3373. Maximize the Number of Target Nodes After Connecting Trees II

Difficulty: Hard

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

There exist two undirected trees with n and m nodes, labeled from [0, n - 1] and [0, m - 1], respectively.

You are given two 2D integer arrays edges1 and edges2 of lengths n - 1 and m - 1, respectively, where edges1[i] = [ai, bi] indicates that there is an edge between nodes ai and bi in the first tree and edges2[i] = [ui, vi] indicates that there is an edge between nodes ui and vi in the second tree.

Node u is target to node v if the number of edges on the path from u to v is even. Note that a node is always target to itself.

Return an array of n integers answer, where answer[i] is the maximum possible number of nodes that are target to node i of the first tree if you had to connect one node from the first tree to another node in the second tree.

Note that queries are independent from each other. That is, for every query you will remove the added edge before proceeding to the next query.

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]]
• Output: [8,7,7,8,8]
• Explanation:
  • 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 4 from the second tree.
  • For i = 2, connect node 2 from the first tree to node 7 from the second tree.
  • For i = 3, connect node 3 from the first tree to node 0 from the second tree.
  • For i = 4, connect node 4 from the first tree to node 4 from the second tree.

Example 2:

• Input: edges1 = [[0,1],[0,2],[0,3],[0,4]], edges2 = [[0,1],[1,2],[2,3]]
• Output: [3,6,6,6,6]
• Explanation: For every i, connect node i of the first tree with any node of the second tree.

Constraints:

• 2 <= n, m <= 105
• edges1.length == n - 1
• edges2.length == m - 1
• edges1[i].length == edges2[i].length == 2
• edges1[i] = [ai, bi]
• 0 <= ai, bi < n
• edges2[i] = [ui, vi]
• 0 <= ui, vi < m
• The input is generated such that edges1 and edges2 represent valid trees.

Hint:

1. Compute an array even where even[u] is the number of nodes at an even distance from node u, for every u of the first tree.
2. Compute an array odd where odd[u] is the number of nodes at an odd distance from node u, for every u of the second tree.
3. answer[i] = even[i]+ max(odd[1], odd[2], …, odd[m - 1])

Solution:

We need to maximize the number of target nodes for each node in the first tree after connecting it to any node in the second tree. A node is considered a target to another node if the path between them has an even number of edges.

Approach

1. Problem Analysis:

   • Target Node Definition: A node u is a target to node v if the number of edges on the path from u to v is even. This includes the node itself (distance 0).
   • Connection Strategy: For each node i in the first tree, we connect it to some node j in the second tree. The goal is to choose j such that the number of target nodes for i is maximized.

2. Key Insight:

   • Bipartite Coloring: Trees are bipartite graphs, meaning nodes can be colored with two colors such that adjacent nodes have different colors. The parity of the path length between two nodes depends on their colors:
     • If two nodes have the same color, the path length between them is even.
     • If they have different colors, the path length is odd.
   • Maximizing Target Nodes:
     • First Tree (Tree1): The number of target nodes for i within Tree1 is the count of nodes in Tree1 that have the same color as i.
     • Second Tree (Tree2): By connecting i to a node j in Tree2, the number of target nodes in Tree2 for i is the count of nodes in Tree2 that have the opposite color of j. To maximize this, we choose j such that the count of opposite-colored nodes in Tree2 is maximized.

3. Algorithm:

   • Tree2 Processing:
     • Perform BFS starting from node 0 to color all nodes in Tree2 (0 or 1).
     • Count nodes with color 0 (count0) and color 1 (count1). The maximum of these counts (base = max(count0, count1)) gives the maximum possible target nodes from Tree2 for any node in Tree1.
   • Tree1 Processing:
     • Perform BFS starting from node 0 to color all nodes in Tree1.
     • Count nodes with color 0 (cnt0) and color 1 (cnt1).
   • Result Calculation: For each node i in Tree1:
     • If i has color 0, the number of target nodes in Tree1 is cnt0.
     • If i has color 1, the number of target nodes in Tree1 is cnt1.
     • The total target nodes for i is the sum of target nodes in Tree1 and base.

Let's implement this solution in PHP: 3373. Maximize the Number of Target Nodes After Connecting Trees II

<?php
/**
 * @param Integer[][] $edges1
 * @param Integer[][] $edges2
 * @return Integer[]
 */
function maxTargetNodes($edges1, $edges2) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// === Test ===

$edges1 = [[0,1],[0,2],[2,3],[2,4]];
$edges2 = [[0,1],[0,2],[0,3],[2,7],[1,4],[4,5],[4,6]];
print_r(maxTargetNodes($edges1, $edges2)); // Expected: [8,7,7,8,8]

$edges1 = [[0,1],[0,2],[0,3],[0,4]];
$edges2 = [[0,1],[1,2],[2,3]];
print_r(maxTargetNodes($edges1, $edges2)); // Expected: [3,6,6,6,6]
?>

Explanation:

1. Tree2 Processing:

   • We build an adjacency list for Tree2 and perform BFS starting from node 0 to color nodes. Nodes at an even distance from node 0 are colored 0, and those at an odd distance are colored 1.
   • We count nodes colored 0 (count0) and 1 (count1). The maximum of these counts (base) represents the maximum target nodes achievable from Tree2 when connecting any node in Tree1 to Tree2.

2. Tree1 Processing:

   • Similarly, we build an adjacency list for Tree1 and perform BFS starting from node 0 to color nodes.
   • We count nodes colored 0 (cnt0) and 1 (cnt1).

3. Result Calculation:

   • For each node i in Tree1, if i is colored 0, the target nodes in Tree1 are cnt0; otherwise, they are cnt1.
   • The total target nodes for i is the sum of target nodes in Tree1 and base (the maximum target nodes from Tree2).

This approach efficiently leverages bipartite properties of trees to compute the solution in linear time relative to the number of nodes and edges.

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