Maximum Score of a Node Sequence

Maximum Score of a Node Sequence

1. What is this problem about?

The Maximum Score of a Node Sequence interview question is a graph optimization problem. You are given an undirected graph where each node has a score. You need to find a sequence of four distinct nodes $(a, b, c, d)$ such that there are edges between $(a, b)$, $(b, c)$, and $(c, d)$. The goal is to maximize the sum of the scores of these four nodes.

This is essentially looking for the highest-scoring path of length 3 (4 nodes) in the graph.

2. Why is this asked in interviews?

Google asks the Maximum Score of a Node Sequence coding problem to test a candidate's ability to optimize a combinatorial search on a graph. A brute-force check of all paths of length 3 would be $O(V * E)$, which is too slow. It evaluates if you can recognize that for any edge $(b, c)$, you just need to find the "best" neighbors $a$ of $b$ and $d$ of $c$ that are distinct from each other.

3. Algorithmic pattern used

The Array, Graph, Sorting, Enumeration interview pattern is key.

1. For each node, store its top 3 neighbors with the highest scores. (Top 3 is enough to guarantee finding distinct nodes $a$ and $d$).
2. Iterate through every existing edge $(b, c)$ in the graph.
3. For each edge, try all pairs of nodes $(a, d)$ where $a$ is in the top 3 neighbors of $b$, and $d$ is in the top 3 neighbors of $c$.
4. Ensure all four nodes $(a, b, c, d)$ are distinct.
5. Calculate the score score[a] + score[b] + score[c] + score[d] and track the maximum.

4. Example explanation

Edge (B, C).

• B's top neighbors: A (score 100), X (score 50), Y (score 30).
• C's top neighbors: D (score 90), B (score 80 - ignore), Z (score 40).

Try pairs:

1. (A, D): Both distinct from B and C. (A, B, C, D) is valid. Score = 100 + B + C + 90.
2. (A, Z): Valid. Score = 100 + B + C + 40. The algorithm checks these combinations for every edge.

5. Common mistakes candidates make

In the Maximum Score of a Node Sequence coding problem, a frequent mistake is not storing enough neighbors. If you only store the top 1 neighbor, that neighbor might be the other end of the edge (e.g., $a$ could be $c$). Storing the top 3 ensures that even if one neighbor is $b$ or $c$ or $d$, you still have options. Another error is iterating over nodes instead of edges, which is often less efficient for path-based problems.

6. Interview preparation tip

For graph problems involving paths of fixed length, think about "fixing" the middle edge or middle node. This often reduces the search space significantly. Also, pre-sorting or pre-filtering neighbor lists is a powerful technique for optimizing graph traversals.