Maximize the Number of Target Nodes After Connecting Trees

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Difficulty: Medium

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Topics: Tree, Depth-First Search, Breadth-First Search

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Introduction

In this tutorial, we will explore a problem involving two undirected trees and how to maximize the number of target nodes after connecting them. This problem is a great exercise in understanding tree structures and graph traversal techniques such as Depth-First Search (DFS) and Breadth-First Search (BFS).

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Prerequisites

Basic understanding of tree data structures

Familiarity with graph traversal algorithms, specifically DFS and BFS

Basic knowledge of PHP programming language

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Problem Statement

We have two undirected trees with n and m nodes, respectively. Each tree has distinct labels ranging from 0 to n - 1 for the first tree and 0 to m - 1 for the second tree.

You are given two 2D integer arrays edges1 and edges2, where:

edges1[i] = [ai, bi] indicates an edge between nodes ai and bi in the first tree.

edges2[i] = [ui, vi] indicates an edge between nodes ui and vi in the second tree.

Additionally, you are given an integer k. A node u is considered target to node v if the number of edges on the path from u to v is less than or equal to k. Notably, a node is always a target to itself.

Your task is to return an array of n integers answer, where answer[i] represents the maximum possible number of nodes that are target to node i of the first tree after connecting it to a node in the second tree.

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Examples

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Example 1:

Input: edges1 = [[0,1],[0,2],[2,3],[2,4]], edges2 = [[0,1],[0,2],[0,3],[2,7],[1,4],[4,5],[4,6]], k = 2

Output: [9,7,9,8,8]

Explanation:\n
- For i = 0, connect node 0 from the first tree to node 0 from the second tree.
- For i = 1, connect node 1 from the first tree to node 0 from the second tree.
- For i = 2, connect node 2 from the first tree to node 4 from the second tree.
- For i = 3, connect node 3 from the first tree to node 4 from the second tree.
- For i = 4, connect node 4 from the first tree to node 4 from the second tree.
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Example 2:

Input: edges1 = [[0,1],[0,2],[0,3],[0,4]], edges2 = [[0,1],[1,2],[2,3]], k = 1

Output: [6,3,3,3,3]

Explanation:\n
- For every i, connect node i of the first tree with any node of the second tree.
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Constraints

2 <= n, m <= 1000

edges1.length == n - 1

edges2.length == m - 1

edges1[i].length == edges2[i].length == 2

0 <= ai, bi < n

0 <= ui, vi < m

The input is generated such that edges1 and edges2 represent valid trees.

0 <= k <= 1000

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Hints

For each node u in the first tree, find the number of nodes at a distance of at most k from node u.

For each node v in the second tree, find the number of nodes at a distance of at most k - 1 from node v.

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Solution Overview

To solve this problem, we need to maximize the number of target nodes for each node in the first tree after connecting it to a node in the second tree. The solution involves efficiently calculating the number of nodes within the specified distance in both trees and then combining these counts optimally.

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Approach

Problem Analysis: We need to compute, for each node in the first tree, the maximum number of target nodes achievable by connecting it to any node in the second tree. The target nodes include nodes from both trees that are within a distance k from the node in the first tree.

Intuition:\n
- For any node i in the first tree, the number of target nodes within the first tree is the count of nodes within distance k from i.
- When connecting i to a node j in the second tree, the number of target nodes in the second tree is the count of nodes within distance k-1 from j.
- To maximize the total target nodes for i, we should choose j in the second tree that maximizes the count of nodes within distance k-1 from it.

Source: Original Article