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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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 32
Sublinear algorithms
Algorithms to (approximately) solve problems on large data sets usingresources that are sublinear in the size of the input.
Algorithm
4
56 20
Tradeoff between resources and quality of approximation.
Our focus: Property testing
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 2 / 32
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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 3 / 32
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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 3 / 32
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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 3 / 32
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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 3 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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
ε, n
(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
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 4 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 5 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 5 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 5 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 5 / 32
Erasure-resilient property testing
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 6 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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
Reject w.h.p iff is ε-far from P
Motivation: Some function values could beprotected for privacy, or erased by an adversary.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 7 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 8 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 8 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 8 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 8 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 9 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 10 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 10 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
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].
1 2 n
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 11 / 32
Our resultsErasure-resilient testers for monotonicity, Lipschitz property and convexity
For functions f : [n]→ Rα-erasure-resilient testers for monotonicity, Lipschitz property andconvexity.
O( 11−α) factor query complexity overhead for all α ∈ (0, 1).
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
(εd
).
Hard example for our algorithms.
- Our algorithms will not detect violations if α = Ω( ε√d).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 12 / 32
Our resultsErasure-resilient testers for monotonicity, Lipschitz property and convexity
For functions f : [n]→ Rα-erasure-resilient testers for monotonicity, Lipschitz property andconvexity.
O( 11−α) factor query complexity overhead for all α ∈ (0, 1).
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
(εd
).
Hard example for our algorithms.
- Our algorithms will not detect violations if α = Ω( ε√d).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 12 / 32
Our resultsErasure-resilient testers for monotonicity, Lipschitz property and convexity
For functions f : [n]→ Rα-erasure-resilient testers for monotonicity, Lipschitz property andconvexity.
O( 11−α) factor query complexity overhead for all α ∈ (0, 1).
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
(εd
).
Hard example for our algorithms.
- Our algorithms will not detect violations if α = Ω( ε√d).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 12 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 13 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 14 / 32
Parameterized function property testing
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 15 / 32
Property testing[Rubinfeld Sudan 96, Goldreich Goldwasser 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
ε, n
Tester for P
Oracle(f)
x f(x)Accept w.h.p. iff ∈ P
Reject w.h.p iff is ε-far from P
Convention: Complexity measuredin terms of the domain size n.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 16 / 32
Parameterized property testing[Pallavoor Raskhodnikova Varma 17]
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
ε, n, r
(r - Extra parameter of f)
Tester for P
Oracle(f)
x f(x)Accept w.h.p. iff ∈ P
Reject w.h.p iff is ε-far from P
Complexity expressed in terms of r.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 17 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 18 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 18 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 18 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 19 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 20 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 20 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 20 / 32
Ongoing work and future directions
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 21 / 32
Ongoing work
Separating erasures from corruptions.
I Is tolerant testing harder than erasure-resilient testing in general?
Adversarial erasures vs. random erasures
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 22 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 23 / 32
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!
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 24 / 32
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!
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 24 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 25 / 32
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].
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 25 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 26 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 26 / 32
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.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 26 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥
60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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
13p2
⊥60
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 27 / 32
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)
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 28 / 32
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)
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 28 / 32
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)
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 28 / 32
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)
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 28 / 32
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)
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 28 / 32
Query lemma
Claim
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
).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 29 / 32
Query lemma
Claim
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
).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 29 / 32
Query lemma
Claim
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
).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 29 / 32
Query lemma
Claim
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
).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 29 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Expected number of queries to a single levelClaim: E [# of queries to one level of binary search] = O
(1
1−α
).
Level k:
1 2 n
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
1− α.
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 30 / 32
Analysis overview
1 Array is sorted =⇒ Tester accepts.
2 Array is ε-far from sorted =⇒ Detects violation w.p. ≥ ε in aniteration.
I In 2/ε independent iterations:
Pr[tester does not detect a violation] ≤ (1− ε)2/ε ≤ 13 .
Tester rejects with high constant probability.
3 The expected number of queries made is O(
lognε(1−α)
).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 31 / 32
What we showed
Theorem
There exists an α-erasure-resilient ε-tester for sortedness of n-length
arrays that makes O(
lognε(1−α)
)queries for all α, ε ∈ (0, 1).
Nithin M. Varma Advisor: Dr. Sofya RaskhodnikovaFast and fault-resilient sublinear algorithmsNovember 26, 2017 32 / 32