Conference Paper
Deterministic HistoryIndependent Strategies for Storing Information on WriteOnce Memories.
Conference: Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 913, 2007, Proceedings
Source: DBLP

Chapter: Optimal Monotone Encodings
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ABSTRACT: Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n,k)monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, ..., n} to r bits. Monotone encodings are relevant to the study of tamperproof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of ksuperimposed families, which we call αfraction kmultiuser tracing ((k, α)FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)FUT families and by constructing an (n,k)monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)monotone encoding of length 2logn + O(1), which is optimal up to an additive constant.06/2008: pages 258270;  [Show abstract] [Hide abstract]
ABSTRACT: We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multichannel conflict resolution and data security. We prove close upper and lower bounds on the size of superselectors and we provide efficient algorithms for their constructions. Albeit our bounds are very general, when they are instantiated on the combinatorial structures that are particular cases of superselectors (e.g., (p,k,n)selectors, (d,\ell)listdisjunct matrices, MUT_k(r)families, FUT(k, a)families, etc.) they match the best known bounds in terms of size of the structures (the relevant parameter in the applications). For appropriate values of parameters, our results also provide the first efficient deterministic algorithms for the construction of such structures.10/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We formalize a realistic model for computations over massive data sets. The model, referred to as the {\em adversarial sketch model}, unifies the wellstudied sketch and data stream models together with a cryptographic flavor that considers the execution of protocols in "hostile environments", and provides a framework for studying the complexity of many tasks involving massive data sets. The adversarial sketch model consists of several participating parties: honest parties, whose goal is to compute a predetermined function of their inputs, and an adversarial party. Computation in this model proceeds in two phases. In the first phase, the adversarial party chooses the inputs of the honest parties. These inputs are sets of elements taken from a large universe, and provided to the honest parties in an online manner in the form of a sequence of insert and delete operations. Once an operation from the sequence has been processed it is discarded and cannot be retrieved unless explicitly stored. During this phase the honest parties are not allowed to communicate. Moreover, they do not share any secret information and any public information they share is known to the adversary in advance. In the second phase, the honest parties engage in a protocol in order to compute a predetermined function of their inputs. In this paper we settle the complexity (up to logarithmic factors) of two fundamental problems in this model: testing whether two massive data sets are equal, and approximating the size of their symmetric difference. We construct explicit and efficient protocols with sublinear sketches of essentially optimal size, polylogarithmic update time during the first phase, and polylogarithmic communication and computation during the second phase. Our main technical contribution is an explicit and deterministic encoding scheme that enjoys two seemingly conflicting properties: incrementality and high distance, which may be of independent interest.SIAM J. Comput. 01/2011; 40:18451870.
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