arXiv subjects

David Peleg

Publications and source records attributed to David Peleg.

At least 37 records · Page 2Linked to original sources

arXiv2020

Singularly Optimal Randomized Leader Election

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses $O(n)$ messages with high probability and runs in $O(\log^2 n)$ time (with high probability) to elect a unique leader. The $O(n)$ message complexity should be contrasted with the $Ω(n \log n)$ lower bounds for the deterministic message complexity of leader election algorithms (regardless of time), proven by Korach, Moran, and Zaks (TCS, 1989) for asynchronous algorithms and by Afek and Gafni (SIAM J. Comput., 1991) for synchronous networks. Hence, our result also separates the message complexities of randomized and deterministic leader election. More importantly, our (randomized) time complexity of $O(\log^2 n)$ for obtaining the optimal $O(n)$ message complexity is significantly smaller than the long-standing $\tildeΘ(n)$ time complexity obtained by Afek and Gafni and by Singh (SIAM J. Comput., 1997) for message optimal (deterministic) election in asynchronous networks. In synchronous complete networks, Afek and Gafni showed an essentially singularly optimal deterministic algorithm with $O(\log n)$ time and $O(n \log n)$ messages. Ramanathan et al. (Distrib. Comput. 2007) used randomization to improve the message complexity, and showed a randomized algorithm with $O(n)$ messages and $O(\log n)$ time (with failure probability $O(1 / \log^{Ω(1)}n)$). Our second result is a tightly singularly optimal randomized algorithm, with $O(1)$ time and $O(n)$ messages, for this setting, whose time bound holds with certainty and message bound holds with high probability.

cs.DC↗

arXiv2019

Graph Realizations: Maximum and Minimum Degree in Vertex Neighborhoods

The classical problem of degree sequence realizability asks whether or not a given sequence of $n$ positive integers is equal to the degree sequence of some $n$-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper, we initiate the study of neighborhood degree profiles. We focus on the natural problem of realizing maximum and minimum neighborhood degrees. More specifically, we ask the following question: Given a sequence $D$ of $n$ non-negative integers $0\leq d_1\leq \cdots \leq d_n$, does there exist a simple graph with vertices $v_1,\ldots, v_n$ such that for every $1\le i \le n$, the maximum (resp. minimum) degree in the neighborhood of $v_i$ is exactly $d_i$? We provide in this work various results for both maximum as well as minimum neighborhood degree for general $n$ vertex graphs. Our results are first of its kind that studies extremal neighborhood degree profiles. For maximum neighborhood degree profiles, we provide a {\em complete realizability criteria}. In comparison, we observe that the minimum neighborhood profiles are not so well-behaved, for these our necessary and sufficient conditions for realizability {\em differ by a factor of at most two}.

cs.DS↗

arXiv2019

Efficiently Realizing Interval Sequences

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n)$ satisfying the condition $a_i \leq d_i \leq b_i$, for each $i \in [1,n]$. There is a characterisation (also implying an $O(n)$ verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length $p$ non-graphic sequence to a graphic one.

cs.DS↗

arXiv2019

Hotelling Games with Random Tolerance Intervals

The classical Hotelling game is played on a line segment whose points represent uniformly distributed clients. The $n$ players of the game are servers who need to place themselves on the line segment, and once this is done, each client gets served by the player closest to it. The goal of each player is to choose its location so as to maximize the number of clients it attracts. In this paper we study a variant of the Hotelling game where each client $v$ has a tolerance interval, randomly distributed according to some density function $f$, and $v$ gets served by the nearest among the players eligible for it, namely, those that fall within its interval. (If no such player exists, then $v$ abstains.) It turns out that this modification significantly changes the behavior of the game and its states of equilibria. In particular, it may serve to explain why players sometimes prefer to "spread out", rather than to cluster together as dictated by the classical Hotelling game. We consider two variants of the game: symmetric games, where clients have the same tolerance range to their left and right, and asymmetric games, where the left and right ranges of each client are determined independently of each other. We fully characterize the Nash equilibria of the 2-player game. For at least 3 players, we characterize a specific class of strategy profiles, referred to as canonical profiles, and show that these profiles are the only ones that may yield Nash equilibria in our game. Moreover, the canonical profile, if exists, is uniquely defined for every $n$ and $f$. In both the symmetric and asymmetric setting, we give necessary and sufficient conditions for the canonical profile to be a Nash equilibrium, and demonstrate their application for several distributions.

