nimheap

nimheap is a data structure that combines features of a binary heap and a Nim game strategy. It is primarily used in competitive programming and algorithm analysis for solving problems related to impartial games. The core idea is to represent the game state as a collection of heaps, similar to the game of Nim, and then use heap-based operations to efficiently determine winning and losing positions.

The nimheap structure typically stores elements in a way that allows for quick calculation of the Nim-sum, which is the bitwise XOR of the sizes of all heaps. This Nim-sum is a crucial concept in the Sprague-Grundy theorem, which states that every impartial game under the normal play convention is equivalent to a Nim heap of a certain size (its nim-value or Grundy number).

In practice, a nimheap might be implemented using an array or a balanced binary search tree, depending on the specific requirements for insertion, deletion, and querying of heap sizes. The efficiency of nimheap operations directly impacts the overall performance of algorithms that rely on it. For instance, if the game involves frequent modifications to the heaps, a data structure that supports logarithmic time updates would be preferred.

The nimheap is particularly useful when the game state can be decomposed into independent subgames, each of which can be represented as a heap. By calculating the Nim-sum of the nim-values of these subgames, one can determine if the current player has a winning strategy. If the Nim-sum is zero, the current position is a losing position; otherwise, it is a winning position.