# Dense Subgraphs on Dynamic Networks

## Atish Das Sarma, Ashwin Lall, Danupon Nanongkai, and Amitabh Trehan

Abstract
In distributed networks, it is often useful for the nodes to be aware of dense subgraphs, e.g., such a dense subgraph could reveal dense substructures in otherwise sparse graphs (e.g. the World Wide Web or social networks); these might reveal community clusters or dense regions for possibly maintaining good communication infrastructure. In this work, we address the problem of self-awareness of nodes in a dynamic network with regards to graph density, i.e., we give distributed algorithms for maintaining dense subgraphs that the member nodes are aware of. The only knowledge that the nodes need is that of the dynamic diameter $D$, i.e., the maximum number of rounds it takes for a message to traverse the dynamic network. For our work, we consider a model where the number of nodes are fixed, but a powerful adversary can add or remove a limited number of edges from the network at each time step. The communication is by broadcast only and follows the CONGEST model. Our algorithms are continuously executed on the network, and at any time (after some initialization) each node will be aware if it is part (or not) of a particular dense subgraph. We give algorithms that $(2+\epsilon)$-approximate the densest subgraph and $(3 + \epsilon)$-approximate the at-least-k-densest subgraph (for a given parameter k). Our algorithms work for a wide range of parameter values and run in $O(D \log_{1+\epsilon} n)$ time. Further, a special case of our results also gives the first fully decentralized approximation algorithms for densest and at-least-k-densest subgraph problems for static distributed graphs.

Guest: Amitabh Trehan
Host: Shantanu Das

# “Tri, Tri again”: Finding Triangles and Small Subgraphs in a Distributed Setting

## Danny Dolev, Christoph Lenzen, and Shir Peled

Abstract:
Let $G = (V,E)$ be an n-vertex graph and $M_d$ a d-vertex graph, for some constant d. Is $M_d$ a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to $O(log n)$ bits. A simple deterministic algorithm that requires $O(n^{(d-2)/d}/ \log n)$ communication rounds is presented. For the special case that $M_d$ is a triangle, we present a probabilistic algorithm that requires an expected $O(n^{1/3}/(t^{2/3} + 1))$ rounds of communication, where t is the number of triangles in the graph, and $O(\min\{n^{1/3} \log^{2/3}n/(t^{2/3} + 1), n^{1/3}\})$ with high probability. We also present deterministic algorithms specially suited for sparse graphs.

Guest: Christoph Lenzen
Host: Shantanu Das

# Position Discovery for a System of Bouncing Robots

## Jurek Czyzowicz, Leszek Gasieniec, Adrian Kosowski, Evangelos Kranakis, Oscar Morales Ponce, and Eduardo Pacheco

Abstract:
A collection of n anonymous mobile robots is deployed on a unit-perimeter ring or a unit-length line segment. Every robot starts moving at constant speed, and bounces each time it meets any other robot or segment endpoint, changing its walk direction. We study the problem of position discovery, in which the task of each robot is to detect
the presence and the initial positions of all other robots. The robots cannot communicate or perceive information about the environment in any way other than by bouncing. Each robot has a clock allowing it to observe the times of its bounces. The robots have no control on their walks, which are determined by their initial positions and the starting directions. Each robot executes the same position detection algorithm, which receives input data in real-time about the times of the bounces, and terminates when the robot is assured about the existence and the positions of all the robots.
Some initial configuration of robots are shown to be infeasible — no position detection algorithm exists for them. We give complete characterizations of all infeasible initial configurations for both the ring and the segment, and we design optimal position detection algorithms for all feasible configurations. For the case of the ring, we show that all robot configurations in which not all the robots have the same initial direction are feasible. We give a position detection algorithm working for all feasible configurations. The cost of our algorithm depends on the number of robots starting their movement in each direction.

Guest: Jurek Czyzowicz
Host: Shantanu Das

# The Strong At-Most-Once Problem

## Sotirios Kentros, Chadi Kari, Aggelos Kiayias

Abstract: The at-most-once problem in shared memory asks for the completion of a number of tasks by a set of independent processors while adhering to “at most once” semantics. At-most-once algorithms are evaluated in terms of effectiveness, which is a measure that expresses the total number of tasks completed at-most-once in the worst case. Motivated by the lack of deterministic solutions with high effectiveness, we study the feasibility of (a close variant of) this problem. The strong at most once problem is solved by an at-most-one algorithm when all tasks are performed if no participating processes crash during the execution of the algorithm. We prove that the strong at-most-once problem has consensus number 2. This explains, via impossibility, the lack of wait-free deterministic solutions with high effectiveness for the at most once problem using only read/write atomic registers. We then present the first k-adaptive effectiveness optimal randomized solution for the strong at-most-once problem, that has optimal expected work for a non-trivial number of participating processes. Our solution also provides the first k-adaptive randomized solution for the Write-All problem, a dual problem to at-most-once.

Guest: Sotiris Kentros

Host: Yvonne-Anne Pignolet

# Randomized Distributed Decision

## Pierre Fraigniaud, Amos Korman, Merav Parter, David Peleg

Abstract: The paper tackles the power of randomization in the context of locality by analyzing the ability to`boost’ the success probability of deciding a distributed language. The main outcome of this analysis is that the distributed computing setting contrasts significantly with the sequential one as far as randomization is concerned. Indeed, we prove that in some cases, the ability to increase the success probability for deciding distributed languages is rather limited. Informally, a (p,q)-decider for a language L is a distributed randomized algorithm which accepts instances in L with probability at least p and rejects instances outside of L with probability at least q. It is known that every hereditary language that can be decided in t rounds by a (p,q)-decider, where p^2+q>1, can actually be decided deterministically in O(t) rounds. In one of our results we give evidence supporting the conjecture that the above statement holds for all distributed languages. This is achieved by considering the restricted case of path topologies. We then turn our attention to the range below the aforementioned threshold, namely, the case where p^2+q\leq1. We define B_k(t) to be the set of all languages decidable in at most t rounds by a (p,q)-decider, where p^{1+1/k}+q>1. It is easy to see that every language is decidable (in zero rounds) by a (p,q)-decider satisfying p+q=1. Hence, the hierarchy B_k provides a spectrum of complexity classes between determinism and complete randomization. We prove that all these classes are separated: for every integer k\geq 1, there exists a language L satisfying L\in B_{k+1}(0) but L\notin B_k(t) for any t=o(n). In addition, we show that B_\infty(t) does not contain all languages, for any t=o(n). Finally, we show that if the inputs can be restricted in certain ways, then the ability to boost the success probability becomes almost null.

Guest: Merav Parter

Host: Yvonne-Anne Pignolet