Nithin M. Varma Advisor: Dr. Sofya Raskhodnikova November ... · Nithin M. Varma Advisor: Dr. Sofya...

Fast and fault-resilient sublinear algorithms

Nithin M. Varma

Advisor: Dr. Sofya Raskhodnikova

November 26, 2017

Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 1 / 32

Sublinear algorithms

Algorithms to (approximately) solve problems on large data sets usingresources that are sublinear in the size of the input.

Algorithm

456 20

Tradeoff between resources and quality of approximation.

Our focus: Property testing

Algorithm

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Algorithm

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Algorithm

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Algorithm

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Talk outline

Brief overview of property testing

Erasure-resilient property testing

Joint work with Kashyap Dixit, Sofya Raskhodnikova, and AbhradeepThakurta. Published in the proceedings of ICALP 2016.

Under review at SIAM Journal of Computing.

Parameterized property testing of functions

Joint work with Ramesh Krishnan S. Pallavoor and SofyaRaskhodnikova. Published in the proceedings of ITCS 2017.

Accepted to ACM Transactions on Computation Theory.

Current and future directions

Talk outline

Property testing[Rubinfeld and Sudan ‘96, Goldreich, Goldwasser and Ron ‘98]

Decide w.p. ≥ 2/3 if a function f satisfies a property P or is ε-far fromP.

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

(n: Size of the domain)

Tester for P

Oracle(f)

x f(x)Accept w.h.p. iff ∈ P

Reject w.h.p iff is ε-far from P

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

x f(x)

Accept w.h.p. iff ∈ P

-3 -2 -1 0 2 7 12 19 27

Green square

12 -far from

green square

Tester for P

Oracle(f)

Properties that we study

1 Monotonicity, Lipschitz property and convexity of real-valuedfunctions.

2 Widely studied in property testing[GGLRS00, DDGLRRS99, EKKRV00, FLNRRS02, PRR03, F04, PRR06, BGJRW09, BCGM12,

BBM12, CS13a, CS13b, JR13, CST14, BerRY14, BlaRY14, CDST15, KMS15, CDJS15, BB16].

A f : [n]→ R is

monotone if x < y =⇒ f(x) ≤ f(y) for all x, y ∈ [n].

c-Lipschitz if |f(x)− f(y)| ≤ c · |x− y| for all x, y ∈ [n].

convex if x < y < z =⇒ f(y)−f(x)y−x ≤ f(z)−f(y)

z−y for all x, y, z ∈ [n].

A f : [n]→ R is

Erasure-resilient property testing[Dixit Raskhodnikova Thakurta Varma 16]

Decide w.p. 2/3 if a function f with ≤ α fraction erased pointssatisfies a property P or is ε-far from P (w.r.t nonerased points).

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

ε, α, n Erasure-resilienttester for P

Oracle(f)

x f(x) or ⊥ Accept w.h.p.if f ∈ P

Motivation: Some function values could beprotected for privacy, or erased by an adversary.

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

x f(x) or ⊥

Accept w.h.p.if f ∈ P

-3 -2 -1 0 2 7 12 19 27⊥ ⊥ ⊥

Green square

12 -far from

green square

Oracle(f)

Features of our model

Function values could be erased adversarially (worst case).

Tester does not know the locations of erasures in advance.

Falls in between standard property testing and tolerant propertytesting [Parnas, Ron and Rubinfeld 06].

I Standard testing : All function values are present and correct.

I Tolerant testing : Some function values are adversarially corrupted.

I Erasure-resilient testing : Some function values are adversariallyerased.

Overview of our results

Black box transformation of some ‘simple’ standard testers toefficient erasure-resilient testers.

Efficient erasure-resilient testers in cases where our transformationdoes not apply.

Existence of a property that has an ‘efficient’ standard tester andno ‘efficient’ erasure-resilient tester.

Our results: A black box transformation

Makes uniform testers for extendable properties α-erasure-resilient.

