arXiv subjects

Yuval Emek

Publications and source records attributed to Yuval Emek.

At least 37 records · Page 2Linked to original sources

arXiv2020

Multicast Communications in Tree Networks with Heterogeneous Capacity Constraints

A widely studied problem in communication networks is that of finding the maximum number of communication requests that can be scheduled concurrently, subject to node and/or link capacity constraints. In this paper, we consider the problem of finding the largest number of multicast communication requests that can be serviced simultaneously by a network of tree topology, subject to heterogeneous capacity constraints. This problem generalizes the following two problems studied in the literature: a) the problem of finding a largest induced $k$-colorable subgraph of a chordal graph, b) the maximum multi-commodity flow problem in tree networks. The problem is already known to be NP-hard and to admit a $c$-approximation ($c \approx 1.58$) in the case of homogeneous capacity constraints. We first show that the problem is much harder to approximate in the heterogeneous case. We then use a generalization of a classical algorithm to obtain an $M$-approximation where $M$ is the maximum number of leaves of the subtrees representing the multicast communications. Surprisingly, the same algorithm, though in various disguises, is used in the literature at least four times to solve related problems (though the analysis is different). The special case of the problem where instances are restricted to unicast communications in a star topology network is known to be polynomial-time solvable. We extend this result and show that the problem can be solved in polynomial time for a set of paths in a tree that share a common vertex.

cs.DS↗

arXiv2019

Low Diameter Graph Decompositions by Approximate Distance Computation

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of such decompositions inherently rely on the subtractive form of the triangle inequality. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 13), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the Congest, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 96) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only O(log n) times, which is of interest for capacitated problems and simulating Congest algorithms on the tree into which the graph is embedded.

cs.DC↗

arXiv2019

Message Reduction in the Local Model is a Free Lunch

A new \emph{spanner} construction algorithm is presented, working under the \emph{LOCAL} model with unique edge IDs. Given an $n$-node communication graph, a spanner with a constant stretch and $O (n^{1 + \varepsilon})$ edges (for an arbitrarily small constant $\varepsilon > 0$) is constructed in a constant number of rounds sending $O (n^{1 + \varepsilon})$ messages whp. Consequently, we conclude that every $t$-round LOCAL algorithm can be transformed into an $O (t)$-round LOCAL algorithm that sends $O (t \cdot n^{1 + \varepsilon})$ messages whp. This improves upon all previous message-reduction schemes for LOCAL algorithms that incur a $\log^{Ω(1)} n$ blow-up of the round complexity.

cs.DC↗

arXiv2019

Bayesian Generalized Network Design

We study network coordination problems, as captured by the setting of generalized network design (Emek et al., STOC 2018), in the face of uncertainty resulting from partial information that the network users hold regarding the actions of their peers. This uncertainty is formalized using Alon et al.'s Bayesian ignorance framework (TCS 2012). While the approach of Alon et al. is purely combinatorial, the current paper takes into account computational considerations: Our main technical contribution is the development of (strongly) polynomial time algorithms for local decision making in the face of Bayesian uncertainty.

cs.GT↗

arXiv2019

Deterministic Leader Election in Programmable Matter

Addressing a fundamental problem in programmable matter, we present the first deterministic algorithm to elect a unique leader in a system of connected amoebots assuming only that amoebots are initially contracted. Previous algorithms either used randomization, made various assumptions (shapes with no holes, or known shared chirality), or elected several co-leaders in some cases. Some of the building blocks we introduce in constructing the algorithm are of interest by themselves, especially the procedure we present for reaching common chirality among the amoebots. Given the leader election and the chirality agreement building block, it is known that various tasks in programmable matter can be performed or improved. The main idea of the new algorithm is the usage of the ability of the amoebots to move, which previous leader election algorithms have not used.

cs.DC↗

arXiv2019

Hierarchical b-Matching

A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a $b$-matching every vertex $v$ has an associated bound $b_v$, and a maximum $b$-matching is a maximum set of edges, such that every vertex $v$ appears in at most $b_v$ of them. We study an extension of this problem, termed {\em Hierarchical b-Matching}. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. In an {\em Hierarchical b-matching} we look for a maximum set of edges, that will obey all bounds (that is, no vertex $v$ participates in more than $b_v$ edges, then all the vertices in one subset do not participate in more that that subset's bound of edges, and so on hierarchically). We propose a polynomial-time algorithm for this new problem, that works for any number of levels of this hierarchical structure.

