arXiv subjects

Sankeerth Rao

Publications and source records attributed to Sankeerth Rao.

5 recordsLinked to original sources

arXiv2020

Probabilistic Existence of Large Sets of Designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t$-$(n,k,λ)$ combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a $t$-$(n,k,λ)$ design exist but, in fact, with positive probability there exists a large set of such designs -- that is, a partition of the set of $k$-subsets of $[n]$ into $t$-designs $t$-$(n,k,λ)$ designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large $n$, large sets of $t$-$(n,k,λ)$ designs exist whenever $k > 12t$ and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all $k > 12t$.

math.CO↗

arXiv2019

Communication and Memory Efficient Testing of Discrete Distributions

We study distribution testing with communication and memory constraints in the following computational models: (1) The {\em one-pass streaming model} where the goal is to minimize the sample complexity of the protocol subject to a memory constraint, and (2) A {\em distributed model} where the data samples reside at multiple machines and the goal is to minimize the communication cost of the protocol. In both these models, we provide efficient algorithms for uniformity/identity testing (goodness of fit) and closeness testing (two sample testing). Moreover, we show nearly-tight lower bounds on (1) the sample complexity of any one-pass streaming tester for uniformity, subject to the memory constraint, and (2) the communication cost of any uniformity testing protocol, in a restricted `one-pass' model of communication.

cs.LG↗

arXiv2019

Torus polynomials: an algebraic approach to ACC lower bounds

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in this framework, implying that ACC0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

cs.CC↗

arXiv2017

The independence number of the Birkhoff polytope graph, and applications to maximally recoverable codes

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph $K_{n,n}$ with labels coming from $F_2^d$ , that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n x n grid topologies. Prior to the current work, it was known that d is between $(\log n)^2$ and $n\log n$. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.

math.CO↗

arXiv2017

Lower Bound on the Redundancy of PIR Codes

We prove that the redundancy of a $k$-server PIR code of dimension $s$ is $Ω(\sqrt{s})$ for all $k \ge 3$. This coincides with a known upper bound of $O(\sqrt{s})$ on the redundancy of PIR codes. Moreover, for $k=3$ and $k = 4$, we determine the lowest possible redundancy of $k$-server PIR codes exactly. Similar results were proved independently by Mary Wootters using a different method.

cs.IT↗