Uses the original testers as black box.

O( 11−α) factor query complexity overhead for all α ∈ (0, 1).

Applies to:I Monotonicity over general partial orders

[Fischer Lehman Newman Raskhodnikova Rubinfeld Samorodnitsky 02].I Convexity of black and white images

[Berman Murzabulatov Raskhodnikova 15].I Boolean functions having at most k alternations in values.

Our results: A black box transformation

Makes uniform testers for extendable properties α-erasure-resilient.

Uses the original testers as black box.

Applies to:I Monotonicity over general partial orders

[Fischer Lehman Newman Raskhodnikova Rubinfeld Samorodnitsky 02].I Convexity of black and white images

[Berman Murzabulatov Raskhodnikova 15].I Boolean functions having at most k alternations in values.

A limitation of our black box transformationExample: Testing sortedness of n-length arrays

Every uniform tester requires Ω(√n) queries.

Better (optimal) tester that makes O(log n) queries [Ergun Kannan

Kumar Rubinfeld Vishwanathan 00].

Random search index

⊥ 4 5 · · · · 1020····1514

I Perform a binary search on [1..n] for a random search index.I Query the points along the search path.I Reject if there are array values that violate sortedness.

Just one erasure breaks this tester.

Some of the known optimal testers for monotonicity, Lipschitz propertyand convexity also break on a constant number of erasures.

Random search index

⊥ 4 5 · · · · 1020····1514

Random search index

⊥ 4 5 · · · · 1020····1514

Random search index

⊥ 4 5 · · · · 1020····1514

Random search index

⊥ 4 5 · · · · 1020····1514

Random search index

⊥ 4 5 · · · · 1020····1514

Our resultsErasure-resilient testers for monotonicity, Lipschitz property and convexity

For functions f : [n]→ Rα-erasure-resilient testers for monotonicity, Lipschitz property andconvexity.

For functions f : [n]d → R

α-erasure-resilient testers for monotonicity and Lipschitz property.

O( 11−α) factor query complexity overhead for α = O

Hard example for our algorithms.

- Our algorithms will not detect violations if α = Ω( ε√d).

Our resultsSeparation of erasure-resilient testing from standard testing

There exists a property P and a constant c > 0 such that

standard testing of P with O(1ε

)queries

α-erasure-resilient testing of P needs Ω(nc) queries ∀ constant α.

Erasure-resilient testing is much harder than standard testing ingeneral.

Summary of our contributions

Definition of a model of property testing that accounts for erasuresin the input data.

A black box transformation from some simple property testers toerasure-resilient testers.

Efficient erasure-resilient testers for monotonicity, Lipschitzproperty and convexity.

A strong separation between testing in the presence of erasuresand testing in the absence of erasures.

Parameterized function property testing

Property testing[Rubinfeld Sudan 96, Goldreich Goldwasser Ron 98]

-3 -2 -1 0 2 7 12 19 27

Tester for P

Oracle(f)

Convention: Complexity measuredin terms of the domain size n.

Parameterized property testing[Pallavoor Raskhodnikova Varma 17]

-3 -2 -1 0 2 7 12 19 27

ε, n, r

(r - Extra parameter of f)

Tester for P

Oracle(f)

Complexity expressed in terms of r.

Motivation for parameterization

Find input parameters tailored to the combinatorics of specificproblems.

Worst-case lower bounds in terms of n may not capture thefine-grained complexity of the problem.

I Sortedness testing of real-valued arrays requires Ω(log n) queries[Fischer 04].

I Sortedness testing of Boolean arrays needs only Θ(1) queries.

Parameters tailored to problems

Image diameter: The image diameter of a function f : D → R ismaxx,y∈D |f(y)− f(x)|.

I Studied for testing Lipschitz property [Jha Raskhodnikova 13].

Image size: The image size of a function f is the number ofdistinct values f takes.

I Suitable for sortedness, monotonicity, and convexity [Pallavoor

Raskhodnikova Varma 17].

