# 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

# A Distributed Algorithm for Gathering Many Fat Mobile Robots in the Plane

## Chrysovalandis Agathangelou, Chryssis Georgiou, and Marios Mavronicolas

Abstract: We revisit the problem of gathering autonomous robots in the plane. In particular, we consider non-transparent unit-disc robots (i.e., fat) in an asynchronous setting with vision as the only means of coordination and robots only make local decisions. We use a state-machine representation to formulate the gathering problem and develop a distributed algorithm that solves the problem for any number of fat robots. The main idea behind the algorithm is to enforce the robots to reach a configuration in which all the following hold: (a) The robots’ centers form a convex hull in which all robots are on the convex, (b) Each robot can see all other robots, and (c) The configuration is connected: every robot touches another robot and all robots form together a connected formation.
We show that starting from any initial configuration, the fat robots eventually reach such a configuration and terminate yielding a solution to the gathering problem.

Guest: Chrysovalandis Agathangelou
Host: Shantanu Das

# 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