Lower Bounds for Distinct Elements in the Message Passing Model

David Woodruff and Qin Zhang

This interview introduces the results of two papers, one published at SODA 2014, one at DISC 2013.

——– SODA 2014 ————
Title: An Optimal Lower Bound for Distinct Elements in the Message Passing Model

Abstract: In the message-passing model of communication, there are k players each with their own private input, who try to compute or approximate a function of their inputs by sending messages to one another over private channels. We consider the setting in which each player holds a subset S_i of elements of a universe of size n, and their goal is to output a (1+\eps)-approximation to the total number of distinct elements in the union of the sets S_i with constant probability, which can be amplified by independent repetition. This problem has applications in data mining, sensor networks, and network monitoring. We resolve the communication complexity of this problem up to a constant factor, for all settings of n, k and \eps, by showing a lower bound of \Omega(k \cdot \min(n, 1/\eps^2) + k \log n) bits. This improves upon previous results, which either had non-trivial restrictions on the relationships between the values of n, k and \eps, or were suboptimal by logarithmic factors, or both.

———- DISC 2013 —————–
Title: When Distributed Computation is Communication Expensive

Abstract: We consider a number of fundamental statistical and graph problems in the message-passing model, where we have k machines (sites), each holding a piece of data, and the machines want to jointly solve a problem defined on the union of the k data sets. The communication is point-to-point, and the goal is to minimize the total communication among the k machines. This model captures all point-to-point distributed computational models with respect to minimizing communication costs. Our analysis shows that exact computation of many statistical and graph problems in this distributed setting requires a prohibitively large amount of communication, and often one cannot improve upon the communication of the simple protocol in which all machines send their data to a centralized server. Thus, in order to obtain protocols that are communication-efficient, one has to allow approximation, or investigate the distribution or layout of the data sets.
Guest: Qin Zhang
Host: Yvonne-Anne Pignolet

Convergence in (Social) Influence Networks

Silvio Frischknecht, Barbara Keller, and Roger Wattenhofer

Abstract:
We study the convergence of influence networks, where each node changes its state according to the majority of its neighbors. Our main result is a new bound on the convergence time in the synchronous model, solving the classic “Democrats and Republicans” problem. Furthermore, we give a bound for the sequential model in which the sequence of steps is given by an adversary and a bound for the sequential model in which the sequence of steps is given by a benevolent process.

Guest: Barbara Keller

Host: Shantanu Das

A Super-Fast Distributed Algorithm for Bipartite Metric Facility Location

James Hegeman and Sriram Pemmaraju

Abstract:  The facility location problem consists of a set of facilities F, a set of clients C, an opening cost f_i associated with each facility x_i, and a connection cost D(x_i,y_j) between each facility x_i and client y_j. The goal is to find a subset of facilities to open, and to connect each client to an open facility, so as to minimize the total facility opening costs plus connection costs. This paper presents the first expected-sub-logarithmic-round distributed O(1)-approximation algorithm in the CONGEST model for the metric facility location problem on the complete bipartite network with parts F and C. Our algorithm has an expected running time of O((log log n)^3) rounds, where n = |F| + |C|. This result can be viewed as a continuation of our recent work (ICALP 2012) in which we presented the first sub-logarithmic-round distributed O(1)-approximation algorithm for metric facility location on a clique network. The bipartite setting presents several new challenges not present in the problem on a clique network. We present two new techniques to overcome these challenges. (i) In order to deal with the problem of not being able to choose appropriate probabilities (due to lack of adequate knowledge), we design an algorithm that performs a random walk over a probability space and analyze the progress our algorithm makes as the random walk proceeds. (ii) In order to deal with a problem of quickly disseminating a collection of messages, possibly containing many duplicates, over the bipartite network, we design a probabilistic hashing scheme that delivers all of the messages in expected-O(log log n) rounds.

