Mohsen Ghaffari

Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding

We study coding schemes for error correction in interactive communications. Such interactive coding schemes simulate any n-round interactive protocol using N rounds over an adversarial channel that corrupts up to ρN transmissions. Important performance measures for a coding scheme are its maximum tolerable error rate ρ, communication complexity N, and computational complexity. We give the first coding scheme for the standard setting which performs optimally in all three measures: Our randomized non-adaptive coding scheme has a near-linear computational complexity and tolerates any error rate δ <; 1/4 with a linear N = Θ(n) communication complexity. This improves over prior results [1]-[4] which each performed well in two of these measures. We also give results for other settings of interest, namely, the first computationally and communication efficient schemes that tolerate ρ <; 2/7 adaptively, ρ <; 1/3 if only one party is required to decode, and ρ <; 1/2 if list decoding is allowed. These are the optimal tolerable error rates for the respective settings. These coding schemes also have near linear computational and communication complexity. These results are obtained via two techniques: We give a general black-box reduction which reduces unique decoding, in various settings, to list decoding. We also show how to boost the computational and communication efficiency of any list decoder to become near linear1.

MST in Log-Star Rounds of Congested Clique

We present a randomized algorithm that computes a Minimum Spanning Tree (MST) in O(log* n) rounds, with high probability, in the Congested Clique model of distributed computing. In this model, the input is a graph on n nodes, initially each node knows only its incident edges, and per round each two nodes can exchange O(log n) bits. Our key technical novelty is an O(log* n) Graph Connectivity algorithm, the heart of which is a (recursive) forest growth method, based on a combination of two ideas: a sparsity-sensitive sketching aimed at sparse graphs and a random edge sampling aimed at dense graphs. Our result improves significantly over the O(log log log n) algorithm of Hegeman et al. [PODC 2015] and the O(log log n) algorithm of Lotker et al. [SPAA 2003; SICOMP 2005].

Maximal independent sets in multichannel radio networks

We present new upper bounds for fundamental problems in multichannel wireless networks. These bounds address the benefits of dynamic spectrum access, i.e., to what extent multiple communication channels can be used to improve performance. In more detail, we study a multichannel generalization of the standard graph-based wireless model without collision detection, and assume the network topology satisfies polynomially bounded independence. Our core technical result is an algorithm that constructs a maximal independent set (MIS) in <i>O</i>(^{log2 <i>n</i>}/<i>F</i>)+ Õ(log<i>n</i>) rounds, in networks of size <i>n</i> with <i>F</i> channels, where the Õ-notation hides polynomial factors in log log <i>n</i>. Moreover, we use this MIS algorithm as a subroutine to build a constant-degree connected dominating set in the same asymptotic time. Leveraging this structure, we are able to solve global broadcast and leader election within <i>O</i>(<i>D</i> + ^{log2 <i>n</i>}/<i>F</i> + Õ(log<i>n</i>) rounds, where <i>D</i> is the diameter of the graph, and <i>k</i>-message multi-message broadcast in <i>O</i>(<i>D</i> + <i>k</i> + ^{log2 <i>n</i>}/<i>F</i>}\big)+Õ(log<i>n</i>) rounds for unrestricted message size (with a slow down of only a log factor on the <i>k</i> term under the assumption of restricted message size). In all five cases above, we prove: (a) our results hold with high probability (i.e., at least 1--1/<i>n</i>); (b) our results are within polyloglog factors of the relevant lower bounds for multichannel networks; and (c) our results beat the relevant lower bounds for single channel networks. These new (near) optimal algorithms significantly expand the number of problems now known to be solvable faster in multichannel versus single channel wireless networks.

A delaunay triangulation architecture supporting churn and user mobility in MMVEs

This article proposes a new distributed architecture for update message exchange inmassively multi-user virtual environments (MMVE). MMVE applications require delivery of updates among various locations in the virtual environment. The proposed architecture here exploits the location addressing of geometrical routing in order to alleviate the need for IP-specific queries. However, the use of geometrical routing requires careful choice of overlay to achieve high performance in terms of minimizing the delay. At the same time, the MMVE is dynamic, in sense that users are constantly moving in the 3D virtual space. As such, our architecture uses a distributed topology control scheme that aims at maintaining the requires QoS to best support the greedy geometrical routing, despite user mobility or churn. We will further prove the functionality and performance of the proposed scheme through both theory and simulations.

