From clutters to matroids

Peer ReviewedPostprint (published version

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

An Optimal Single-Path Routing Algorithm in the Datacenter Network DPillar

DPillar has recently been proposed as a server-centric datacenter network and is combinatorially related to (but distinct from) the well-known wrapped butterfly network. We explain the relationship between DPillar and the wrapped butterfly network before proving that the underlying graph of DPillar is a Cayley graph; hence, the datacenter network DPillar is node-symmetric. We use this symmetry property to establish a single-path routing algorithm for DPillar that computes a shortest path and has time complexity O(k), where k parameterizes the dimension of DPillar (we refer to the number of ports in its switches as n). Our analysis also enables us to calculate the diameter of DPillar exactly. Moreover, our algorithm is trivial to implement, being essentially a conditional clause of numeric tests, and improves significantly upon a routing algorithm earlier employed for DPillar. Furthermore, we provide empirical data in order to demonstrate this improvement. In particular, we empirically show that our routing algorithm improves the average length of paths found, the aggregate bottleneck throughput, and the communication latency. A secondary, yet important, effect of our work is that it emphasises that datacenter networks are amenable to a closer combinatorial scrutiny that can significantly improve their computational efficiency and performance

A time- and message-optimal distributed algorithm for minimum spanning trees

This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in $\tilde{O}(D + \sqrt{n})$ time and exchanges $\tilde{O}(m)$ messages (both with high probability), where $n$ is the number of nodes of the network, $D$ is the diameter, and $m$ is the number of edges. This is the first distributed MST algorithm that matches \emph{simultaneously} the time lower bound of $\tilde{\Omega}(D + \sqrt{n})$ [Elkin, SIAM J. Comput. 2006] and the message lower bound of $\Omega(m)$ [Kutten et al., J.ACM 2015] (which both apply to randomized algorithms). The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound {\em does not} work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires \emph{both} $\tilde{\Omega}(D + \sqrt{n})$ rounds and $\Omega(m)$ messages

Connectivity Lower Bounds in Broadcast Congested Clique

We prove three new lower bounds for graph connectivity in the 1-bit broadcast congested clique model, BCC(1). First, in the KT-0 version of BCC(1), in which nodes are aware of neighbors only through port numbers, we show an ?(log n) round lower bound for Connectivity even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the ?(log n) lower bound result extends to the KT-1 version of the BCC(1) model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. This result substantially improves upon the existing ?(log^* n) deterministic lower bound (Jurdzi?ski et el., SIROCCO 2018) for this problem. Since nodes know IDs of their neighbors in the KT-1 model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem Partition in which Alice and Bob are given two set partitions on [n] and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-1 Connectivity lower bound holds only for deterministic algorithms, in our third result we extend this ?(log n) KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related ConnectedComponents problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of Connectivity in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for Connectivity in the BCC(1) model

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum