On Proximity Oblivious Testing

Oded Goldreich - Weizmann Institute of ScienceDana Ron – Tel Aviv University

Property Testing: informal definition

A relaxation of decision problems:For a fixed property P and any object O,determine whether O has property Por is far from having property P (i.e., O is far from any other object having P).

Focus: sub-linear time algorithms – performing the task by inspecting the object at few locations.

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Property Testing: The standard (one-sided error) definition

A property P = n Pn , where Pn is a set of functions with domain Dn.

The (standard) tester gets explicit input n and , and oracle access to a function with domain Dn.• If f Pn then Pr[Tf(n,) accepts] = 1.

• If f is -far from Pn then Pr[Tf(n,) rejects] > 2/3. (Distance is defined as fraction of disagreements.)Focus: query complexity q(n,)=q() ( « |

Dn| )

Terminology: is called the proximity parameter.

How does a tester use the proximity parameter

Some testers use the proximity parameter merely to determine the number of times that a basic test is performed, where the basic test is oblivious of the proximity parameter.

We call such basic tests Proximity Oblivious Testers.

Example: the [Blum,Luby,Rubinfeld] (BLR) linearity tester

On input n, (and access to f),repeat the following basic test (1/) times:

1. Select uniformly x,y in Dn

2. If f(x) + f(y) f(x+y) then reject.If any basic test rejects then Reject o.w. Accept.

Proximity Oblivious Testing

A property P = n Pn ’ where Pn is a set of functions with domain Dn.A P.O. Tester (POT) gets explicit input n (but not ),and oracle access to a function f with domain Dn.• If f Pn then Pr[Tf(n) accepts] = 1.• If f Pn then Pr[Tf(n) rejects] (P(f)), where P(f) denotes the distance of f from P and : (0,1] (0,1] is the “detection rate”

Focus: constant query complexity q(n)=q

Note: A standard tester can be obtained by repeating the POT (i.e., on prox. par. , repeat (1/()) times).

Questions Concerning POTs

1. Which “testable” properties have POTs?2. How does the complexity of the standard

tester obtained by repeating the POT compare to the complexity of the best possible standard tester?

Motivational discussion: Property testing relates local views to global properties - POTs take this to an extreme (how does constant-size view relate to distance to property).Study of this subclass of testers (those obtained from POTs) may shed light on property testing at large.POTs appeared (implicitly) mainly for Algebraic Properties (e.g., linearity and low-degree polynomials). Here we focus on Graph Properties (in two standard models).

Models used for Testing Graph Properties

1 2 … d

1 2 … d

1

n

Bounded-Degree Graphs Model(graph is represented by n incidence lists of size d)• Queries: Who is i’th neighbor of v?• Distance: Fraction of modifications in lists (among dn entries)• Suitable: (Almost)-regular sparse graphs (in particular, constant-degree graphs)

Dense Graphs Model(graph is represented by n x n adjacency matrix)• Queries: Is (u,v) E ? • Distance: Fraction of matrix modifications (among n2 entries)• Suitable: Dense graphs

v

1u

G=(V,E) is represented by a function fG :[n][n]{0,1}.

G=(V,E) is represented by a function fG :[n][d][n].

Our Results

Dense graphs model:

- Give constant-query POTs for several natural graph properties and prove matching lower bounds.

- Give example of natural property where there is no constant-query POT.

- Characterize class of graph properties that have constant-query POTs: show that equal properties that correspond to induced subgraph freeness. (Note: quite restricted compared to standard testers as characterized by [Alon, Fischer, Newman, Shapire](Bounded-degree graphs model:

- Characterize class of graph properties that have constant-query POTs: show that equal properties that correspond to certain generalized notion of subgraph freeness (includes induces/non-induces subgraph freeness, but also degree regularity (non-hereditary)).

This talk

The dense graphs model: Two simple examples

Recall: in this model a graph G=(V,E) isrepresented by a function fG:

[n][n]{0,1}.

