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```									Proximity Oblivious Testing

Oded Goldreich
Weizmann Institute of Science

Joint work with Dana Ron
Property Testing: informal definition

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

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Focus: sub-linear time algorithms – performing the
task by inspecting the object at few locations.
Property Testing: the standard (one-sided error) def’n

A property P = n Pn , where Pn is a set of functions
with domain Dn.
The tester gets explicit input n and ,
• If f  Pn then Prob[Tf(n,) accepts] = 1.
• If f is -far from Pn then Prob[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 in order 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 BLR (linearity) tester.
On input (prox.par.)  and oracle f,
repeat the following test O(1/  ) times:
1. Select uniformly x,y in Dn
2. Accept iff f(x)+f(y)=f(x+y).
Proximity Oblivious Testing: the basic definition

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 ),
• If f  Pn then Prob[Tf(n) accepts] = 1.
• If f  Pn then Prob[Tf(n) rejects] > (P(f)),
where : (0,1] (0,1]            (is the “detection rate”)
and P(f) denotes the distance of f from P.

N.B.: A standard tester is obtained by repeating the POT
(i.e., on prox. par. , repeat O(1/()) times).

Focus: constant query complexity q(n)=q ( « | Dn |)

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 .
These questions are studied mainly in two standard models
of testing graph properties:
(i) the adjacency matrix model and (ii) the bounded-degree model.

Example: the BLR (linearity) tester.
The complexity of the (std.) tester obtained
by repeating the POT equals (up to a constant)
the complexity of the best possible standard tester.
PART 1:      In the adjacency matrix model

A graph G=(V,E) is represented
by a function g:[N][N]{0,1}
(i.e., g(u,v)=1 iff (u,v) is an edge in G).

This (representation) determines:
1.   The type of queries: adjacency queries
2.   The distance measure: #differences/N2
The adjacency matrix model: two simple examples

A graph G=(V,E) is represented
by a function g:[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], and accept iff the
induced subgraph is a biclique (i.e., has an even number of edges).
Example 2: analysis of the 3-query POT
Select s[N] arbitrarily, and random u,v[N], and accept iff the
induced subgraph is a biclique (i.e., has an even number of edges).

Analysis technique:
s                                        s induces a partition,
u and v check it.
(s)

[N] \ (s)
Suppose that the graph is -far
from Biclique. Then

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

induced subgraph            induced subgraph
has 1 or 3 edges             has a single edge
Example 3: triangle-freeness [AFKS, 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 “far from straightforward”.
Example 4: testing bipartiteness

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

THM: Bipartitness has no O(1)-query POT.
PF: A graph can be -far from Bipartiteness still
all its O(1)-vertex induced subgraphs may be bipartite.
E.g., an odd-length super-cycle consisting of (1/√)
(equal-sized) independent sets such that each adjacent
pairs of sets is connected by a complete bipartite graph.

Conclusion: easily testable properties may not have POTs.
Characterization of graph properties having a POT

THM (oversimplified): Property P has an O(1)-query POT
iff P equals the set of F-free graphs, where F is a fixed set of
O(1)-size graphs.
PF idea: Given a POT , we derive a canonical POT (a la [GT]), which
yields a characterization of P in terms of forbidden subgraphs (equiv.,
allowed induced subgraphs). In the other direction, use [AFKS].

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

THM (actual): Property P = N PN has a O(1)-query POT iff for some
constant c and every N, it holds that PN equals the set of FN -free graphs,
where FN is a set of c-size graphs.
Example 5: testing Clique Collection (CC)

Recall that CC is efficiently testable with Õ(1/) queries
[GR], and even Õ(-4/3) non-adaptive queries suffice.

THM: CC has a 3-query POT with ()=O(2),
and no O(1)-query POT can do better.
PF (of the lower bound): Consider a collection of
1/4 balanced bicliques, each of size 4N. This graph is
-far from CC while rejecting it requires hitting some
biclique at least three times.

