Erasures versus errors in local decoding and property testing

Raskhodnikova, Sofya; Ron-Zewi, Noga; Varma, Nithin

doi:10.1002/rsa.21031

Cited by 7 publications

(7 citation statements)

References 84 publications

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“…Theorem 1.6 and Theorem 1.3 together imply that for the property of monotonicity, nonadaptive tolerant testing is strictly harder than nonadaptive ER testing, and also significantly less efficient than adaptive tolerant testing. Our results make progress towards answering the open question raised by Raskhodnikova, Ron-Zewi, and Varma [RRV19] on the existence of natural properties for which one can show a separation between tolerant testing and ER testing in terms of query complexity.…”

Section: Separating Erasure-resilient Testing From Tolerant Testingmentioning

confidence: 75%

“…Our nonadaptive erasure-resilient tester for monotonicity with complexity O(log n) and our lower bound on the query complexity of nonadaptive algorithms for LIS estimation imply that nonadaptive tolerant testing is strictly harder than nonadaptive erasure-resilient testing for the natural property of monotonicity, thereby making progress towards solving an open question raised by Raskhodnikova, Ron-Zewi, and Varma [RRV19].…”

Section: Introductionmentioning

confidence: 88%

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New Sublinear Algorithms and Lower Bounds for LIS Estimation

Newman,

Varma

2020

Preprint

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Estimating the length of the longest increasing subsequence (LIS) in an array is a problem of fundamental importance. Despite the significance of the LIS estimation problem and the amount of attention it has received, there are important aspects of the problem that are not yet fully understood. There are no better lower bounds for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, and also provide novel algorithms that complement our lower bound.Specifically, for every constant ∈ (0, 1), every nonadaptive algorithm that outputs an estimate of the length of the LIS in an array of length n to within an additive error of n has to make log Ω(log(1/ )) n queries. Next, we design nonadaptive LIS estimation algorithms whose complexity decreases as the the number of distinct values, r, in the array decreases. We first present a simple algorithm that makes Õ(r/ 3 ) queries and approximates the length of the LIS with an additive error bounded by n. This algorithm has better complexity than the best previously known adaptive algorithm (Saks and Seshadhri; 2017) for the same problem when r poly log(n). We then use our algorithm to construct a nonadaptive algorithm with query complexity Õ( √ r •poly(1/λ)) that, for an array in which the LIS is of length at least λn, outputs a multiplicative Ω(λ)-approximation to the length of the LIS. These algorithms bridge the gap between the constant-query LIS estimation for Boolean-valued arrays, and the O(poly log n)query (adaptive) algorithm for the general case (Saks and Seshadhri; 2017). Our algorithm also improves upon state of the art nonadaptive algorithm (Rubinstein, Seddighin, Song, and Sun; for LIS estimation (for r = n) in terms of approximation guarantee.Finally, we present a O(log n)-query nonadaptive erasure-resilient tester for monotonicity. Our result implies that lower bounds on erasure-resilient testing of monotonicity does not give good lower bounds for LIS estimation. It also implies that nonadaptive tolerant testing is strictly harder than nonadaptive erasure-resilient testing for the natural property of monotonicity, thereby making progress towards solving an open question (Raskhodnikova, Ron-Zewi, and Varma; 2019).

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Section: Separating Erasure-resilient Testing From Tolerant Testingmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 88%

New Sublinear Algorithms and Lower Bounds for LIS Estimation

Newman,

Varma

2020

Preprint

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“…Similarly to the tolerant testing scenario, PCPPs were also used in [DRTV18] to show that there exists a property of boolean strings of length n that has a tester with query complexity independent of n, but for any constant α > 0, every α-erasure-resilient tester is required to query Ω(n c ) many bits for some c > 0, thereby establishing a separation between the models. Later, in [RRV19] PCPP constructions were used to provide a separation between the erasure-resilient testing model and the tolerant testing model.…”

Section: Our Resultsmentioning

confidence: 99%

“…For the erasure resilient model, in addition to the separation between that model and the standard testing model, [DRTV18] designed efficient erasure-resilient testers for important properties, such as monotonicity and convexity. Shortly after, in [RRV19] a separation between the erasure-resilient testing model and the tolerant testing model was established. The last separation requires an additional construction (outside PCPPs), which remains an obstacle to obtaining better than polynomial separations.…”

Section: Related Workmentioning

confidence: 99%

Hard properties with (very) short PCPPs and their applications

Ben‐Eliezer¹,

Fischer²,

Levi³

et al. 2019

Preprint

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We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ℓ, we construct a property P (ℓ) ⊆ {0, 1} n satisfying the following: Any testing algorithm for P (ℓ) requires Ω(n) many queries, and yet P (ℓ) has a constant query PCPP whose proof size is O(n • log (ℓ) n), where log (ℓ) denotes the ℓ times iterated log function (e.g., log (2) n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n • polylog n).As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ℓ, we construct a property that has a constant-query tester, but requires Ω(n/ log (ℓ) (n)) queries for every tolerant or erasure-resilient tester.

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“…Erasure-resilient sublinear-time algorithms, in the context of testing properties of functions, were first investigated by Dixit et al [DRTV18], and further studied by Raskhodnikova et al [RRV19], Pallavoor et al [PRW20], and Ben-Eliezer et al [BFLR20].…”

Section: Related Workmentioning

confidence: 99%

Erasure-Resilient Sublinear-Time Graph Algorithms

Levi¹,

Pallavoor²,

Raskhodnikova³

et al. 2020

Preprint

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We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. '06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms '08) and Eden et al. (ICALP '17). We conclude with a discussion of our model and open questions raised by our work.

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Erasures versus errors in local decoding and property testing

Cited by 7 publications

References 84 publications

New Sublinear Algorithms and Lower Bounds for LIS Estimation

New Sublinear Algorithms and Lower Bounds for LIS Estimation

Hard properties with (very) short PCPPs and their applications

Erasure-Resilient Sublinear-Time Graph Algorithms

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