Raphael Yuster Haifa University Uri Zwick Tel Aviv University Finding cycles using rectangular matrix multiplication and dynamic programming … Raphael Yuster Haifa University Uri Zwick Tel Aviv University

Matrix multiplication  = Authors Complexity folklore n3 Strassen (1969) n2.81 Coppersmith, Winograd (1990) n2.38 ??? n

Rectangular Matrix multiplication b a  = b a a c a b  = c b M(a,b,c) = M(b,c,a) = M(c,a,b) = ...

Rectangular Matrix multiplication b c c  = a a There are also improved specially designed rectangular matrix multiplication algorithms

Chain Matrix Product Let Ai be an ni ni+1 matrix. Compute A1A2…Ak. Matrix multiplication is associative, so all association orders give the same result, but may have vastly different cost. Easy to solve using dynamic programming.

Chain Matrix Product (cont.) Let P(i,j) be the cost of computing Ai…Aj. Complexity of naïve solution is O(k3). For M(a,b,c)=abc, there is a much more complicated O(k log k) algorithm.

Does the graph contain a triangle? Finding triangles A triangle A graph Does the graph contain a triangle? Yes!

Algorithms for finding triangles Let A be the adjacency matrix of a graph G. Then, G contains a triangle iff A2AT≠0. Authors Running time folklore mn n < n2.38 Itai-Rodeh ’78 m3/2 AYZ ’97 m2 /(+1) < m1.41

Finding triangles in O(m2 /(+1)) time [AYZ ’97] Let  be a parameter. . High degree vertices: vertices of degree  . Low degree vertices: vertices of degree < . There are at most 2m/ high degree vertices =

A graph G contains a Ck iff Ak-1AT≠0 ? Finding longer cycles A graph G contains a Ck iff Ak-1AT≠0 ? We want simple cycles!

Color coding [AYZ ’95] Assign each vertex v a random number c(v) from {0,1,...,k-1}. Remove all edges (u,v) for which c(v)≠c(u)+1 (mod k). All cycles of length k in the graph are now simple. If a graph contains a Ck then with a probability of at least k - k it still contains a Ck after this process. An improved version works with probability 2 - O(k). Can be derandomized at a logarithmic cost.

Finding C2k in O(m2-1/k) time [AYZ ’97] Let  be a parameter. . High degree vertices: vertices of degree  . Low degree vertices: vertices of degree < . There are at most 2m/ high degree vertices = Finding paths of length k Finding a cycle passing through a high degree vertex

Algorithms for finding a Ck Authors Running time Monien ’85 mn AYZ ’95 n < n2.38 AYZ ’97 m2-1/k/2 [AYZ ’97]: “We have not been able to use fast matrix multiplication to obtain faster algorithms for finding Ck, for k4, in sparse graphs.”

Finding even cycles even faster [YZ ’97] In undirected graphs, a C2k, for any fixed k, can be found in O(n2) time! We are still working on: Finding odd cycles עוד יותר מהר

Algorithms for finding a C4 Authors Running time Monien ’85 mn AYZ ’95 n < n2.38 AYZ ’97 m3/2 Eisenbrand, Grandoni ’03 m 2-2/ n1/ YZ ’03 m(4-1)/(2+1)<m1.48

Finding C4’s in O(m 2-2/ n1/) time [Eisendbrand, Grandoni ’03] Let  be a parameter. . High degree vertices: vertices of degree  . Low degree vertices: vertices of degree < . There are at most 2m/ high degree vertices

Finding C4’s in O(m 2-2/ n1/) time [EG ’03]

Finding C4’s in O(m(4-1)/(2+1)) time [YZ ’03] Let  be a parameter. High degree vertices:  ≤ deg(v) Medium degree vertices: 1/2 ≤ deg(v) <  Low degree vertices: deg(v) < 1/2 There are at most 2m/ high degree vertices. There are at most 2m/1/2 medium degree vertices.

Finding C4’s in O(m(4-1)/(2+1)) time [YZ ’03]

Where do we go from here? Why just three degree classes? How many cases will we have for C5, C6, …?

The general case Partition the vertices into log n degree classes: Vi={ vV | 2i ≤ deg(v) ≤ 2i+1 } Number of cycle classes is only (log n)k. What is the most efficient way of handling each class?

Handling a given cycle class Choose i and j. Find all paths of length j-i that pass through vertices of degrees di ,…, dj. Find all paths of length k-(j-i) that pass through vertices of degrees dj ,…, di. Check whether the graph contains a cycle from the class.

Finding paths We get a chain matrix product problem! Degree: Size of class: We get a chain matrix product problem! We can use the sparsity of the matrices.

It is convenient to express everything as a power of m. Finding paths (cont.) It is convenient to express everything as a power of m. Degree: Size of class:

Finding cycles Theorem: There is an time algorithm for finding Ck’s in directed graphs.

Finding C5’s Requires a non-constant number of degree classes. Theorem: Requires a non-constant number of degree classes. The worst-case running time is obtained for regular graphs.

If the conjecture is true, then c6<1.65. Finding C6’s Conjecture: If the conjecture is true, then c6<1.65.

The conjecture holds for k=3,5. Finding odd cycles Conjecture: for odd k3. The conjecture holds for k=3,5.

Open problems Is it possible to extend the technique presented to obtain improved algorithms? Find a feasible way for rigorously computing c6, c7, etc. Other applications?