Dynamic Programming

What are the two properties need for a dynamic programming solution to be efficient?

Optimal substructure property Overlapping Sub-problems property

What are the 7 matrices defined in Strassens algorithm?

P1 = A11(B12 - B22) P2 = (A11 + A12)B22 P3 = (A21 + A22) B11 P4 = A22(B21 - B11) P5 = (A11 + A22)(B11 + B22) P6 = (A12 - A22)(B21 + B22) P7 = (A11 - A21)(B11 + B12)

What is dynamic programming?

Stores the solutions to smaller instances of a problem in an array or table so that they can be easily accessed when solving a bigger problem. Useful when the solution to a given problem uses the same sub-problems multiple times

What is tabulation?

A bottom-up strategy, start with the base-case then build our solution up step by step until we have calculated the main problem

What is memoization?

A top-down strategy, we start from the desired problem and recursively work backwards, filling in the smaller problems as needed

What is Strassen's algorithm?

A trick that allows a 4x4 matrix C, calculated by the product of two 4x4 matrices A and B, to be calculating using 7 matrix products instead of 8

What is Karatsuba multiplication?

Karatsuba reduces the cost of multiplication by converting it into sum of numbers. Karatsuba recursively breaks down two integers into 4 integers, bottom and upper. Then, karatsuba calls itself again for the product of the two upper and bottom integers, A and B. Karatsuba then calls itself for the product of (upper 1 + upper 2) + (bottom1 + bottom2), C. With the three products, we then add A

What is a sub-sequence of a string?

Let u and v be strings. The u is a sub-sequence of v if one can obtain u by deleting zero or more symbols from v while keeping the order of the remaining symbols the same

Which problems can dynamic programming be applied to?

To optimization problems, this strategy can succeed where greedy algorithms fail