#2242

Hard

LeetCode Problem Workspace

Maximum Score of a Node Sequence

Find the maximum score of a valid node sequence in an undirected graph with given node scores and edges.

Problem Statement

You are given an undirected graph with n nodes numbered from 0 to n - 1. Alongside this, you are provided a 0-indexed integer array scores of length n, where scores[i] represents the score of node i. A sequence of nodes is valid if it consists of exactly 4 nodes where each consecutive pair is connected by an edge from the given graph.

The goal is to find the maximum score of a valid node sequence. If no valid sequence of length 4 exists, return -1. You are also given a list of edges, where each edge connects two nodes. The nodes in a sequence must be connected by edges, and the sequence must contain exactly 4 nodes. If there are multiple sequences with the same maximum score, return any one of them.

Examples

Example 1

Input: scores = [5,2,9,8,4], edges = [[0,1],[1,2],[2,3],[0,2],[1,3],[2,4]]

Output: 24

The figure above shows the graph and the chosen node sequence [0,1,2,3]. The score of the node sequence is 5 + 2 + 9 + 8 = 24. It can be shown that no other node sequence has a score of more than 24. Note that the sequences [3,1,2,0] and [1,0,2,3] are also valid and have a score of 24. The sequence [0,3,2,4] is not valid since no edge connects nodes 0 and 3.

Example 2

Input: scores = [9,20,6,4,11,12], edges = [[0,3],[5,3],[2,4],[1,3]]

Output: -1

The figure above shows the graph. There are no valid node sequences of length 4, so we return -1.

Constraints

n == scores.length
4 <= n <= 5 * 104
1 <= scores[i] <= 108
0 <= edges.length <= 5 * 104
edges[i].length == 2
0 <= ai, bi <= n - 1
ai != bi
There are no duplicate edges.

Solution Approach

Graph Traversal

Start by exploring valid node sequences using depth-first or breadth-first search to traverse the graph. For each potential node sequence, ensure each pair of consecutive nodes is connected by an edge.

Edge Triplet Consideration

Since each valid sequence consists of exactly 4 nodes, identify triplets of edges that form valid paths in the graph. Calculate the score for each valid triplet and track the maximum score encountered.

Optimization with Sorting

Sort the nodes and edges by score values to prioritize higher score nodes and minimize unnecessary computations. This can help in reducing the search space while maximizing the score calculation.

Complexity Analysis

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

The time and space complexity depends on the chosen approach for traversal and edge checking. Optimizing the search space and using sorting techniques can reduce the complexity by limiting the number of possible edge combinations to check.

What Interviewers Usually Probe

Can the candidate identify and efficiently explore valid edge triplets?
Does the candidate consider the edge constraints and node connectivity in each sequence?
How well does the candidate optimize the solution for large inputs, considering the constraints?

Common Pitfalls or Variants

Common pitfalls

Failing to properly account for edge connections when checking for valid node sequences.
Not optimizing the search space for large graphs, leading to excessive computational complexity.
Misunderstanding the requirement of a sequence length of exactly 4 nodes, leading to incorrect solutions.

Follow-up variants

Consider variations where the sequence length is changed to 3 or 5 nodes.
Extend the problem to include weighted edges and explore how this changes the sequence scoring.
Implement a solution for sequences with dynamic edge connectivity instead of static edges.

FAQ

What is the goal of the Maximum Score of a Node Sequence problem?

The goal is to find the maximum score of a valid node sequence of exactly 4 nodes in a graph where each consecutive pair is connected by an edge.

How do you ensure a node sequence is valid in this problem?

A sequence is valid if all consecutive nodes in the sequence are connected by an edge, and the sequence must be exactly 4 nodes long.

How does GhostInterview help with the Maximum Score of a Node Sequence problem?

GhostInterview offers step-by-step guidance in identifying valid node sequences, optimizing the search for maximum scores, and considering all relevant edge combinations.

What if there is no valid sequence of length 4?

If no valid sequence exists, the solution should return -1 as specified in the problem.

Can this problem be solved using a greedy approach?

A greedy approach could be attempted, but careful consideration of all valid edge triplets and node connections is necessary to ensure correctness.

terminal

Solution

Solution 1

#### Python3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

class Solution:
    def maximumScore(self, scores: List[int], edges: List[List[int]]) -> int:
        g = defaultdict(list)
        for a, b in edges:
            g[a].append(b)
            g[b].append(a)
        for k in g.keys():
            g[k] = nlargest(3, g[k], key=lambda x: scores[x])
        ans = -1
        for a, b in edges:
            for c in g[a]:
                for d in g[b]:
                    if b != c != d != a:
                        t = scores[a] + scores[b] + scores[c] + scores[d]
                        ans = max(ans, t)
        return ans

Continue Practicing

#2234 maximum total beauty of the gardens #2171 removing minimum number of magic beans #2122 recover the original array