieve its neighbors from the adjacency list.
4
Step 4: Sort the neighbors' values in descending order.
5
Step 5: Select the top 'k' neighbors (or all neighbors if there are fewer than 'k') and sum their values. Add this sum to the center node's value to get the star sum for this center node.
6
Step 6: Update the maximum star sum encountered so far.
7
Step 7: After iterating through all nodes, return the maximum star 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 + m)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(n + m)

📝

Why?

The algorithm iterates through all nodes and, for each node, sorts its neighbors. The maximum number of neighbors a node can have is n-1, so the sorting operation inside the loop contributes O(n log n) to the time complexity across all nodes. Building the adjacency list takes O(m) time. The space complexity is dominated by the adjacency list which stores the neighbors of each node.

📖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

Heap (Priority Queue)

Instead of sorting the neighbors, a min-heap (priority queue) of size 'k' can be maintained. Insert positive neighbor values into the heap. If the heap size exceeds 'k', remove the smallest element. The star sum can then be calculated using the elements in the heap.

Time: O(m log k), where m is the number of edges. Inserting into and removing from the heap takes O(log k) time. This occurs at most m times while building the adjacency list and iterating through neighbors.Space: O(n + k), where n is the number of nodes and k is the maximum allowed neighbors. The heap has a size of k and adjacency list take O(n) space.

Trade-offs: This approach might be slightly faster if k is much smaller than n, as log(k) will be smaller than log(n). However, if k is close to n, the sorting approach might be more efficient due to the lower overhead of the sorting algorithm.

✨

Key Insights

1Insight 1: The core idea is to iterate through each node and treat it as the potential center of a star graph.
2Insight 2: For each potential center, we need to consider its neighbors and select the 'k' largest positive neighbors to maximize the star sum. Negative neighbors always decrease the sum, so we only include positive neighbors in our adjacency list.
3Insight 3: Sorting the neighbors' values allows us to easily pick the top 'k' largest neighbors.

📚

Key Terms Explained

For Students

Array

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

Graph

Nodes connected by edges. Like a social network or city map!

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: A node has no neighbors. The solution correctly handles this case by simply using the node's value as the star sum.
Edge case 2: A node has neighbors, but all neighbor values are non-positive. In this case, the star sum is just the value of the center node.
Edge case 3: k = 0. The star graph only contains the center node, and the star sum is just the value of the center node.
Edge case 4: All values are negative. The code correctly identifies the maximum star sum, which will be the largest value node (least negative).
Edge case 5: Empty edges array. The code correctly calculates the sum for each node based on itself, as no neighbors will be added.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Including negative neighbor values in the star sum without proper consideration. This will always result in a suboptimal star sum.

✗Common mistake 2: Not handling the case where the number of neighbors is less than 'k'. The solution should only sum up to the available number of neighbors.

✗Common mistake 3: Not initializing the maximum sum to a sufficiently small value (e.g., negative infinity) to handle cases where all values are negative.

✗Common mistake 4: Not freeing dynamically allocated memory (C implementation) leading to memory leaks.

✅

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: Social Network Analysis. Finding influential users (nodes) by maximizing the 'sum' of their connections' (neighbors) influence scores (values). 'k' could represent a limit on the number of connections to consider for calculating influence.

🏦Real-world scenario 2: Recommendation Systems. Identifying the best items (nodes) to recommend to a user by maximizing the sum of the ratings (values) of similar items (neighbors) the user has already interacted with. 'k' can be the limited number of recommended items.

💡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.

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: November 5, 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 13

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

@CodeNinjaJul 6

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

@TechLeadJun 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 Star Sum of a Graph"?

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?