cs.GT↗

arXiv2019

Hotelling Games with Multiple Line Faults

The Hotelling game consists of n servers each choosing a point on the line segment, so as to maximize the amount of clients it attracts. Clients are uniformly distributed along the line, and each client buys from the closest server. In this paper, we study a fault-prone version of the Hotelling game, where the line fails at multiple random locations. Each failure disconnects the line, blocking the passage of clients. We show that the game admits a Nash equilibrium if and only if the rate of faults exceeds a certain threshold, and calculate that threshold approximately. Moreover, when a Nash equilibrium exists we show it is unique and construct it explicitly. Hence, somewhat surprisingly, the potential occurrence of failures has a stabilizing effect on the game (provided there are enough of them). Additionally, we study the social cost of the game (measured in terms of the total transportation cost of the clients), which also seems to benefit in a certain sense from the potential presence of failures.

cs.GT↗

arXiv2018

Wireless Expanders

This paper introduces an extended notion of expansion suitable for radio networks. A graph $G=(V,E)$ is called an $(α_w, β_w)$-{wireless expander} if for every subset $S \subseteq V$ s.t. $|S|\leq α_w \cdot |V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $β_w \cdot |S|$ vertices in $V\backslash S$ adjacent in $G$ to exactly one vertex in $S'$. The main question we ask is the following: to what extent are ordinary expanders also good {wireless} expanders? We answer this question in a nearly tight manner. On the positive side, we show that any $(α, β)$-expander with maximum degree $Δ$ and $β\geq 1/Δ$ is also a $(α_w, β_w)$ wireless expander for $β_w = Ω(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\}))$. Thus the wireless expansion is smaller than the ordinary expansion by at most a factor logarithmic in $\min\{Δ/ β, Δ\cdot β\}$, which depends on the graph \emph{average degree} rather than maximum degree; e.g., for low arboricity graphs, the wireless expansion matches the ordinary expansion up to a constant. We complement this positive result by presenting an explicit construction of a "bad" $(α, β)$-expander for which the wireless expansion is $β_w = O(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\})$. We also analyze the theoretical properties of wireless expanders and their connection to unique neighbor expanders, and demonstrate their applicability: Our results yield improved bounds for the {spokesmen election problem} that was introduced in the seminal paper of Chlamtac and Weinstein (1991) to devise efficient broadcasting for multihop radio networks. Our negative result yields a significantly simpler proof than that from the seminal paper of Kushilevitz and Mansour (1998) for a lower bound on the broadcast time in radio networks.

cs.DS↗

arXiv2018

Fault-Tolerant Hotelling Games

The $n$-player Hotelling game calls for each player to choose a point on the line segment, so as to maximize the size of his Voronoi cell. This paper studies fault-tolerant versions of the Hotelling game. Two fault models are studied: line faults and player faults. The first model assumes that the environment is prone to failure: with some probability, a disconnection occurs at a random point on the line, splitting it into two separate segments and modifying each player's Voronoi cell accordingly. A complete characterization of the Nash equilibria of this variant is provided for every $n$. Additionally, a one to one correspondence is shown between equilibria of this variant and of the Hotelling game with no faults. The second fault model assumes the players are prone to failure: each player is removed from the game with i.i.d. probability, changing the payoffs of the remaining players accordingly. It is shown that for $n \geq 3$ this variant of the game has no Nash equilibria.

cs.GT↗

arXiv2017

Assortative Mixing Equilibria in Social Network Games

It is known that individuals in social networks tend to exhibit homophily (a.k.a. assortative mixing) in their social ties, which implies that they prefer bonding with others of their own kind. But what are the reasons for this phenomenon? Is it that such relations are more convenient and easier to maintain? Or are there also some more tangible benefits to be gained from this collective behaviour? The current work takes a game-theoretic perspective on this phenomenon, and studies the conditions under which different assortative mixing strategies lead to equilibrium in an evolving social network. We focus on a biased preferential attachment model where the strategy of each group (e.g., political or social minority) determines the level of bias of its members toward other group members and non-members. Our first result is that if the utility function that the group attempts to maximize is the degree centrality of the group, interpreted as the sum of degrees of the group members in the network, then the only strategy achieving Nash equilibrium is a perfect homophily, which implies that cooperation with other groups is harmful to this utility function. A second, and perhaps more surprising, result is that if a reward for inter-group cooperation is added to the utility function (e.g., externally enforced by an authority as a regulation), then there are only two possible equilibria, namely, perfect homophily or perfect heterophily, and it is possible to characterize their feasibility spaces. Interestingly, these results hold regardless of the minority-majority ratio in the population. We believe that these results, as well as the game-theoretic perspective presented herein, may contribute to a better understanding of the forces that shape the groups and communities of our society.

