Proximity Oblivious Testing

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Proximity Oblivious Testing

• Oded Goldreich
• Weizmann Institute of Science

Joint work with Dana Ron
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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 whether O is far from having
property P (i.e., far from any other object
having P).
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Property Testing the standard (one-sided error)
defn

• A property P ?n Pn , where Pn is a set of
  functions with domain Dn.
• The tester gets explicit input n and ?,
• and oracle access to a function with domain Dn.
• If f ? Pn then ProbTf(n,?) accepts 1.
• If f is ?-far from Pn then ProbTf(n,?)
  rejects gt 2/3.

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
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
• Select uniformly x,y in Dn
• Accept iff f(x)f(y)f(xy).

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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 ?),
• and oracle access to a function with domain Dn.
• If f ? Pn then ProbTf(n) accepts 1.
• If f ? Pn then ProbTf(n) rejects gt
  ?(?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 )
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Questions addressed in this work

• 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 .
• 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.
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PART 1 In the adjacency matrix model
A graph G(V,E) is represented by a function
gN?N?0,1 (i.e., g(u,v)1 iff (u,v) is
an edge in G).
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The adjacency matrix model two simple examples
A graph G(V,E) is represented by a function
gN?N?0,1.
Example 1 Clique. The property of being a clique
has a trivial two-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).
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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 consider an induced
partition.
s
?(s)
N \ ?(s)
Suppose that the graph is ?-far from Biclique.
Then
edges in same side non-edges between sides
gt ?N2 induced subgraph induced
subgraph has 1 or 3 edges has a
single edge
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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.
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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 A
graph can be ?-far from Bipartiteness still
all its O(1)-vertex induced subgraphs may be
bipartite. E.g., a super-cycle 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.
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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.
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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.
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Example 6 testing c-Clique Collection (c-CC)
Recall that c-CC is testable with Õ(1/?) queries
GR, even non-adaptively!
THM For every c?2, the property c-CC has a
(c1)-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 a large clique
of size (1-sqrt(?))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.
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PART 2 In the bounded-degree model
A graph G(V,E) of degree bound d, is
represented by a function gN?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).
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The bounded-degree model preliminaries to the
characterization

• DEF (generalized subgraph freeness) graphs with
  vertices marked full, semi-full, and partial such
  that a disallowed mapping of F(n,EF) to
  G(N,E) satisfies
• for full vertex v, map(neigh(v))
  neigh(map(v))
• for semi-full vertex v, map(neigh(v))
  neigh(map(v)) ? map(n)
• for partial vertex v, map(neigh(v)) ?
  neigh(map(v))
• E.g., induced (resp., non-induced) graph-freeness
  corresponds to the special case of using only
  semi-full (resp., partial) markings.

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Generalized subgraph freeness non-propagation

• DEF (abbrev.) a disallowed mapping of F(n,
  EF) to G(N,E) satisfies
• for full vertex v, map(neigh(v))
  neigh(map(v))
• for semi-full vertex v, map(neigh(v))
  neigh(map(v)) ? map(n)
• for partial vertex v, map(neigh(v)) ?
  neigh(map(v)).

Def F is non-propagating if there exists
?(0,1?(0,1 such that if every mapping of every
marked graph in F to the graph G uses a vertex in
B, then G is ?(B/N)-close to being F-free.

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

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The bounded-degree model characterization
Def F is non-propagating if there exists
?(0,1?(0,1 such that if every mapping of every
marked graph in F to the graph G uses a vertex in
B, then G is ?(B/N)-close to being F-free.

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

THM (ov. 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 ?
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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 witness, 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?
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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.h
  tml
• A companion paper is available at
  http//www.wisdom.weizmann.ac.il/oded/p_testAA.ht
  ml

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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.