#2497

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Maximum Star Sum of a Graph

Find the maximum star sum in a graph with specific node values and edges, using greedy algorithms to select the optimal neighbors.

array greedy graph sorting heap priority queue

Problem Statement

You are given a graph with n nodes, where each node has a specific value represented in an array. The graph is undirected, and a list of edges represents the connections between nodes. A star graph is formed by selecting one node as the center and connecting it to its neighbors, forming a star-shaped structure.

The challenge is to maximize the star sum by selecting one center node and at most k neighbors. The neighbors selected should contribute to the maximum sum, making the solution rely on greedy selection and validation of the optimal combination of nodes.

Examples

Example 1

Input: vals = [1,2,3,4,10,-10,-20], edges = [[0,1],[1,2],[1,3],[3,4],[3,5],[3,6]], k = 2

Output: 16

The above diagram represents the input graph. The star graph with the maximum star sum is denoted by blue. It is centered at 3 and includes its neighbors 1 and 4. It can be shown it is not possible to get a star graph with a sum greater than 16.

Example 2

Input: vals = [-5], edges = [], k = 0

Output: -5

There is only one possible star graph, which is node 0 itself. Hence, we return -5.

Constraints

n == vals.length
1 <= n <= 105
-104 <= vals[i] <= 104
0 <= edges.length <= min(n * (n - 1) / 2, 105)
edges[i].length == 2
0 <= ai, bi <= n - 1
ai != bi
0 <= k <= n - 1

Solution Approach

Identify Potential Centers

Begin by considering each node as a potential center for the star graph. The sum for each potential center will be the value of the node itself, plus the values of its neighbors that increase the total sum.

Greedy Neighbor Selection

For each node, sort its neighbors by the value they add to the sum. Select the top k neighbors that provide the largest positive contribution to the center node’s value, ensuring the sum is maximized.

Sum Calculation and Validation

After selecting the optimal neighbors, calculate the total sum for each potential star graph and validate it against other possible centers. Return the maximum sum found after testing all options.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the number of edges and nodes. Sorting neighbors for each node adds a complexity of O(n log n), but a more efficient approach may depend on careful neighbor selection. The space complexity is O(n) due to the storage of the graph and neighbor information.

What Interviewers Usually Probe

The candidate should recognize that greedy choices and local optimization are key to maximizing the star sum.
Look for understanding of graph traversal and how to handle neighbors efficiently without unnecessary computations.
Candidates should demonstrate the ability to implement the greedy approach and validate the sum correctly.

Common Pitfalls or Variants

Common pitfalls

Forgetting that not all neighbors should be included—only those that increase the sum.
Overcomplicating the selection of neighbors or failing to efficiently sort and choose the top k neighbors.
Misunderstanding the problem constraints and attempting to include all neighbors instead of limiting the number.

Follow-up variants

Allowing more than k neighbors to be selected may change the approach and solution complexity.
Instead of maximizing the sum, trying to minimize it by selecting the least beneficial neighbors.
Expanding to weighted graphs where edges also contribute to the sum would increase the complexity of the problem.

FAQ

What is the greedy approach in the Maximum Star Sum of a Graph problem?

The greedy approach involves selecting the node with the largest value and choosing the top k neighbors that maximize the total sum of the star graph.

How do I know which neighbors to include in the star graph?

Include only the neighbors whose values positively contribute to the total sum of the graph. This is determined by sorting the neighbors by their value and selecting the top k.

What is a star graph in this context?

A star graph is a subgraph where one node acts as the center, and it connects to one or more neighbors, forming a star-like structure.

How does the Maximum Star Sum of a Graph relate to greedy algorithms?

The problem relies on greedy algorithms by selecting neighbors that maximize the sum, ensuring that only the most beneficial nodes are included in the final star graph.

What are the key challenges in solving the Maximum Star Sum of a Graph?

The main challenges include efficiently selecting the right neighbors, ensuring that only beneficial neighbors are included, and handling large graphs within time constraints.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maxStarSum(self, vals: List[int], edges: List[List[int]], k: int) -> int:
        g = defaultdict(list)
        for a, b in edges:
            if vals[b] > 0:
                g[a].append(vals[b])
            if vals[a] > 0:
                g[b].append(vals[a])
        for bs in g.values():
            bs.sort(reverse=True)
        return max(v + sum(g[i][:k]) for i, v in enumerate(vals))

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