1 CSE 417: Algorithms and Computational Complexity Winter 2001 Lecture 13 Instructor: Paul Beame

2 All-pairs shortest paths  If no negative-weight edges and sparse graphs run Dijsktra from each vertex O(nm log n)  What about other cases?

3 Floyd’s Algorithm Idea  Interior vertices in a path  Dijsktra’s Algorithm  at each step always computed shortest paths that only had interior vertices from a set S at each step  Floyd’s Algorithm  slowly add interior vertices on fixed schedule rather than depending on the graph v w interior vertices

4 Floyd’s Algorithm  Vertices V={v 1,...,v n }  Let D k [i,j] = length of shortest path from v i to v j that only allows {v 1,...,v k } as interior vertices  Note  D 0 [i,j] = weight of edge (v i,v j ) if (v i,v j )  E  if (v i,v j )  E  D n [i,j] = length of shortest path from v i to v j

5 Floyd’s Algorithm vivi vertices from v 1,...,v k vivi vivi vertices from v 1,...,v k-1 vivi Computing D k [i,j] Case 1: v k not used D k [i,j] = D k-1 [i,j]

6 Floyd’s Algorithm vjvj vkvk vertices from v 1,...,v k-1 Case 2: v k used vertices from v 1,...,v k-1 vivi

7 Floyd’s Algorithm vjvj vkvk vertices from v 1,...,v k-1 Case 2: v k used vertices from v 1,...,v k-1 vivi not on shortest path since no negative cycles

8 Floyd’s Algorithm vjvj vkvk vertices from v 1,...,v k-1 Case 2: v k used vivi vertices from v 1,...,v k-1 D k [i,j] = D k-1 [i,k] + D k-1 [k,j]

9 Floyd’s Algorithm  Floyd(G) D 0  weight matrix of G for k=1 to n do for i=1 to n do for i=1 to n do D k [i,j]  min{ D k-1 [i,j], D k-1 [i,k]+D k-1 [k,j]} endfor endfor endfor O(n 3 ) time O(n 2 ) storage by maintaining array for last two values of k

10 Multiplying Faster  On the first HW you analyzed our usual algorithm for multiplying numbers   (n 2 ) time  We can do better!  We’ll describe the basic ideas by multiplying polynomials rather than integers  Advantage is we don’t get confused by worrying about carries at first

11 Note on Polynomials  These are just formal sequences of coefficients so when we show something multiplied by x k it just means shifted k places to the left

12 Polynomial Multiplication  Given:  Degree n-1 polynomials P and Q  P = a n-1 x n-1 + a n-2 x n-2 + … + a 2 x 2 + a 1 x + a 0  Q = b n-1 x n-1 + b n-2 x n-2 + … + b 2 x 2 + b 1 x + b 0  Compute:  Degree 2n-2 Polynomial P  Q  P  Q = a n-1 b n-1 x 2n-2 + (a n-1 b n-2 + a n-2 b n-1 ) x 2n (a n-1 b i+1 +a n-2 b i a i+1 b n-1 ) x n+i a 0 b 0  Obvious Algorithm:  Compute all a i b j and collect terms   (n 2 ) time

13 Naive Divide and Conquer  Assume n is a power of 2  P = (a n-1 x n/2-1 + a n-2 x n/2-2 + … + a n/2 ) x n/2 + (a n/2-1 x n/ a 2 x 2 + a 1 x + a 0 ) = P 1 x n/2 + P 0  Q = Q 1 x n/2 + Q 0  P  Q = (P 1 x n/2 +P 0 )(Q 1 x n/2 +Q 0 ) = P 1 Q 1 x n +(P 1 Q 0 +P 0 Q 1 )x n/2 +P 0 Q 0  4 sub-problems of size n/2 + plus linear combining  T(n)=4T(n/2)+cn  Solution T(n) = O(n 2 )

14 Karatsuba’s Algorithm  A better way to compute the terms  Compute  P 0 Q 0  P 1 Q 1  (P 1 +P 0 )(Q 1 +Q 0 ) which is P 1 Q 1 +P 1 Q 0 +P 0 Q 1 +P 0 Q 0  Then  P 0 Q 1 +P 1 Q 0 = (P 1 +P 0 )(Q 1 +Q 0 ) - P 0 Q 0 - P 1 Q 1  3 sub-problems of size n/2 plus O(n) work  T(n) = 3 T(n/2) + cn  T(n) = O(n  ) where  = log 2 3 =