Example 1: Clique. The property of being a clique has a trivial single-query POT with ()=.

Example 2: BiClique. The property of being a biclique has a three-query POT with ()=.

Select s[n] arbitrarily, and random u,v[n]. Accept iff the induced subgraph is a biclique (i.e., has an even number of edges).

Example 2 continuedPOT: Select s[n] arbitrarily, and random u,v[n].Accept iff the induced subgraph is a biclique (i.e., has an even number of edges).

Analysis technique:

s induces a partition,u and v check it.

#edges in same side + #non-edges between sides N2

Suppose that the graph is atdistance from Biclique. Then:

s

(s)[n] \ (s)

induced subgraph has 1 or 3 edges

induced subgraphhas 1 edge

x

w.p. over u,v Get:

()=

Example 3: Triangle-Freeness [Alon,Fischer,Krivelevitch,Szegedy], [Alon]

THM: -freeness has a 3-query POT with ()=1/Tower(1/), but no O(1)-query POT with ()=poly().

The point is that being -far from -freeness means that n2 edges must be omitted to obtain a -free graph, but this does not mean that the graph has n3 (nor poly()n3 ) triangles.

Conclusion: easy testability and POT-ness are not straightforward (what seems easy is not necessarily so).

Example 4: testing bipartiteness

Thm: Bipartitness has no O(1)-query POT.

Recall that Bipartitness is efficiently testable with poly(1/) queries.

Conclusion: easily testable properties may not have POTs.

Pf: Consider an odd-length super-cycle consisting of (1/1/2) (equal-sized) independent sets, with complete bipartite graphs between each adjacent pair. The graph is -far from bipartite, but no O(1)-size subgraph gives evidence

Characterization of graph properties that have a POT

Thm: Property P has an O(1)-query POT iff P equals the set of F-free graphs for some F that is a fixed set of O(1)-size graphs. (To be precise, P= n Pn and Pn equals the set of Fn-free graphs.)

Defn: For a graph G and a set of graphs F, we say that G is F-free if no induced subgraph of G belongs to F.

Proof builds on [Goldreich Trevisan] and [Alon,Fischer,Krivelevitch,Szegedy].Examples: Clique I2-free,

Bi-Clique { , } –freeNote: the (detection) function () is not necessarily polynomial, and may be e.g. a tower.

Example 5: testing Clique Collection (CC)

Thm: CC has a 3-query POT with ()=(2), and no O(1)-query POT can do better.

A graph G belongs to CC if it consists of a union of cliques (of any number and size).

CC is efficiently testable with Õ(1/) queries (by a (std.) adaptive tester) and even Õ(-4/3) non-adaptive queries suffice [GR].

Conclusion: The (std.) tester obtained by repeating the best POT may have significantly higher complexity than the best standard tester.

Example 6: Testing c-Clique Collection (c-CC)

Thm: For every c2, the property c-CC has a (c+1)-query POT with ()=(c/2), and no O(1)-query POT can do better.

A graph G belongs to c-CC if it consists of a union of c cliques (of any size), for a constant c.

c-CC is efficiently testable with Õ(1/) queries (by a (std.) non-adaptive tester) [GR].

Conclusion: The (std.) tester obtained by repeating the best POT may have tremendously higher complexity than the best standard tester.

Summary and Open Problems Initiate study of Proximity Oblivious Testers in context of graph properties.

Give positive and negative results in two standard models of testing graph properties, and in particular provide characterization in each model.

Several conclusions in dense graphs model: - Easy testability and POT-ness are not straightforward (what seems easy is not necessarily so). - Easily testable properties may not have POTs. - The (std.) tester obtained by repeating the best POT may have significantly higher complexity than the best standard tester.

In dense graphs model: for what sets F does F-freeness have poly() detection probability? (For single graphs F have answer in [Alon&Shapire] ).

In bounded-degree model: issue of “propogation” (Teaser…)

Thanks