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Finding cycles using rectangular matrix

multiplication and dynamic programming

- Raphael Yuster Haifa Univ. - Oranim
- Uri Zwick Tel Aviv University

Matrix multiplication

?

Authors Complexity

folklore n3

Strassen (1969) n2.81

Coppersmith, Winograd (1990) n2.38

??? n?

Rectangular Matrix multiplication

c

c

b

?

a

a

b

a

c

a

?

b

c

b

M(a,b,c) M(b,c,a) M(c,a,b) ...

Rectangular Matrix multiplication

c

b

c

a

?

a

There are also improved specially designed

rectangular matrix multiplication algorithms

Chain Matrix Product

- Let Ai be an ni? ni1 matrix.
- Compute A1A2Ak.
- 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 AiAj.

Complexity of naïve solution is O(k3). For

M(a,b,c)abc, there is a much more complicated

O(k log k) algorithm.

Finding triangles

A triangle

A graph

Does the graph contain a triangle?

Yes!

Algorithms for finding trianglesLet A be the

adjacency matrix of a graph G. Then, G contains

a triangle iff A2?AT?0.

Authors Running time

folklore mn

folklore n? lt n2.38

Itai-Rodeh 78 m3/2

AYZ 97 m2? /(?1) lt m1.41

Finding triangles in O(m2? /(?1)) time AYZ 97

? m(?-1) /(?1)

Let ? be a parameter.

. High degree vertices vertices of degree

? ?.Low degree vertices vertices of degree lt ?.

There are at most 2m/? high degree vertices

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

? m1/k

Let ? be a parameter.

. High degree vertices vertices of

degree ? ?.Low degree vertices vertices of

degree lt ?.

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? lt n2.38

AYZ 97 m2-1/?k/2?

AYZ 97We 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 onFinding odd cycles ???

???? ???

Algorithms for finding a C4

Authors Running time

Monien 85 mn

AYZ 95 n? lt n2.38

AYZ 97 m3/2

Eisenbrand, Grandoni 03 m 2-2/? n1/?

YZ 03 m(4?-1)/(2?1)ltm1.48

Finding C4s 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 lt ?.

There are at most 2m/? high degree vertices

Finding C4s in O(m 2-2/? n1/?) time EG 03

Finding C4s 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) lt ? Low degree vertices

deg(v) lt ?1/2

There are at most 2m/? high degree

vertices.There are at most 2m/?1/2 medium degree

vertices.

Finding C4s 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) 2i1 - 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

Degree

Sizeof class

We get a chain matrix product problem! We can use

the sparsity of the matrices.

Finding paths (cont.)

It is convenient to express everything as a power

of m.

Degree

Sizeof class

Finding cycles

Theorem There is an time

algorithm for finding Cks

in directed graphs.

Finding C5s

Theorem

Requires a non-constant number of degree

classes. The worst-case running time is obtained

for regular graphs.

Finding C6s

Conjecture

If the conjecture is true, then c6lt1.65.

Finding odd cycles

Conjecture

for odd k?3.

The conjecture holds for k3,5.

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?