On Proximity Oblivious Testing

1. On Proximity Oblivious Testing Oded Goldreich - Weizmann Institute of Science Dana Ron – Tel Aviv University

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

3. Property Testing: The standard (one-sided error) definition • A propertyP = nPn, where Pnis 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 complexityq(n,)=q() ( « |Dn| ) Terminology:is called theproximity parameter.

4. 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 testsProximity 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(x+y) then reject. • If any basic test rejects then Reject o.w. Accept.

5. Proximity Oblivious Testing • A property P = nPn ’ 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 Pand : (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

6. 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 possiblestandard 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. POTsappeared (implicitly) mainly for Algebraic Properties (e.g., linearity and low-degree polynomials). Here we focus on Graph Properties(in two standard models).

7. u 1 • Bounded-Degree Graphs Model • (graph is represented bynincidence lists of size d) • Queries: Who is i’th neighbor ofv? • Distance: Fraction of modifications in lists (among dn entries) • Suitable:(Almost)-regularsparse graphs (in particular, constant-degree graphs) 1 2 … d 1 2 … d 1 n 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 n2entries) • 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].

8. This talk 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)).

9. The dense graphs model: Two simple examples Recall: in this model a graph G=(V,E) is represented 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).

10. s (s) [n] \ (s) x 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,uand vcheck it. Suppose that the graph is atdistance  fromBiclique. Then: #edges in same side + #non-edges between sides N2 w.p. over u,v Get:()= induced subgraph has 1 or 3 edges induced subgraphhas1edge

11. Example 3: Triangle-Freeness [Alon,Fischer,Krivelevitch,Szegedy], [Alon] THM:-freeness has a 3-query POT with ()=1/Tower(1/), but noO(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).

12. 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 noO(1)-size subgraph gives evidence Conclusion:easily testable properties may not have POTs.

13. Examples:Clique I2-free, Bi-Clique{,}–free 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= nPnand Pnequals the set of Fn-free graphs.) Proof builds on [Goldreich Trevisan] and [Alon,Fischer,Krivelevitch,Szegedy]. Note: the (detection) function()is not necessarily polynomial, andmay be e.g. a tower.

14. 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 noO(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.

15. 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 (c+1)-query POT with ()=(c/2), and noO(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.

16. 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 significantlyhigher 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…)

17. Thanks