Number of Ways to Assign Edge Weights II

The problem asks for determining the number of ways to assign weights to tree edges such that the total path cost between nodes is calculated efficiently. Using dynamic programming and binary-tree traversal, we can calculate path costs by tracking states that represent edge weight assignments and their impact on node-to-node path sums. This problem requires efficient handling of multiple queries and state transitions.

Problem Statement

You are given a tree with n nodes, labeled from 1 to n, rooted at node 1. The tree is represented by a 2D array of edges, where each entry edges[i] = [ui, vi] indicates an undirected edge between nodes ui and vi. All edges initially have a weight of 0. You must assign a weight of either 1 or 2 to each edge in the tree.

For each query, you are asked to compute the total weight of the path between two nodes u and v. The cost of the path is the sum of the weights of the edges in the path connecting the two nodes. Your task is to efficiently calculate this for multiple queries.

Examples

Example 1

Input: edges = [[1,2]], queries = [[1,1],[1,2]]

Output: [0,1]

Example 2

Input: edges = [[1,2],[1,3],[3,4],[3,5]], queries = [[1,4],[3,4],[2,5]]

Output: [2,1,4]

Constraints

• 2 <= n <= 105
• edges.length == n - 1
• edges[i] == [ui, vi]
• 1 <= queries.length <= 105
• queries[i] == [ui, vi]
• 1 <= ui, vi <= n
• edges represents a valid tree.

Solution Approach

Binary Tree Traversal and State Tracking

Use DFS or BFS for tree traversal to compute and track the state of each node's edge weights and their contribution to the path sums. This will allow us to efficiently calculate the path costs in a tree structure.

Dynamic Programming with States

Apply dynamic programming to track the current state of each node's edge weight assignments. Specifically, track chainLength and sumParity states to minimize redundant calculations and avoid recalculating path costs for each query.

Efficient Query Handling

Once the states have been computed, store results for quick access during query evaluation. This allows for efficient answering of up to 105 queries in constant time.

Complexity Analysis

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

The time complexity depends on the traversal and dynamic programming approach. Traversing the tree takes O(n) time, and handling each query is O(1) after preprocessing. Thus, the overall time complexity is O(n + q), where n is the number of nodes and q is the number of queries.

What Interviewers Usually Probe

• Look for the candidate's understanding of dynamic programming and tree traversal techniques.
• Assess if the candidate can efficiently track states to handle multiple queries.
• Evaluate the candidate's ability to optimize the solution for large inputs, such as n and q up to 10^5.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to account for the initial state of edges before assigning weights.
• Not optimizing the query handling after preprocessing, leading to slower solutions.
• Misunderstanding the role of dynamic programming in this problem, resulting in inefficient solutions.

Follow-up variants

• Adding constraints where the tree has different types of edge weights.
• Allowing edge weights to be chosen from a larger set of values (e.g., 1, 2, 3).
• Modifying the problem to use directed graphs instead of trees.

FAQ

How can I efficiently handle the multiple queries in the 'Number of Ways to Assign Edge Weights II' problem?

Once you preprocess the tree using dynamic programming to track edge weight states, you can answer each query in constant time.

What are the key patterns involved in solving the 'Number of Ways to Assign Edge Weights II' problem?

This problem involves binary-tree traversal and dynamic programming to track edge weights and efficiently compute path costs.

Can I apply this approach to other tree-related dynamic programming problems?

Yes, this approach of state tracking and dynamic programming can be applied to other problems involving tree traversal and path cost computation.

What is the time complexity of the solution for 'Number of Ways to Assign Edge Weights II'?

The time complexity is O(n + q), where n is the number of nodes and q is the number of queries, due to the preprocessing step and constant time query handling.

How does dynamic programming help in solving this problem?

Dynamic programming helps by tracking states of the edge weights and their contribution to path costs, reducing redundant calculations and optimizing query handling.