cs.DS↗

arXiv2018

Selecting a Leader in a Network of Finite State Machines

This paper studies a variant of the \emph{leader election} problem under the \emph{stone age} model (Emek and Wattenhofer, PODC 2013) that considers a network of $n$ randomized finite automata with very weak communication capabilities (a multi-frequency asynchronous generalization of the \emph{beeping} model's communication scheme). Since solving the classic leader election problem is impossible even in more powerful models, we consider a relaxed variant, referred to as \emph{$k$-leader selection}, in which a leader should be selected out of at most $k$ initial candidates. Our main contribution is an algorithm that solves $k$-leader selection for bounded $k$ in the aforementioned stone age model. On (general topology) graphs of diameter $D$, this algorithm runs in $\tilde{O}(D)$ time and succeeds with high probability. The assumption that $k$ is bounded turns out to be unavoidable: we prove that if $k = ω(1)$, then no algorithm in this model can solve $k$-leader selection with a (positive) constant probability.

cs.DC↗

arXiv2018

Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency

In a generalized network design (GND) problem, a set of resources are assigned to multiple communication requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource specific cost function. For example, a request may be to establish a virtual circuit, thus contributing to the load on each edge in the circuit. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current paper advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) we present a generic approximation framework that yields approximation results for a much wider family of requests in both directed and undirected graphs; (2) our framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) our framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current paper is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources' technology and does not grow with the number of resources, the number of requests, or their weights. The design of our framework relies heavily on Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.

cs.GT↗

arXiv2017

Dynamic Networks of Finite State Machines

Like distributed systems, biological multicellular processes are subject to dynamic changes and a biological system will not pass the survival-of-the-fittest test unless it exhibits certain features that enable fast recovery from these changes. In particular, a question that is crucial in the context of biological cellular networks, is whether the system can keep the changing components \emph{confined} so that only nodes in their vicinity may be affected by the changes, but nodes sufficiently far away from any changing component remain unaffected. Based on this notion of confinement, we propose a new metric for measuring the dynamic changes recovery performance in distributed network algorithms operating under the \emph{Stone Age} model (Emek \& Wattenhofer, PODC 2013), where the class of dynamic topology changes we consider includes inserting/deleting an edge, deleting a node together with its incident edges, and inserting a new isolated node. Our main technical contribution is a distributed algorithm for maximal independent set (MIS) in synchronous networks subject to these topology changes that performs well in terms of the aforementioned new metric. Specifically, our algorithm guarantees that nodes which do not experience a topology change in their immediate vicinity are not affected and that all surviving nodes (including the affected ones) perform $\mathcal{O}((C + 1) \log^{2} n)$ computationally-meaningful steps, where $C$ is the number of topology changes; in other words, each surviving node performs $\mathcal{O}(\log^{2} n)$ steps when amortized over the number of topology changes. This is accompanied by a simple example demonstrating that the linear dependency on $C$ cannot be avoided.

cs.DC↗

arXiv2017

Stable Secretaries

We define and study a new variant of the secretary problem. Whereas in the classic setting multiple secretaries compete for a single position, we study the case where the secretaries arrive one at a time and are assigned, in an on-line fashion, to one of multiple positions. Secretaries are ranked according to talent, as in the original formulation, and in addition positions are ranked according to attractiveness. To evaluate an online matching mechanism, we use the notion of blocking pairs from stable matching theory: our goal is to maximize the number of positions (or secretaries) that do not take part in a blocking pair. This is compared with a stable matching in which no blocking pair exists. We consider the case where secretaries arrive randomly, as well as that of an adversarial arrival order, and provide corresponding upper and lower bounds.

cs.GT↗

arXiv2017

Exploring an Infinite Space with Finite Memory Scouts

Consider a small number of scouts exploring the infinite $d$-dimensional grid with the aim of hitting a hidden target point. Each scout is controlled by a probabilistic finite automaton that determines its movement (to a neighboring grid point) based on its current state. The scouts, that operate under a fully synchronous schedule, communicate with each other (in a way that affects their respective states) when they share the same grid point and operate independently otherwise. Our main research question is: How many scouts are required to guarantee that the target admits a finite mean hitting time? Recently, it was shown that $d + 1$ is an upper bound on the answer to this question for any dimension $d \geq 1$ and the main contribution of this paper comes in the form of proving that this bound is tight for $d \in \{ 1, 2 \}$.

math.PR↗

arXiv2016

Online Matching: Haste makes Waste!

This paper studies a new online problem, referred to as \emph{min-cost perfect matching with delays (MPMD)}, defined over a finite metric space (i.e., a complete graph with positive edge weights obeying the triangle inequality) $\mathcal{M}$ that is known to the algorithm in advance. Requests arrive in a continuous time online fashion at the points of $\mathcal{M}$ and should be served by matching them to each other. The algorithm is allowed to delay its request matching commitments, but this does not come for free: the total cost of the algorithm is the sum of metric distances between matched requests \emph{plus} the sum of times each request waited since it arrived until it was matched. A randomized online MPMD algorithm is presented whose competitive ratio is $O (\log^{2} n + \log Δ)$, where $n$ is the number of points in $\mathcal{M}$ and $Δ$ is its aspect ratio. The analysis is based on a machinery developed in the context of a new stochastic process that can be viewed as two interleaved Poisson processes; surprisingly, this new process captures precisely the behavior of our algorithm. A related problem in which the algorithm is allowed to clear any unmatched request at a fixed penalty is also addressed. It is suggested that the MPMD problem is merely the tip of the iceberg for a general framework of online problems with delayed service that captures many more natural problems.

