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Testing Acyclicity of Directed Graphs in Sublinear Time

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Michael Bender - SUNY Stony Brook

Dana Ron - Tel Aviv University

For a fixed property Pand any object O,

determine whether O has property P

or whether O is farfrom having propertyP(i.e., far from any other object having P ).

Task should be performed by querying the object(in as few places as possible).

- Initially defined by Rubinfeld and Sudan in the context of Program Testing (of algebraic functions).
- Goldreich Goldwasser and Ron initiated study of testing properties of (undirected) graphs.
- Body of work deals with properties of functions, graphs, strings, sets of points...

Property Testings is a relaxation of exactly deciding the property P, and can be viewed as approximately deciding the property.

We desire testing algorithms that are (much) more efficient than corresponding exact decision procedures (in particular, sub-linear in size of object).

Useful both when exact decision cannot be done efficiently, (so must perform approximation), and when have relatively fast procedure but want very fast procedure.

A testing algorithm for graph property P is given query access to the tested graph G, and a distance parametere.

If G has property P then the algorithm should accept with probability at least 2/3.

If G ise-far from having property P then the algorithm should reject with probability at least 2/3.

In order that testing be well define, need to determine:

- Representation of graphs (type of queries performed by testing algorithm).
- Notion of distance (what does it mean to be e-far from having the property).

Consider two (standard) representations:

- Adjacency Matrix
- Incidence Lists

v

w

u

v

If

Then M[u,v]=1

otherwise M[u,v]=0

u

1

0

Queries = Probes into matrix

Distance to having property = fraction of entries that should be modified (among all N2 ) so that obtain property. Say thate-far if fraction is greater thane.

Assume graph has bounded degree d

i

If

v

u

i

Then L[u,i]=v

v

u

May also consider having lists of incoming edges.

Queries = Probes into lists.

Distance to having property = fraction of entries that should be modified (among all d N ) so that obtain property. Say thate-far if fraction is greater thane.

Thm1. There is an algorithm for testing acyclicity of directed graphs in the adjacency-matrix model that has query complexity and running time .

Thm2. Testing acyclicity of directed graphs in the incidence-lists representation requires queries (for constant d ande), even when possible to perform queries on incoming edges.

Acyclicity Testing Algorithm

- Uniformly and independently select vertices in the graph.
- Obtain the subgraph induced by selected vertices, by querying the existence of an edge between every pair of vertices selected.
- If the induced subgraph contains a cycle thenreject, otherwise, accept.

If graph Gis acyclic then clearly always accepted.From now on assume G is e-far from acyclic. Will prove that G is rejected with probability at least 2/3.

Proof Idea:1. Show that G contains large subset W s.t. every vertex in W has many outgoing edges to other vertices in W.2. Show that w.h.p. sufficiently large uniform sample from W induces a subgraph containing acycle.Correctness of alg’ follows since w.h.p. sample selected by alg’ contains sufficiently large subsample from W.

Lem1. If G is e-far from acyclic, then there exists a set W of vertices having size at least such that every v in W has at least outgoing edges to W.

Proof: Prove contrapositive statement: assume no such W exists, show that G is e-close to acyclic - define ordering of vertices with few edges violating this ordering.

Order vertices from last to first.At each step let Z denote remaining vertices. As long as , by assumption have v in Zwith at most (e/2)N outgoing edges to Z. Let v be next vertex in (reverse) order. When order Z arbitrarily.

Lem2. Let W be such that every v in W has at least e’|W| outgoing edges to W. Then w.h.p. a sample of size in W contains a cycle.

Proof idea: (Simple) probabilistic argument shows that w.h.p., every vertex v in sample has at least one outgoing edge to another vertex u in sample. That is , there is no sink vertex, implying a cycle.

Proof idea for lower bound of on query complexity: Show that cannot distinguish in fewer queries btwn families -

N1/3

N/2

N2/3

Undirected

Directed

Simple argument:

Thm1:

Adj’

Thm2:

[GR]:

Inc’

What is real complexity of testing acyclicity in incidence-lists model? We conjecture that actually higher than our lower bound.

Is there any sublinear-query algorithm for problem?