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The game can be reduced to a theoretical observation: Alice will win if the XOR of all elements of the array (before any move) is already 0 or if the length of the array is even. This is because, under optimal play, a XOR of 0 or an even number of moves ensures Alice's win. When it's an even number, Bob will be the one forced to make a move that reduces the XOR to 0 if possible.
Time Complexity: O(n), where n is the length of nums.
Space Complexity: O(1), as no extra space is used beyond a few variables.
1def xorGame(nums):
2 return reduce(lambda x, y: x ^ y, nums) == 0 or len(nums) % 2 == 0The solution checks if the XOR of the entire list is already zero or if the list length is even. If either condition is met, Alice wins. We use Python's reduce function to calculate the XOR of the nums list.
This approach builds upon understanding the subproblems of removing each number and calculating the XOR of the remaining elements. Even though its complexity is higher, this method helps us understand more granular moves and outcomes if not assuming mathematical shortcuts.
Time Complexity: O(2^n) due to the exponential number of XOR states possible.
Space Complexity: O(n), largely dictated by function call stack depth.
1function xorGame
The dynamic programming strategy in JavaScript implements a recursive decision-making process with memoized state identifiers for checking all potential outcomes and the XOR modifications per move.