cs.DS↗

arXiv2015

Space-Constrained Interval Selection

We study streaming algorithms for the interval selection problem: finding a maximum cardinality subset of disjoint intervals on the line. A deterministic 2-approximation streaming algorithm for this problem is developed, together with an algorithm for the special case of proper intervals, achieving improved approximation ratio of 3/2. We complement these upper bounds by proving that they are essentially best possible in the streaming setting: it is shown that an approximation ratio of $2 - ε$ (or $3 / 2 - ε$ for proper intervals) cannot be achieved unless the space is linear in the input size. In passing, we also answer an open question of Adler and Azar \cite{AdlerAzar03} regarding the space complexity of constant-competitive randomized preemptive online algorithms for the same problem.

cs.DS↗

arXiv2014

Semi-Streaming Set Cover

This paper studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph $G = (V, E)$ whose edges arrive one-by-one and the goal is to construct an edge cover $F \subseteq E$ with the objective of minimizing the cardinality (or cost in the weighted case) of $F$. We consider a parameterized relaxation of this problem, where given some $0 \leq ε< 1$, the goal is to construct an edge $(1 - ε)$-cover, namely, a subset of edges incident to all but an $ε$-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight trade-off between $ε$ and the approximation ratio: We design a semi-streaming algorithm that on input graph $G$, constructs a succinct data structure $\mathcal{D}$ such that for every $0 \leq ε< 1$, an edge $(1 - ε)$-cover that approximates the optimal edge \mbox{($1$-)cover} within a factor of $f(ε, n)$ can be extracted from $\mathcal{D}$ (efficiently and with no additional space requirements), where \[ f(ε, n) = \left\{ \begin{array}{ll} O (1 / ε), & \text{if } ε> 1 / \sqrt{n} \\ O (\sqrt{n}), & \text{otherwise} \end{array} \right. \, . \] In particular for the traditional set cover problem we obtain an $O(\sqrt{n})$-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by $ε$) of matching lower bounds.

cs.DS↗

arXiv2013

Ants: Mobile Finite State Machines

Consider the Ants Nearby Treasure Search (ANTS) problem introduced by Feinerman, Korman, Lotker, and Sereni (PODC 2012), where $n$ mobile agents, initially placed at the origin of an infinite grid, collaboratively search for an adversarially hidden treasure. In this paper, the model of Feinerman et al. is adapted such that the agents are controlled by a (randomized) finite state machine: they possess a constant-size memory and are able to communicate with each other through constant-size messages. Despite the restriction to constant-size memory, we show that their collaborative performance remains the same by presenting a distributed algorithm that matches a lower bound established by Feinerman et al. on the run-time of any ANTS algorithm.

cs.DC↗

arXiv2012

The Price of Matching Selfish Vertices

We analyze the setting of minimum-cost perfect matchings with selfish vertices through the price of anarchy (PoA) and price of stability (PoS) lens. The underlying solution concept used for this analysis is the Gale-Shapley stable matching notion, where the preferences are determined so that each player (vertex) wishes to minimize the cost of her own matching edge.

cs.CG↗

arXiv2012

Signaling Schemes for Revenue Maximization

Signaling is an important topic in the study of asymmetric information in economic settings. In particular, the transparency of information available to a seller in an auction setting is a question of major interest. We introduce the study of signaling when conducting a second price auction of a probabilistic good whose actual instantiation is known to the auctioneer but not to the bidders. This framework can be used to model impressions selling in display advertising. We study the problem of computing a signaling scheme that maximizes the auctioneer's revenue in a Bayesian setting. While the general case is proved to be computationally hard, several cases of interest are shown to be polynomially solvable. In addition, we establish a tight bound on the minimum number of signals required to implement an optimal signaling scheme and show that at least half of the maximum social welfare can be preserved within such a scheme.

cs.GT↗

arXiv2012

Stone Age Distributed Computing

The traditional models of distributed computing focus mainly on networks of computer-like devices that can exchange large messages with their neighbors and perform arbitrary local computations. Recently, there is a trend to apply distributed computing methods to networks of sub-microprocessor devices, e.g., biological cellular networks or networks of nano-devices. However, the suitability of the traditional distributed computing models to these types of networks is questionable: do tiny bio/nano nodes "compute" and/or "communicate" essentially the same as a computer? In this paper, we introduce a new model that depicts a network of randomized finite state machines operating in an asynchronous environment. Although the computation and communication capabilities of each individual device in the new model are, by design, much weaker than those of a computer, we show that some of the most important and extensively studied distributed computing problems can still be solved efficiently.

cs.DC↗