Minimum Weighted Subgraph With the Required Paths II

This problem involves finding the minimum weighted subtree that allows paths between two source nodes and a destination node. The key approach is binary-tree traversal and tracking state across queries. Efficient solutions typically leverage techniques like binary lifting for faster traversal.

Problem Statement

You are given an undirected weighted tree with n nodes, numbered from 0 to n - 1. It is represented by a 2D integer array edges of length n - 1, where edges[i] = [ui, vi, wi] indicates that there is an edge between nodes ui and vi with weight wi. Additionally, you are given a 2D integer array queries, where queries[j] = [src1j, src2j, destj].

Return an array answer of length equal to queries.length, where answer[j] is the minimum total weight of a subtree such that it is possible to reach destj from both src1j and src2j using edges in this subtree.

Examples

Example 1

Input: edges = [[0,1,2],[1,2,3],[1,3,5],[1,4,4],[2,5,6]], queries = [[2,3,4],[0,2,5]]

Output: [12,11]

The blue edges represent one of the subtrees that yield the optimal answer.

answer[0] : The total weight of the selected subtree that ensures a path from src1 = 2 and src2 = 3 to dest = 4 is 3 + 5 + 4 = 12 . answer[1] : The total weight of the selected subtree that ensures a path from src1 = 0 and src2 = 2 to dest = 5 is 2 + 3 + 6 = 11 .

Example 2

Input: edges = [[1,0,8],[0,2,7]], queries = [[0,1,2]]

Output: [15]

Constraints

• 3 <= n <= 105
• edges.length == n - 1
• edges[i].length == 3
• 0 <= ui, vi < n
• 1 <= wi <= 104
• 1 <= queries.length <= 105
• queries[j].length == 3
• 0 <= src1j, src2j, destj < n
• src1j, src2j, and destj are pairwise distinct.
• The input is generated such that edges represents a valid tree.

Solution Approach

Binary Tree Traversal

Start by traversing the tree with a Depth-First Search (DFS) or Breadth-First Search (BFS) to explore paths between nodes. This helps to track which edges contribute to paths between the sources and the destination for each query.

Binary Lifting

Implement binary lifting, a technique for optimizing LCA (Lowest Common Ancestor) queries. This allows you to efficiently find common ancestors and traverse paths more quickly, crucial for large input sizes.

State Tracking and Path Minimization

Track the current state of each path during the traversal. For each query, minimize the weight of the subtree by calculating the minimal total weight of edges needed to connect the required nodes.

Complexity Analysis

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

The time and space complexity depend on the approach. DFS/BFS with binary lifting can handle large input sizes in logarithmic time for each query. The complexity of the final solution can be roughly O(n log n) for preprocessing with O(log n) per query for LCA queries and path minimization.

What Interviewers Usually Probe

• Understanding how to apply DFS/BFS with binary lifting is key to solving this problem efficiently.
• The candidate should demonstrate awareness of tree traversal and path optimization strategies.
• Look for candidates who focus on time complexity optimization when handling large input sizes.

Common Pitfalls or Variants

Common pitfalls

• Not efficiently handling large trees and queries, leading to timeouts or inefficient solutions.
• Overlooking the need to minimize the weight of the subtree for each query, which can lead to incorrect answers.
• Failing to apply binary lifting correctly, causing unnecessary recalculations of paths.

Follow-up variants

• Optimizing the algorithm for a higher number of nodes and queries.
• Extending the problem to support dynamic updates to the tree or queries.
• Handling multiple types of tree structures beyond binary trees, such as general graphs.

FAQ

What is the main approach for solving the Minimum Weighted Subgraph With the Required Paths II problem?

The problem can be solved using binary-tree traversal combined with techniques like binary lifting to optimize path queries and state tracking for efficient subtree weight calculation.

How does binary lifting help in solving the Minimum Weighted Subgraph With the Required Paths II?

Binary lifting optimizes the process of finding the Lowest Common Ancestor (LCA) of nodes, which is crucial for minimizing the path weights between source and destination nodes in the tree.

What is the time complexity of the best solution for the Minimum Weighted Subgraph With the Required Paths II?

The optimal time complexity is O(n log n) for preprocessing with O(log n) per query, depending on the approach used for tree traversal and binary lifting.

What are some common mistakes when solving the Minimum Weighted Subgraph With the Required Paths II?

Common mistakes include not optimizing the tree traversal, overlooking the importance of minimizing subtree weights, and failing to apply binary lifting correctly.

How can GhostInterview assist with solving the Minimum Weighted Subgraph With the Required Paths II problem?

GhostInterview provides hints and solutions for implementing efficient tree traversal, binary lifting, and state tracking techniques, which are essential for solving this problem efficiently.