1 / 8

Algorithm Design Methodologies

Algorithm Design Methodologies. Divide & Conquer Dynamic Programming Backtracking. Optimization Problems. Dynamic programming is typically applied to optimization problems In such problems, there are many feasible solutions

gavin-johnson
Download Presentation

Algorithm Design Methodologies

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algorithm Design Methodologies • Divide & Conquer • Dynamic Programming • Backtracking

  2. Optimization Problems • Dynamic programming is typically applied to optimization problems • In such problems, there are many feasible solutions • We wish to find a solution with the optimal (maximum or minimum value) • Examples: Minimum spanning tree, shortest paths

  3. Matrix Chain Multiplication • To multiply two matrices A (p by q) and B (q by r) produces a matrix of dimensions (p by r) and takes p * q * r “simple” scalar multiplications

  4. Matrix Chain Multiplication Given a chain of matrices to multiply: A1 * A2 * A3 * A4 we must decide how we will paranthesize the matrix chain: • (A1*A2)*(A3*A4) • A1 * (A2 * (A3*A4)) • A1 * ((A2*A3) * A4) • (A1 * (A2*A3)) * A4 • ((A1*A2) * A3) * A4

  5. Matrix Chain Multiplication • We define m[i,j] as the minimum number of scalar multiplications needed to compute Ai..j • Thus, the cheapest cost of multiplying the entire chain of n matrices is A [1,n] • If i <> j, we know m[i,j] = m[i,k] + m[k+1,j] + p[i-1]*p[k]*p[j] for some value of k  [i,j)

  6. Elements of Dynamic Programming • Optimal Substructure • Overlapping Subproblems

  7. Optimal Substructure • This means the optimal solution for a problem contains within it optimal solutions for subproblems. • For example, if the optimal solution for the chain A1*A2*…*A6 is ((A1*(A2*A3))*A4)*(A5*A6) then this implies the optimal solution for the subchain A1*A2*….*A4 is ((A1*(A2*A3))*A4)

  8. Overlapping Subproblems • Dynamic programming is appropriate when a recursive solution would revisit the same subproblems over and over • In contrast, a divide and conquer solution is appropriate when new subproblems are produced at each recurrence

More Related