arXiv subjects

Amos Korman

Publications and source records attributed to Amos Korman.

43 records · Page 3Linked to original sources

arXiv2011

Local Distributed Decision

A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. This paper introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well.

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arXiv2010

Approximating the Statistics of various Properties in Randomly Weighted Graphs

Consider the setting of \emph{randomly weighted graphs}, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, properties of weighted graphs typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some properties albeit the problem of computing them in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the \emph{diameter} of a given weighted graph, yet, computing the \emph{expected} diameter of a given randomly weighted graph is \SharpP{}-hard even if the edge weights are identically distributed. In this paper, we define a family of properties of weighted graphs and show that for each property in this family, the problem of computing the \emph{$k^{\text{th}}$ moment} (and in particular, the expected value) of the corresponding random variable in a given randomly weighted graph $G$ admits a \emph{fully polynomial time randomized approximation scheme (FPRAS)} for every fixed $k$. This family includes fundamental properties of weighted graphs such as the diameter of $G$, the \emph{radius} of $G$ (with respect to any designated vertex) and the weight of a \emph{minimum spanning tree} of $G$.

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arXiv2009

An Optimal Labeling Scheme for Ancestry Queries

An ancestry labeling scheme assigns labels (bit strings) to the nodes of rooted trees such that ancestry queries between any two nodes in a tree can be answered merely by looking at their corresponding labels. The quality of an ancestry labeling scheme is measured by its label size, that is the maximal number of bits in a label of a tree node. In addition to its theoretical appeal, the design of efficient ancestry labeling schemes is motivated by applications in web search engines. For this purpose, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of a simple interval-based ancestry scheme with label size $2\log_2 n$ bits (Kannan et al., STOC '88), a considerable amount of work was devoted to improve the bound on the size of a label. The current state of the art upper bound is $\log_2 n + O(\sqrt{\log n})$ bits (Abiteboul et al., SODA '02) which is still far from the known $\log_2 n + Ω(\log\log n)$ bits lower bound (Alstrup et al., SODA '03). In this paper we close the gap between the known lower and upper bounds, by constructing an ancestry labeling scheme with label size $\log_2 n + O(\log\log n)$ bits. In addition to the optimal label size, our scheme assigns the labels in linear time and can support any ancestry query in constant time.

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arXiv2009

Compact Ancestry Labeling Schemes for Trees of Small Depth

An {\em ancestry labeling scheme} labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its {\em label size}, that is the maximal number of bits stored in a label, taken over all $n$-node trees. The design of ancestry labeling schemes finds applications in XML search engines. In the context of these applications, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of an original scheme of size $2\log n$ bits, a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is $\log n + O(\sqrt{\log n})$ bits which is still far from the known $\log n + Ω(\log\log n)$ lower bound. Moreover, the hidden constant factor in the additive $O(\sqrt{\log n})$ term is large, which makes this term dominate the label size for typical current XML trees. In attempt to provide good performances for real XML data, we rely on the observation that the depth of a typical XML tree is bounded from above by a small constant. Having this in mind, we present an ancestry labeling scheme of size $\log n+2\log d +O(1)$, for the family of trees with at most $n$ nodes and depth at most $d$. In addition to our main result, we prove a result that may be of independent interest concerning the existence of a linear {\em universal graph} for the family of forests with trees of bounded depth.

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arXiv2009

On the Additive Constant of the k-server Work Function Algorithm

We consider the Work Function Algorithm for the k-server problem. We show that if the Work Function Algorithm is c-competitive, then it is also strictly (2c)-competitive. As a consequence of [Koutsoupias and Papadimitriou, JACM 1995] this also shows that the Work Function Algorithm is strictly (4k-2)-competitive.

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arXiv2006

Labeling Schemes with Queries

We study the question of ``how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels''. Let $f$ be a function on pairs of vertices. An $f$-labeling scheme for a family of graphs $\cF$ labels the vertices of all graphs in $\cF$ such that for every graph $G\in\cF$ and every two vertices $u,v\in G$, the value $f(u,v)$ can be inferred by merely inspecting the labels of $u$ and $v$. This paper introduces a natural generalization: the notion of $f$-labeling schemes with queries, in which the value $f(u,v)$ can be inferred by inspecting not only the labels of $u$ and $v$ but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one {\em query}) enables us to reduce the label size of many labeling schemes significantly.

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arXiv2006

General Compact Labeling Schemes for Dynamic Trees

Let $F$ be a function on pairs of vertices. An {\em $F$- labeling scheme} is composed of a {\em marker} algorithm for labeling the vertices of a graph with short labels, coupled with a {\em decoder} algorithm allowing one to compute $F(u,v)$ of any two vertices $u$ and $v$ directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study {\em distributed dynamic} labeling schemes. This paper investigates labeling schemes for dynamic trees. This paper presents a general method for constructing labeling schemes for dynamic trees. Our method is based on extending an existing {\em static} tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Our resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. Informally, for any function $k(n)$ and any static $F$-labeling scheme on trees, we present an $F$-labeling scheme on dynamic trees incurring multiplicative overhead factors (over the static scheme) of $O(\log_{k(n)} n)$ on the label size and $O(k(n)\log_{k(n)} n)$ on the amortized message complexity. In particular, by setting $k(n)=n^ε$ for any $0<ε<1$, we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for all the above mentioned functions.

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