Virtual Trees Method - Codeforces

This lecture explores the concept of virtual trees, key properties, their efficient construction, and application using dynamic programming (DP) on trees to solve specific problems.

Introduction

• Virtual Trees: Useful for tree-related problems involving a subset of vertices while maintaining the structure with their pairwise lowest common ancestors (LCAs).
• Efficiency: Transitions from tree T with N vertices to a substructure dependent on the size of the selected set X, accelerating algorithms, especially for DP computations.

Definition of a Virtual Tree

• Rooted Tree T: Given tree.
• Subset X: Selected vertices.
• Virtual Tree A(X): Includes all LCAs of pairs from X. Edges are between vertices if one is an ancestor of the other in T.

Key Properties and Their Proofs

Property 1: Size Estimation

• Statement: For any set X, (|A(X)| \leq 2|X| - 1).
• Proof: Utilizes depth-first search (DFS), showing each LCA equals a vertex in X or an LCA from earlier pairs.
• Result: Recursive relation leads to the final size estimate.
• Special Case: Holds exact in perfect binary trees with X as leaves.

Property 2: Representation through Sequential LCAs

• Statement: Vertices ordered by DFS, virtual tree is the union of X and LCAs of consecutive vertices.
• Proof: Recursive expansion demonstrates required representation.

Construction of a Virtual Tree

• Complexity: (O(|X|(\log|X| + \log N))) with preprocessing (O(N\log N)).

Construction Steps

1. Preprocessing: DFS or methods like binary lifting for entry times and LCAs.
2. Vertex Sorting: Sort X vertices by DFS (entry times).
3. Adding LCAs: Compute LCAs for consecutive vertices, union with X.
4. Tree Construction: Traverse and maintain a stack to restore the tree structure efficiently.

Application: Counting Subtrees in a Colored Tree

Problem Statement

• Task: Count subsets S of vertices where induced graph G[S] is a tree, all degree 1 vertices have the same color.

Solution Idea

• Break by Color: For each color, build a virtual tree for X (vertices with color c), perform DP to count valid subtrees.
• Efficiency: Virtual trees reduce problem size significantly, making DP feasible.

Complexity

• Total complexity: (O(N\log N + cC(|X_c|(\log|X_c| + \log N))) = O(N\log N)).

Solution Implementation

• C++ Code: Provided with comments explaining main algorithm components.
• Preprocessing & LCA computation: Using DFS and binary lifting.
• Virtual Tree Construction: Using sorted vertices and stack-based traversal.
• DP on Virtual Tree: Combines subtree results to count valid subtrees.
• Result Collection: Summing results for each color, output under modulo 998244353.

Conclusion

• Virtual Trees: Effective for reducing problem size in tree-related computations.
• Application: Demonstrated significant acceleration in computations for large trees.
• Reference: Based on an editorial for Atcoder problem ABC340G by Nyaan.