On Proximity Oblivious Testing

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On Proximity Oblivious Testing

• Oded Goldreich - Weizmann Institute of Science
• Dana Ron Tel Aviv University

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Property Testing informal definition
A relaxation of decision problems 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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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 PrTf(n,?) accepts 1.
• If f is ?-far from Pn then PrTf(n,?) rejects
  gt 2/3. (Distance is defined as fraction of
  disagreements.)

Focus query complexity q(n,?)q(?) ( Dn )
Terminology ? is called the proximity parameter.
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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
• Select uniformly x,y in Dn
• If f(x) f(y) ? f(xy) then reject.
• If any basic test rejects then Reject o.w. Accept.

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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 PrTf(n) accepts 1.
• If f ? Pn then PrTf(n) rejects ? ?(?P(f)),
  
• where ?P(f) denotes the distance of f from P
  and ? (0,1? (0,1 is the detection rate

Note A standard tester can be obtained by
repeating the POT (i.e., on prox. par. ?, repeat
?(1/?(?)) times).
Focus constant query complexity q(n)q
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Questions Concerning POTs

• Which testable properties have POTs?
• 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).
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Models used for Testing Graph Properties

• 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
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.
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Our Results
This talk
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)).
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The dense graphs model Two simple examples
Recall in this model a graph G(V,E)
is represented by a function fGn?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).
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Example 2 continued
POT 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.
Suppose that the graph is atdistance ? from
Biclique. Then
edges in same side non-edges between sides
? ?N2
w.p. ? ? over u,v
Get?(?)?
induced subgraph has 1 or 3 edges
induced subgraphhas 1 edge
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Example 3 Triangle-Freeness Alon,Fischer,Krivel
evitch,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).
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Example 4 testing bipartiteness
Recall that Bipartitness is efficiently testable
with poly(1/?) queries.
Thm Bipartitness has no O(1)-query POT.
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
Conclusion easily testable properties may not
have POTs.
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Characterization of graph properties that have a
POT
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.
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.)
Proof builds on Goldreich Trevisan and
Alon,Fischer,Krivelevitch,Szegedy.
Note the (detection) function ?(?) is not
necessarily polynomial, and may be e.g. a tower.
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Example 5 testing Clique Collection (CC)
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.
Thm CC has a 3-query POT with ?(?)?(?2), and
no O(1)-query POT can do better.
Conclusion The (std.) tester obtained by
repeating the best POT may have significantly
higher complexity than the best standard tester.
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Example 6 Testing c-Clique Collection (c-CC)
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.
Thm For every c?2, the property c-CC has a
(c1)-query POT with ?(?)?(?c/2), and no
O(1)-query POT can do better.
Conclusion The (std.) tester obtained by
repeating the best POT may have tremendously
higher complexity than the best standard tester.
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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
  AlonShapire ).
• In bounded-degree model issue of propogation
  (Teaser)

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Thanks