Guest: James Hegeman
Host: Yvonne-Anne Pignolet

Sleeping Experts in Wireless Networks

Johannes Dams, Martin Hoefer and Thomas Kesselheim

Abstract: We consider capacity maximization algorithms for wireless networks with changing availabilities of spectrum. There are n sender-receiver pairs (called links) and k channels. We consider an iterative round-based scenario, where in each round the set of channels available to each link changes. Each link independently decides about access to one available channel in order to implement a successful transmission. Transmissions are subject to interference and noise, and we use a general approach based on affectance to define which attempts are successful. This includes recently popular interference models based on SINR.
Our main result is that efficient distributed algorithms from sleeping-expert regret learning can be used to obtain constant-factor approximations if channel availability is stochastic and independently distributed among links. In general, sublinear approximation factors cannot be obtained without the assumption of stochastic independence among links. A direct application of the no-external regret property is not sufficient to guarantee small approximation factors.

Guest: Johannes Dams
Host: Yvonne-Anne Pignolet

Frequency Hopping against a Powerful Adversary

Yuval Emek and Roger Wattenhofer

Abstract:
Frequency hopping is a central method in wireless communication,
offering improved resistance to adversarial interference and
interception attempts, and easy non-coordinated control in dynamic
environments. In this paper, we introduce a new model that supports a
rigorous study of frequency hopping in adversarial settings.
We then propose new frequency hopping protocols that allow a
sender-receiver pair to essentially use the full communication
capacity, despite a powerful adversary that can scan and jam a
significant amount of the ongoing transmissions.
Guest: Yuval Emek

Host: Zvi Lotker


Conflict Resolution and Membership Problem in Beeping Channels

Bojun Huang and Thomas Moscibroda

Abstract: Consider a group of nodes connected through multiple-access channels and the only observable feedback on the channel is a binary value: either one or more nodes have transmitted (busy), or no node has transmitted (idle). The channel model thus described is called Beeping Model and captures computation in hardware using a group of sequential circuit modules connected by a logic-OR gate. It has also been used to study chemical signaling mechanisms between biological cells and carrier-sensing based wireless communication. In this paper, we study the distributed complexity of two fundamental problems in the Beeping Model. In both problems, there is a set of nodes each with a unique identifier i ∈ {1,2,…,n}. A subset of the nodes A ⊆ {1,2,…,n} is called active nodes. In the Membership Problem, every node needs to find out the identifiers of all active nodes. In the Conflict Resolution Problem, the goal is to let every active node use the channel alone (without collision) at least once. We derive two results that characterize the distributed complexity of these problems. First, we prove that in the Beeping Model the two above problems are equally hard. This is in stark contrast to traditional channel models with ternary feedback in which the membership problem is strictly harder than conflict resolution. The equivalence result also leads to a randomized lower bound for conflict resolution, which shows a relative powerlessness of randomization in the beeping model. Secondly, we give a new deterministic algorithm for the problems that achieves the best known parallelization among all practical algorithms.

Guest: Bojun Huang
Host Prof. Zvi Lotker

Braess’s Paradox in Wireless Networks: The Danger of Improved Technology

Michael Dinitz and Merav Parter

Abstract: When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links).  For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decreases the capacity and can in certain cases increase it by Omega(log (Delta Pmax)), where Delta is the ratio of the longest link length to the smallest transmitter-receiver distance and Pmax is the maximum transmission power.  But there is no reason to expect the optimal capacity to be realized in practice, particularly since maximizing the capacity is known to be NP-hard.  In reality, we would expect links to behave as self-interested agents, and thus when introducing a new technology it makes more sense to compare the values reached at game-theoretic equilibria than the optimum values.
In this paper we initiate this line of work by comparing various notions of equilibria (particularly Nash equilibria and no-regret behavior) when using a supposedly “better” technology.  We show a version of Braess’s Paradox for all of them: in certain networks, upgrading technology can actually make the equilibria \emph{worse}, despite an increase in the capacity.  We construct instances where this decrease is a constant factor for power control, interference cancellation, and improvements in the SINR threshold beta, and is Omega(log Delta) when power control is combined with interference cancellation.  However, we show that these examples are basically tight: the decrease is at most O(1)for power control, interference cancellation, and improved beta, and is at most
O(log Delta) when power control is combined with interference cancellation.
Guest: Michael Dinitz
Host: Yvonne-Anne Pignolet

An O(sqrt n) Space Bound for Obstruction-Free Leader Election

George Giakkoupis, Maryam Helmi, Lisa Higham and Philipp Woelfel.

