1. Intro to Computer Algorithms Lecture 17 Phillip G. Bradford Computer Science University of Alabama

2. Announcements • Advisory Board’s Industrial Talk Series • http://www.cs.ua.edu/9IndustrialSeries.shtm • 4-Nov: Bill Engelke, Mercedes-Benz, USA. • Next Research Colloquia: • Prof. Nael Abu-Ghazaleh • 10-Nov @ 11:00am • “Active Routing in Mobile Ad Hoc Networks”

3. Computer Security Research Group • Meets every Friday from 11:00 to 12:00 • In 112 Houser • Computer Security, etc. • Email me to be on the mailing list!

4. CS Story Time • Prof. Jones’ research group • See http://cs.ua.edu/StoryHourSlide.pdf

5. Next Midterm • Tuesday before Thanksgiving ! • 25-November

6. Outline

7. Dynamic Programming • Principle of optimality • Consider any well-formed substructure s of an optimal structure S • Then s is optimal itself. • This allows the combination of well-formed substructures into optimal superstructures

8. Dynamic ProgrammingComputing Binomial Coefficients • For i1 to n do • For j 0 to min(i,k) do • If j=0 or j=k, then • C[i,j]  1 • Else • C[i,j]  C[i-1,j-1] + C[i-1,j] • Endif • Endfor • Endfor • Return C[n,k]

9. What does this cost? • O(nk) • How about (corrected) exercise #6 in the book? Measure rec-calls instead • Omega(nk) for k constant, n varying • For general analyses, Beware of memoization! • Analyses may not reflect running times

10. Warshall’s and Floyd’s Algs. • Adjacency Matrices Columns (to) 1 n 2 1 2 3 Rows (from) n

11. Transitive Closure • When We work with directed graphs • Call them digraphs • Transitive closure T of a digraph is an nxn boolean matrix: • T[i,j]=1 iff there is a non-trivial path between nodes i and j. • T[i,j]=0 iff there is No path between nodes i and j (and i != j).

12. Transitive Closure • All Pairs Shortest Path Problems • Different from Single Source Shortest Paths

13. How to get a Transitive Closure? • How about DFS and BFS? • What about the principle of optimality? • The matrix R(k) is all paths with up to k intermediary nodes • R(0) is the adjacency matrix • R(1) is the adjacency matrix “along with” any paths that are extended by one using node 1. • OR…

14. Transitive Closure via Warshall • R(2) is all shortest paths using nodes 1 and 2 as intermediaries • R(3) is all shortest paths using nodes 1,2, and 3 as intermediaries • … • R(n) is all shortest paths using nodes 1,2,…,n as intermediaries

15. Basis for Warshall’s Algorithm • R(k)i,j = • = R(k-1)i,j OR [ R(k-1)i,k AND R(k-1)k,j ] • If R(k-1)i,j = 1, then R(k)i,j = 1 • Say R(k-1)i,j = 0, so R(k)i,j = 1 • iff R(k-1)i,k = 1 and R(k-1)k,j = 1 • Why?

16. Warshall’s Algorithm • R(0)  Adjacency Matrix • For k  1 to n do • For i 1 to n do • For j  1 do n do • R(k)i,j R(k-1)i,j OR [ R(k-1)i,k AND R(k-1)k,j ] • Endfor • Endfor • Endfor • Return R(n)

17. Floyd’s Algorithm • Weighted (non-directed) graph • No cycles of negative length • Looks like (Min,+) “matrix multiplication”

18. Floyd’s Algorithm • DWeighted Adj. Matrix, augmented by Infinity • For k  1 to n do • For i  1 do n do • For j  1 to n do • D[i,j]  min{ D[i,j], D[i,k] + D[k,j] } • Endfor • Endfor • Endfor • Return D