cs.SI↗

arXiv2017

The Effect of Population Control Policies on Societal Fragmentation

Population control policies are proposed and in some places employed as a means towards curbing population growth. This paper is concerned with a disturbing side-effect of such policies, namely, the potential risk of societal fragmentation due to changes in the distribution of family sizes. This effect is illustrated in some simple settings and demonstrated by simulation. In addition, the dependence of societal fragmentation on family size distribution is analyzed. In particular, it is shown that under the studied model, any population control policy that disallows families of 3 or more children incurs the possible risk of societal fragmentation.

cs.SI↗

arXiv2016

The Domination Game: Proving the 3/5 Conjecture on Isolate-Free Forests

We analyze the domination game, where two players, Dominator and Staller, construct together a dominating set M in a given graph, by alternately selecting vertices into M. Each move must increase the size of the dominated set. The players have opposing goals: Dominator wishes M to be as small as possible, and Staller has the opposite goal. Kinnersley, West and Zamani conjectured that when both players play optimally on an isolate-free forest, there is a guaranteed upper bound for the size of the dominating set that depends only on the size n of the forest. This bound is 3n/5 when the first player is Dominator, and (3n+2)/5 when the first player is Staller. The conjecture was proved for specific families of forests by Kinnersley et al. and later extended by Bujtas. Here we prove it for all isolate-free forests, by supplying an algorithm for Dominator that guarantees the desired bound.

cs.DS↗

arXiv2016

Notions of Connectivity in Overlay Networks

" How well connected is the network? " This is one of the most fundamental questions one would ask when facing the challenge of designing a communication network. Three major notions of connectivity have been considered in the literature, but in the context of traditional (single-layer) networks, they turn out to be equivalent. This paper introduces a model for studying the three notions of connectivity in multi-layer networks. Using this model, it is easy to demonstrate that in multi-layer networks the three notions may differ dramatically. Unfortunately, in contrast to the single-layer case, where the values of the three connectivity notions can be computed efficiently, it has been recently shown in the context of WDM networks (results that can be easily translated to our model) that the values of two of these notions of connectivity are hard to compute or even approximate in multi-layer networks. The current paper shed some positive light into the multi-layer connectivity topic: we show that the value of the third connectivity notion can be computed in polynomial time and develop an approximation for the construction of well connected overlay networks.

cs.DC↗

arXiv2015

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously O(m) messages and O($\sqrt$ n+D) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G's diameter. On the other hand, we show that any MST verification algorithm must send Ω(m) messages and incur Ω($\sqrt$ n + D) time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of Ω(m) messages and Ω($\sqrt$ n+D time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously O(m) messages and O($\sqrt$ n + D) time. Specifically, the best known time-optimal algorithm (using O($\sqrt$ n + D) time) requires O(m + n 3/2) messages, and the best known message-optimal algorithm (using O(m) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.

cs.DC↗

arXiv2015

Distributed Computing on Core-Periphery Networks: Axiom-based Design

Inspired by social networks and complex systems, we propose a core-periphery network architecture that supports fast computation for many distributed algorithms and is robust and efficient in number of links. Rather than providing a concrete network model, we take an axiom-based design approach. We provide three intuitive (and independent) algorithmic axioms and prove that any network that satisfies all axioms enjoys an efficient algorithm for a range of tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the minimality of our axiom set: for networks that satisfy any subset of the axioms, the same efficiency cannot be guaranteed for any deterministic algorithm.

cs.DC↗

arXiv2015

Fault Tolerant BFS Structures: A Reinforcement-Backup Tradeoff

This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms: (a) {\em backup}, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) {\em reinforcement}, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechanisms in a concrete setting, we address the problem of designing a $(b,r)$ {\em fault-tolerant} BFS (or $(b,r)$ FT-BFS for short) structure, namely, a subgraph $H$ of the network $G$ consisting of two types of edges: a set $E' \subseteq E$ of $r(n)$ fault-resistant {\em reinforcement} edges, which are assumed to never fail, and a (larger) set $E(H) \setminus E'$ of $b(n)$ fault-prone {\em backup} edges, such that subsequent to the failure of a single fault-prone backup edge $e \in E \setminus E'$, the surviving part of $H$ still contains an BFS spanning tree for (the surviving part of) $G$, satisfying $dist(s,v,H\setminus \{e\}) \leq dist(s,v,G\setminus \{e\})$ for every $v \in V$ and $e \in E \setminus E'$. We establish the following tradeoff between $b(n)$ and $r(n)$: For every real $ε\in (0,1]$, if $r(n) = {\tildeΘ}(n^{1-ε})$, then $b(n) = {\tildeΘ}(n^{1+ε})$ is necessary and sufficient.

