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Exercise 4

Identify the time and space complexity for both the brute-force solution and the optimized solution.

Solution

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 with memoization 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.

The time and space complexities are identical for the bottom-up and the top-down approaches, both with memoization. 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).

Note that we don't use O(N^2) as we are dealing with two separate values here, rows and columns.

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