Subtree Removal Game with Fibonacci Tree

Subtree Removal Game with Fibonacci Tree

What is this problem about?

The "Subtree Removal Game with Fibonacci Tree" is a specialized competitive programming problem that blends game theory with recursive tree structures. A Fibonacci tree of order n is defined recursively: F(n) has F(n-1) as its left child and F(n-2) as its right child, with base cases F(0) and F(1) being single nodes. In this game, two players take turns removing a subtree from the current tree. The player who cannot move (because the tree is gone) loses. The goal is to determine which player has a winning strategy given a tree of order n.

Why is this asked in interviews?

While extremely niche, companies like Sony or those focusing on mathematical research and optimization ask this to test a candidate's mastery of Game Theory (specifically the Sprague-Grundy theorem) and recursive definitions. It evaluates if a candidate can identify the underlying mathematical pattern (the Grundy value or nim-value) of a complex recursive structure. It also tests the ability to apply dynamic programming to game states where the state space might otherwise be overwhelming.

Algorithmic pattern used

This problem is solved using Game Theory and Dynamic Programming. Specifically, we use the Sprague-Grundy Theorem, which states that any impartial game under the normal play convention is equivalent to a Nim pile of a certain size. We calculate the Grundy value G(n) for a Fibonacci tree of order n. Because a move involves removing a subtree, the game splits into independent subgames. The Grundy value of the whole tree is the XOR sum of the Grundy values of the resulting parts after a removal.

Example explanation

Consider a Fibonacci tree F(2). Its left child is F(1) and its right child is F(0). If a player removes the entire tree from the root, the Grundy value of the remaining state is 0. If they remove a subtree from the left child, they are left with a modified tree. By calculating the Grundy values for small n (0, 1, 2, 3...), a periodic or recursive pattern emerges. For example, if G(0)=1 and G(1)=1, we use these to compute G(2) by looking at all possible reachable Grundy values and taking the Minimum Excluded value (MEX).

Common mistakes candidates make

1. Treating it as a simple tree removal: Forgetting that removing an edge or a node in the middle of a tree (if the game allowed it) is different from removing a subtree.
2. Ignoring XOR logic: In game theory involving multiple independent components, the "sum" of games is handled via the bitwise XOR of their Grundy values, not addition.
3. Recursion Depth: Failing to notice the pattern and trying to simulate the game for large n, which leads to exponential time complexity.

Interview preparation tip

Study the Sprague-Grundy theorem and the concept of MEX (Minimum Excluded value). These are the pillars of solving impartial games. Fibonacci trees are a common way to introduce recursion into math problems; always look for how the property of the tree (e.g., Grundy value) relates to the properties of its children.