cs.DS↗

arXiv2015

Random Preferential Attachment Hypergraphs

The random graph model has recently been extended to a random preferential attachment graph model, in order to enable the study of general asymptotic properties in network types that are better represented by the preferential attachment evolution model than by the ordinary (uniform) evolution lodel. Analogously, this paper extends the random {\em hypergraph} model to a random {\em preferential attachment hypergraph} model. We then analyze the degree distribution of random preferential attachment hypergraphs and show that they possess heavy tail degree distribution properties similar to those of random preferential attachment graphs. However, our results show that the exponent of the degree distribution is sensitive to whether one considers the structure as a hypergraph or as a graph.

cs.SI↗

arXiv2014

Core-Periphery in Networks: An Axiomatic Approach

Recent evidence shows that in many societies worldwide the relative sizes of the economic and social elites are continuously shrinking. Is this a natural social phenomenon? What are the forces that shape this process? We try to address these questions by studying a Core-Periphery social structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society, the periphery. Herein, we present a novel axiom-based model for the forces governing the mutual influences between the elite and the periphery. Assuming a simple set of axioms, capturing the elite's dominance, robustness, compactness and density, we are able to draw strong conclusions about the elite-periphery structure. In particular, we show that a balance of powers between elite and periphery and an elite size that is sub-linear in the network size are universal properties of elites in social networks that satisfy our axioms. We note that the latter is in controversy to the common belief that the elite size converges to a linear fraction of society (most recently claimed to be 1%). We accompany these findings with a large scale empirical study on about 100 real-world networks, which supports our results.

cs.SI↗

arXiv2014

Fault Tolerant Approximate BFS Structures

This paper addresses the problem of designing a {\em fault-tolerant} $(α, β)$ approximate BFS structure (or {\em FT-ABFS structure} for short), namely, a subgraph $H$ of the network $G$ such that subsequent to the failure of some subset $F$ of edges or vertices, the surviving part of $H$ still contains an \emph{approximate} BFS spanning tree for (the surviving part of) $G$, satisfying $dist(s,v,H\setminus F) \leq α\cdot dist(s,v,G\setminus F)+β$ for every $v \in V$. We first consider {\em multiplicative} $(α,0)$ FT-ABFS structures resilient to a failure of a single edge and present an algorithm that given an $n$-vertex unweighted undirected graph $G$ and a source $s$ constructs a $(3,0)$ FT-ABFS structure rooted at $s$ with at most $4n$ edges (improving by an $O(\log n)$ factor on the near-tight result of \cite{BS10} for the special case of edge failures). Assuming at most $f$ edge failures, for constant integer $f>1$, we prove that there exists a (poly-time constructible) $(3(f+1), (f+1) \log n)$ FT-ABFS structure with $O(f n)$ edges. We then consider {\em additive} $(1,β)$ FT-ABFS structures. In contrast to the linear size of $(α,0)$ FT-ABFS structures, we show that for every $β\in [1, O(\log n)]$ there exists an $n$-vertex graph $G$ with a source $s$ for which any $(1,β)$ FT-ABFS structure rooted at $s$ has $Ω(n^{1+ε(β)})$ edges, for some function $ε(β) \in (0,1)$. In particular, $(1,3)$ FT-ABFS structures admit a lower bound of $Ω(n^{5/4})$ edges. Our lower bounds are complemented by an upper bound, showing that there exists a poly-time algorithm that for every $n$-vertex unweighted undirected graph $G$ and source $s$ constructs a $(1,4)$ FT-ABFS structure rooted at $s$ with at most $O(n^{4/3})$ edges.

cs.DS↗

arXiv2013

SINR Diagram with Interference Cancellation

In this paper we study the reception zones of a wireless network in the SINR model with receivers that employ \emph{interference cancellation} (IC), a technique that allows a receiver to decode interfering signals, and \emph{cancel} them from the received signal in order to decode its intended message. We first derive some important topological properties of the diagram describing the reception zones and their connections to \emph{high-order Voronoi diagrams} and other related geometric objects. We then discuss the computational issues that arise when seeking an efficient description of the zones. Our main fundamental result states that although potentially there are exponentially many possible cancellation orderings (and consequently reception cells), in fact there are much fewer nonempty such cells. We prove a (tight) linear bound on the number of cells and provide a polynomial time algorithm to describe the diagram. Moreover, we introduce a novel measure, referred to as the \emph{Compactness Parameter}, which influences the tightness of our bounds. We then utilize the properties established for reception diagrams to devise a logarithmic time algorithm for answering \emph{point-location} queries for networks with IC.

cs.CG↗