Abstract:

We present a deterministic obstruction-free implementation
of leader election from O(n^{1/2}) atomic O(log n)-bit registers in the stan-
dard asynchronous shared memory system with n processes. We provide
also a technique to transform any deterministic obstruction-free algo-
rithm, in which any process can finish if it runs for b steps without
interference, into a randomized wait-free algorithm for the oblivious ad-
versary, in which the expected step complexity is polynomial in n and
b. This transformation allows us to combine our obstruction-free algo-
rithm with the leader election algorithm by Giakkoupis and Woelfel [21],
to obtain a fast randomized leader election (and thus test-and-set) im-
plementation from O(n^{1/2}) O(log n)-bit registers, that has expected step
complexity O(log n) against the oblivious adversary.
Our algorithm provides the first sub-linear space upper bound for obstru-
ction-free leader election. A lower bound of $\Omega(log n)$ has been known
since 1989 [29]. Our research is also motivated by the long-standing open
problem whether there is an obstruction-free consensus algorithm which
uses fewer than n registers.

Distributed Minimum Cut Approximation

Abstract:

We study the problem of computing approximate minimum edge cuts by distributed algorithms. We present two randomized approximation algorithms that both run in a standard synchronous message passing model where in each round, $O(log n)$ bits can be transmitted over every edge (a.k.a. the CONGEST model). The first algorithm is based on a simple and new approach for analyzing random edge sampling, which we call random layering technique. For any any weighted graph and any $\epsilon \in (0, 1)$, the algorithm finds a cut of size at most $O(\epsilon^{-1}\lambda)$ in $O(D) + \tilde{O}(n^{1/2 + \epsilon})$ rounds, where $\lambda$ is the minimum-cut size and the $\tilde{O}$-notation hides poly-logarithmic factors in $n$. In addition, using the outline of a centralized algorithm due to Matula [SODA '93], we present a randomized algorithm to compute a cut of size at most $(2+\epsilon)\lambda$ in $\tilde{O}((D+\sqrt{n})/\epsilon^5)$ rounds for any $\epsilon>0$. The time complexities of our algorithms almost match the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus leading to an answer to an open question raised by Elkin [SIGACT-News '04] and Das Sarma et al. [STOC '11].
To complement our upper bound results, we also strengthen the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which $O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$). For unweighted simple graphs, we show that computing an $\alpha$-approximate minimum cut requires time at least $\tilde{\Omega}(D + \sqrt{n}/\alpha^{1/4})$.

Guest: Mohsen Ghaffari (Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology)

Host: Merav Parter

Faster Rumor Spreading: Breaking the log n Barrier

Chen Avin and Robert Elsässer

Abstract. $O(\log{n})$ rounds has been a well known upper bound for rumor spreading using push&pull in the random phone call model (i.e., uniform gossip in the complete graph). A matching lower bound of $\Omega(\log{n})$ is also known for this special case. Under the assumptions of this model and with a natural addition that nodes can call a partner once they learn its address (e.g., its IP address) we present a new distributed, address-oblivious and robust algorithm that uses push&pull with pointer jumping to spread a rumor to all nodes in only $O(\sqrt{\log{n}})$ rounds, w.h.p. This algorithm can also cope with $F = o( n/2^{\sqrt{\log{n}}} )$ node failures, in which case all but $O(F)$ nodes become informed within $O(\sqrt{\log{n}})$ rounds, w.h.p.

Guest: Chen Avin

Host: Shantanu Das