The cost of radio network broadcast for different models of unreliable links

We study upper and lower bounds for the global and local broadcast problems in the dual graph model combined with different strength adversaries. The dual graph model is a generalization of the standard graph-based radio network model that includes unreliable links controlled by an adversary. It is motivated by the ubiquity of unreliable links in real wireless networks. Existing results in this model [11, 12, 3, 8] assume an offline adaptive adversary - the strongest type of adversary considered in standard randomized analysis. In this paper, we study the two other standard types of adversaries: online adaptive and oblivious. Our goal is to find a model that captures the unpredictable behavior of real networks while still allowing for efficient broadcast solutions. For the online adaptive dual graph model, we prove a lower bound that shows the existence of constant-diameter graphs in which both types of broadcast require Ω(n/ log n) rounds, for network size n. This result is within log-factors of the (near) tight upper bound for the offline adaptive setting. For the oblivious dual graph model, we describe a global broadcast algorithm that solves the problem in O(Dlog n + log2 n) rounds for network diameter D, but prove a lower bound of Ω(√n= log n) rounds for local broadcast in this same setting. Finally, under the assumption of geographic constraints on the network graph, we describe a local broadcast algorithm that requires only O(log2 n logΔ) rounds in the oblivious model, for maximum degree Δ. In addition to the theoretical interest of these results, we argue that the oblivious model (with geographic constraints) captures enough behavior of real networks to render our efficient algorithms useful for real deployments.

Bounds on contention management in radio networks

The local broadcast problem assumes that processes in a wireless network are provided messages, one by one, that must be delivered to their neighbors. In this paper, we prove tight bounds for this problem in two well-studied wireless network models: the classical model, in which links are reliable and collisions consistent, and the more recent dual graph model, which introduces unreliable edges. Our results prove that the Decay strategy, commonly used for local broadcast in the classical setting, is optimal. They also establish a separation between the two models, proving that the dual graph setting is strictly harder than the classical setting, with respect to this primitive.

On the necessity of using Delaunay Triangulation substrate in greedy routing based networks

Large scale decentralized communication systems have motivated a new trend towards online routing where routing decisions are performed based on a limited and localized knowledge of the network. Geometrical greedy routing has been among the simplest and most common online routing schemes. While a geometrical online routing scheme is expected to deliver each packet to the point in the network that is closest to the destination, geometrical greedy routing, when applied over generalized substrate graphs, does not guarantee such delivery as its forwarding decision might deliver packets to a localized minimum instead. This letter investigates the necessary and sufficient conditions of greedy supporting graphs that would guarantee such delivery when used as a greedy routing substrate.

Broadcast throughput in radio networks: routing vs. network coding

The broadcast throughput in a network is defined as the average number of messages that can be transmitted per unit time from a given source to all other nodes when time goes to infinity. Classical broadcast algorithms treat messages as atomic tokens and route them from the source to the receivers by making intermediate nodes store and forward messages. The more recent network coding approach, in contrast, prompts intermediate nodes to mix and code together messages. It has been shown that certain wired networks have an asymptotic network coding gap, that is, they have asymptotically higher broadcast throughput when using network coding compared to routing. Whether such a gap exists for wireless networks has been an open question of great interest. We approach this question by studying the broadcast throughput of the radio network model which has been a standard mathematical model to study wireless communication. We show that there is a family of radio networks with a tight Θ(log log n) network coding gap, that is, networks in which the asymptotic throughput achievable via routing messages is a Θ(log log n) factor smaller than that of the optimal network coding algorithm. We also provide new tight upper and lower bounds showing that the asymptotic worst-case broadcast throughput over all networks with n nodes is Θ(1/log n) messages-per-round for both routing and network coding.

Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a (2&#x03B4;-1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree &#x03B4;, in O(log^8 &#x03B4; &#x22C5; log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were 2^{O(&#x221A;{log n})} by Panconesi and Srinivasan [STOC92] and &#x00D5;(&#x221A;{&#x03B4;}) + O(log^* n) by Fraigniaud, Heinrich, and Kosowski [FOCS16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2&#x03B4;-1)-edge-coloring to poly(loglog n) rounds.The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r &#x2014; where each hyperedge has at most r vertices &#x2014; with n nodes and maximum degree &#x03B4;, this algorithm computes a maximal matching in O(r^5 log^{6+log r } &#x03B4; &#x22C5; log n) rounds.This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkins book, a \big((log &#x03B4;/&#x03B5;)^{O(log 1/&#x03B5;)}\big)-round deterministic algorithm for (1+&#x03B5;)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting &#x03BB;-arboricity graphs with out-degree at most \lceil (1+&#x03B5;)&#x03BB; \rceil, for any constant &#x03B5; 0, hence partially answering Open Problem 10 of Barenboim and Elkins book.

Multi-message broadcast with abstract MAC layers and unreliable links

We study the multi-message broadcast problem using abstract MAC layer models of wireless networks. These models capture the key guarantees of existing MAC layers while abstracting away low-level details such as signal propagation and contention.We begin by studying upper and lower bounds for this problem in a standard abstract MAC layer model---identifying an interesting dependence between the structure of unreliable links and achievable time complexity. In more detail, given a restriction that devices connected directly by an unreliable link are not too far from each other in the reliable link topology, we can (almost) match the efficiency of the reliable case. For the related restriction, however, that two devices connected by an unreliable link are not too far from each other in geographic distance, we prove a new lower bound that shows that this efficiency is impossible. We then investigate how much extra power must be added to the model to enable a new order of magnitude of efficiency. In more detail, we consider an enhanced abstract MAC layer model and present a new multi-message broadcast algorithm that (under certain natural assumptions) solves the problem in this model faster than any known solutions in an abstract MAC layer setting.