Conclusion: The (std.) tester obtained by repeating
the best POT may have significantly higher complexity
than the standard tester.
Example 6: testing c-Clique Collection (c-CC)

Recall that c-CC is testable with Õ(1/) queries [GR],

THM: For every c2, the property c-CC has a (c+1)-query
POT with ()=O(c/2), and no O(1)-query POT can do better.
PF (of the lower bound): Consider a graph consisting of c small
cliques, each of size sqrt()N and one large clique of size (1-c√))N.
This graph is -far from c-CC while rejecting it requires hitting each of
the c small cliques.

Conclusion: The (std.) tester obtained by repeating
the best POT may have tremendously higher complexity
than the standard tester.
PART 2:       In the bounded-degree model

A graph G=(V,E) of degree bound d,
is represented by a function g:[N][d][N]{0}
(i.e., g(u,i)=v iff v is the ith neighbor of u in G
and g(u,i)=0 iff v has less than i neighbors).

This (representation) determines:
1.   The type of queries: incidence queries
2.   The distance measure: #differences/dN
The bounded-degree model:
preliminaries to the characterization

Augment the definition of (induced) subgraph freeness by
referring to the non-existence of external edges that are
incident at certain (marked) vertices.
E.g., standard triangle freeness vs no isolated triangles.
unmarked                       marked

DEF (generalized subgraph freeness): The specified graphs
should not appear as induced subgraphs unless some
marked vertex has an external neighbor.
E.g., this can express degree upper bounds.
Generalized subgraph freeness: non-propagation

DEF (generalized subgraph freeness): The specified
graphs should not appear as induced subgraphs unless
some marked vertex has an external neighbor.

Def: F is non-propagating if there exists :(0,1](0,1]
such that if some vertex set B covers all occurrences in G
of graphs in F, then G is (|B|/N)-close to being F-free.

• Not all sets F are non-propagating.
• For any F with no marked vertices, F is non-propagating.
• Degree-regularity is captured by a non-propagating F.
Note that this is a non-hereditary property.
The bounded-degree model: characterization
Def: F is non-propagating if there exists :(0,1](0,1]
such that if some vertex set B covers all occurrences in G
of graphs in F, then G is (|B|/N)-close to being F-free.

• Not all sets F are non-propagating.
• For any F with no marked vertices, F is non-propagating.
• Degree-regularity is captured by a non-propagating F.
Note that this is a non-hereditary property.

THM (over. sim.): A property P has an O(1)-query POT iff for
some non-propagating F it holds that P equals F-freeness.
OPEN: Can every generalized subgraph freeness property
be captured by F-freeness for some non-propagating F ?
Other Models (of property testing)

THM: If property P is testable by a non-adaptive tester
that (i) makes a number of queries that only depends on
the proximity parameter and (ii) rejects based on a
constant-sized “refutation”, then P has a POT.

Note: strong codeword tests (cf. [GS]) correspond to POTs.
OPEN: Do codes of 1/polylog rate have O(1)-query codeword POT?
The codeword tester of [BS]+[D] is not strong.
The End
The slides of this talk are available at
http://www.wisdom.weizmann.ac.il/~oded/T/pot.ppt

The paper itself is available at
http://www.wisdom.weizmann.ac.il/~oded/p_testPOT.html
A companion paper is available at
http://www.wisdom.weizmann.ac.il/~oded/p_testAA.html
On the companion paper “Algorithmic Aspects
of Property Testing in the Dense Graphs Model”

THM [GT]: If a graph property is testable by q(N,) queries then
it is testable by a canonical tester of query complexity O(q(N,)2).
A canonical tester inspects a random induced subgraph and accepted iff the
inspected graph has a predetermined property.

Me (since 2001): “In this model, there is no room for algorithms --
property testing reduces to sheer combinatorics.”
Me (now): A finer examination (which cares for the quadratic blow-up)
reveals the role of algorithms; as shown in the paper, adaptive algorithms
outperform non-adaptive ones, which in turn outperform canonical testers.

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