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Exercise 6

Identify the time and space complexity of the "Chaos in the Grid with Cats" solutions, both bottom-up and top-down.

Solution

The time and space complexities in this problem are the same as in the previous one, "Chaos in the Grid".

The brute force recursive approach's time complexity without memoization is exponential, O(2^(R + C)), as we have to explore overlapping subproblems many times. R stands for the number of rows, and `C stands for the number of columns.

The space complexity for this approach is O(R + C). This is because the maximum depth of the recursion stack occurs when the recursion follows a path from the top-left to the bottom-right corner, which is R + C - 1 deep.

For the bottom-up approach and the top-down approach with memoization, the time and space complexities are identical. Every grid cell needs to be calculated once, and we are filling a cache object with the same number of values, so both the time and space complexities are O(R * C).

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