Our results

Monotonicity testing

For functions f : [n] 7→ R with image size at most r

Uniform testing with Θ(√r) queries.

I Θ(√n) queries in the standard model [Fischer Lehman Newman

Raskhodnikova Rubinfeld Samorodnitsky 02, Goldreich Goldwasser Lehman Raskhodnikova

Samorodnitsky 00].

Adaptive ε-testing with O( log rε ) queries.

I Θ(log n) queries in the standard model [Ergun Kannan Kumar Rubinfeld

Vishwanathan 00, Fischer 04].

For functions f : [n]d 7→ R with image size at most r

Adaptive ε-testing with O(d log rε ) queries.

I Θ(d log n) queries in the standard model [Chakrabarty Seshadhri 13].

Our results

Samorodnitsky 00].

Our results

Samorodnitsky 00].

Ongoing work and future directions

Ongoing work

Separating erasures from corruptions.

I Is tolerant testing harder than erasure-resilient testing in general?

Adversarial erasures vs. random erasures

Directions for future work

Erasure-resilient testers for other properties.I Linearity, dictatorship, linear threshold functions, nonmonotone

graph properties.

Better erasure-resilient testers for monotonicity and Lipschitzproperty of functions f : [n]d 7→ R.

Fault-resilience in other models of sublinear algorithmsI Local reconstructors for monotonicity and Lipschitz property in the

presence of erasures/corruptions.

Testing clustering of points in Rd with respect to Lp distances.

Other current and past work

Predicting the quality of genome assembly (current)

Joint work with Paul Medvedev, Sofya Raskhodnikova, and Kristoffer Sahlin.

Readability of bipartite graphs

Manuscript.

Joint work with Rayan Chikhi, Vladan Jovicic, Stefan Kratsch, Paul

Medvedev, Martin Milanic and Sofya Raskhodnikova.

Pairwise additive spanners and approximate D-preservers

Published in Siam Journal on Discrete Mathematics. A preliminary versionappeared in the proceedings of ICALP 2013.

Joint work with Kavitha Telikepalli. Done while at TIFR, India.

Thank you!

Other current and past work

Predicting the quality of genome assembly (current)

Joint work with Paul Medvedev, Sofya Raskhodnikova, and Kristoffer Sahlin.

Readability of bipartite graphs

Manuscript.

Joint work with Rayan Chikhi, Vladan Jovicic, Stefan Kratsch, Paul

Medvedev, Martin Milanic and Sofya Raskhodnikova.

Pairwise additive spanners and approximate D-preservers

Published in Siam Journal on Discrete Mathematics. A preliminary versionappeared in the proceedings of ICALP 2013.

Joint work with Kavitha Telikepalli. Done while at TIFR, India.

Thank you!

Erasure-resilient monotonicity tester

Theorem

There exists an α-erasure-resilient ε-tester for monotonicity of

functions f : [n]→ R that makes O(

lognε(1−α)

)queries for all α, ε ∈ (0, 1).

ε-testing of monotonicity using Θ( lognε ) queries [Ergun Kannan Kumar

Rubinfeld Vishwanathan 00, Fischer 04].

Erasure-resilient monotonicity tester

Theorem

There exists an α-erasure-resilient ε-tester for monotonicity of

functions f : [n]→ R that makes O(

lognε(1−α)

ε-testing of monotonicity using Θ( lognε ) queries [Ergun Kannan Kumar

Rubinfeld Vishwanathan 00, Fischer 04].

Monotonicity

A partially erased function f : [n]→ R is monotone, if for all x, y ∈ [n]that are nonerased

x < y =⇒ f(x) ≤ f(y).

𝑥 𝑦

Violation: Two nonerased points x, y ∈ [n] such that x < y andf(x) > f(y).

Monotonicity of functions over [n] ≡ Sortedness of n-length arrays.

Monotonicity

x < y =⇒ f(x) ≤ f(y).

