#1145

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Binary Tree Coloring Game

A two-player game where players color nodes in a binary tree, aiming to outmaneuver each other by choosing adjacent nodes.

tree depth first search binary tree

Problem Statement

Two players play a turn-based game on a binary tree. Given the root of a binary tree and the number of nodes n in the tree, each player takes turns coloring nodes. Initially, Player 1 colors a node x red and Player 2 colors a node y blue. Players must color adjacent uncolored nodes on each turn, choosing from the current node's children or parent.

The game aims for Player 1 to determine whether they can guarantee a win, starting from an optimal move. The problem involves binary tree traversal and state tracking to ensure Player 1 can always secure a victory based on adjacency constraints.

Examples

Example 1

Input: root = [1,2,3,4,5,6,7,8,9,10,11], n = 11, x = 3

Output: true

The second player can choose the node with value 2.

Example 2

Input: root = [1,2,3], n = 3, x = 1

Output: false

Example details omitted.

Constraints

The number of nodes in the tree is n.
1 <= x <= n <= 100
n is odd.
1 <= Node.val <= n
All the values of the tree are unique.

Solution Approach

Tree Traversal and Node Adjacency

The solution involves identifying the relationships between nodes in the tree. A Depth-First Search (DFS) or Breadth-First Search (BFS) can help traverse the tree to track the state of nodes and their adjacency. The goal is to recognize the best move for Player 1, where Player 2's moves are constrained based on adjacency to Player 1's chosen node.

Optimal Move Selection

The optimal move for Player 2 is to pick a node that is adjacent to Player 1's node. Since Player 1 starts the game, the key to guaranteeing a win lies in whether Player 2 can be forced into a corner with no viable adjacent nodes.

State Tracking and Simulation

Tracking the state of nodes (colored or uncolored) is essential for simulating the turns of both players. This allows for real-time decisions to be made about which node to color next, ensuring an efficient solution to check if Player 1 can win under all circumstances.

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 chosen. A DFS or BFS traversal will have a time complexity of O(n), where n is the number of nodes, as each node is visited once. The space complexity is also O(n), primarily for storing the tree structure and the state of nodes during traversal.

What Interviewers Usually Probe

Ensure the candidate demonstrates an understanding of binary tree traversal and the importance of adjacency in game theory.
Look for an approach where the candidate can explain how Player 1's optimal move is determined in relation to Player 2's options.
Evaluate the candidate's ability to simulate turns and state transitions to guarantee a solution with correct time complexity.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the traversal with unnecessary steps that do not contribute to solving the adjacency problem efficiently.
Failing to simulate Player 2's move properly, missing out on how to lock down the adjacent nodes strategically.
Not considering edge cases where Player 1 could lose due to Player 2’s strategic choices or incorrect assumptions about available moves.

Follow-up variants

Modify the problem by increasing the size of the binary tree or adding more complex rules about which nodes can be colored.
Change the initial player to Player 2 and re-evaluate if the strategy changes for Player 1’s optimal move.
Introduce more than two players and evaluate how the game dynamics evolve with additional strategic considerations.

FAQ

How do I determine Player 1's best move in the Binary Tree Coloring Game?

Player 1’s best move is choosing a node that forces Player 2 to be restricted in their choices. This move ensures Player 1 can keep coloring adjacent nodes until a win is achieved.

What tree traversal technique should be used in the Binary Tree Coloring Game?

You should use Depth-First Search (DFS) or Breadth-First Search (BFS) to explore the tree. Both can efficiently track the adjacency of nodes as the game progresses.

What is the time complexity of solving the Binary Tree Coloring Game?

The time complexity is O(n), where n is the number of nodes, as you must visit each node during the tree traversal.

Can the game be won by Player 2 in the Binary Tree Coloring Game?

Yes, but only if Player 2 can make the first move strategically, and Player 1 fails to lock out their possible moves.

How does GhostInterview help with the Binary Tree Coloring Game?

GhostInterview assists by breaking down the problem’s pattern and guiding you through the optimal strategies to secure a win while avoiding common pitfalls.

terminal

Solution

Solution 1: DFS

First, we use DFS to find the node where player 1's colored point $x$ is located, denoted as $node$.

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# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def btreeGameWinningMove(self, root: Optional[TreeNode], n: int, x: int) -> bool:
        def dfs(root):
            if root is None or root.val == x:
                return root
            return dfs(root.left) or dfs(root.right)

        def count(root):
            if root is None:
                return 0
            return 1 + count(root.left) + count(root.right)

        node = dfs(root)
        l, r = count(node.left), count(node.right)
        return max(l, r, n - l - r - 1) > n // 2

Continue Practicing

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