𝑥 𝑦

Monotonicity

x < y =⇒ f(x) ≤ f(y).

𝑥 𝑦

Our erasure-resilient sortedness tester

Input: ε, α ∈ (0, 1); query access to array

Repeat Θ(1/ε) times:1 Sample until you get a nonerased search point t.2 Binary search for t with “uniform” nonerased ‘split points’.3 Reject if there are violations along the search path.

Accept if no violations were found.

⊥⊥ 20t

⊥ 91p1

⊥⊥ 20t

⊥ 91p1

Repeat Θ(1/ε) times:

1 Sample until you get a nonerased search point t.2 Binary search for t with “uniform” nonerased ‘split points’.3 Reject if there are violations along the search path.

⊥⊥ 20t

⊥ 91p1

Repeat Θ(1/ε) times:

1 Sample until you get a nonerased search point t.2 Binary search for t with “uniform” nonerased ‘split points’.3 Reject if there are violations along the search path.

⊥⊥ 20t

⊥ 91p1

Repeat Θ(1/ε) times:1 Sample until you get a nonerased search point t.

2 Binary search for t with “uniform” nonerased ‘split points’.3 Reject if there are violations along the search path.

⊥⊥ 20t

⊥ 91p1

⊥ 20t

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥ 20t

⊥ 91p1

Repeat Θ(1/ε) times:1 Sample until you get a nonerased search point t.2 Binary search for t with “uniform” nonerased ‘split points’.

3 Reject if there are violations along the search path.

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

⊥⊥

⊥ 91p1

Main steps of analysis

1 Array is sorted =⇒ Tester accepts.

2 Array is ε-far from sorted =⇒ Detects violation w.p. ≥ ε in aniteration.

(Witness lemma)

I In Θ(1/ε) independent iterations:

Tester detects a violation with high constant probability.

3 The expected number of queries made is O(

lognε(1−α)

(Query lemma)

(Witness lemma)I In Θ(1/ε) independent iterations:

lognε(1−α)

(Query lemma)

(Witness lemma)

lognε(1−α)

(Query lemma)

(Witness lemma)

lognε(1−α)

(Query lemma)

2 Array is ε-far from sorted =⇒ Detects violation w.p. ≥ ε in aniteration. (Witness lemma)

lognε(1−α)

(Query lemma)

Query lemma

The expected number of queries in one iteration is O(logn1−α

1 Tester traverses a uniformly random search path in a randombinary search tree.

2 The number of levels in a random binary search is O(log n) w.h.p.

3 Expected # of queries to one level of binary search is O(

11−α

Query lemma

11−α

Query lemma

11−α

Query lemma

11−α

Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O

1−α

Level k:

Interval IαI = fraction of erasures in I

Pr [ search point is in I] =# nonerased points in I

Total # nonerased points

≤ |I|(1− αI)n(1− α)

E [# queries to I] =1

1− αI.

∴ E [# of queries to level k] ≤∑

intervals I in level k

|I|n(1− α)

1− α.

1−α

Level k:

≤ |I|(1− αI)n(1− α)

1− αI.

|I|n(1− α)

1− α.

1−α

Level k:

≤ |I|(1− αI)n(1− α)

1− αI.

|I|n(1− α)

1− α.

1−α

Level k:

≤ |I|(1− αI)n(1− α)

1− αI.

|I|n(1− α)

1− α.

1−α

Level k:

≤ |I|(1− αI)n(1− α)

1− αI.

|I|n(1− α)

1− α.

1−α

Level k:

≤ |I|(1− αI)n(1− α)

1− αI.

|I|n(1− α)

1− α.

Analysis overview

I In 2/ε independent iterations:

Pr[tester does not detect a violation] ≤ (1− ε)2/ε ≤ 13 .

Tester rejects with high constant probability.

lognε(1−α)

What we showed

Theorem

There exists an α-erasure-resilient ε-tester for sortedness of n-length

arrays that makes O(

lognε(1−α)

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