Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing $\cal{T}$ is the largest diameter of any tree in $\cal{T}$. A desirable property of a tree packing, that is both sufficient and necessary for leveraging the high connectivity of a graph in distributed communication, is that its diameter is low. Yet, despite extensive research in this area, it is still unclear how to compute a tree packing, whose diameter is sublinear in $|V(G)|$, in a low-diameter graph $G$, or alternatively how to show that such a packing does not exist.
In this paper we provide first non-trivial upper and lower bounds on the diameter of tree packing. First, we show that, for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, there is a tree packing $\cal{T}$ of size $Ω(k)$, diameter $O((101k\log n)^D)$, that causes edge-congestion at most $2$. Second, we show that for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, the diameter of $G[p]$ is $O(k^{D(D+1)/2})$ with high probability, where $G[p]$ is obtained by sampling each edge of $G$ independently with probability $p=Θ(\log n/k)$. This provides a packing of $Ω(k/\log n)$ edge-disjoint trees of diameter at most $O(k^{(D(D+1)/2)})$ each. We then prove that these two results are nearly tight. Lastly, we show that if every pair of vertices in a graph has $k$ edge-disjoint paths of length at most $D$ connecting them, then there is a tree packing of size $k$, diameter $O(D\log n)$, causing edge-congestion $O(\log n)$. We also provide several applications of low-diameter tree packing in distributed computation.
We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum …
We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing …
We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,ω)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$
We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum de…
We present a uniform self-stabilizing algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of an arbitrary positively real-weighted graph. Our algorithm consists in two stages of stabilizing protocols. The first stage is a uniform randomized stabilizing {\em unique naming} protocol, and the second stage is a stabilizing {\em MDST} protocol, designed as a {\em fair composition} of Merlin--Segall's stabilizing protocol and a distributed deterministic stabilizing protocol solving the (MDST) problem. The resulting randomized distributed algorithm presented…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of $(1\pmε)$. This paper considers computing cut sparsifiers of weighted graphs of size $O(n\log (n)/ε^2)$. Our algorithm computes such a sparsifier in time $O(m\cdot\min(α(n)\log(m/n),\log (n)))$, both for graphs with polynomially bounded and unbounded integer weights, where $α(\cdot)$ is the functional inverse of Ackermann's function. This improves upon the state of the art by Benczúr and Karger (SICOMP 2015), which takes $O(m\log^2 (n))$ time. For unbounded we…
Given a set of $n$ points $P$ in the plane, each colored with one of the $t$ given colors, a color-spanning set $S\subset P$ is a subset of $t$ points with distinct colors. The minimum diameter color-spanning set (MDCS) is a color-spanning set whose diameter is minimum (among all color-spanning sets of $P$). Somehow symmetrically, the largest closest pair color-spanning set (LCPCS) is a color-spanning set whose closest pair is the largest (among all color-spanning sets of $P$). Both MDCS and LCPCS have been shown to be NP-complete, but whether they are fixed-parameter tractable (FPT) when $t$ …
How heavy independent sets help to find arborescences with many leaves in DAGs
Institute of Mathematics；Statistics. University of S \ a o Paulo. S \ a o Paulo, Brazil；Center for Mathematics. Computing；Cognition. Federal University of ABC. Santo Andr \'e , S \ a o Paulo, Brazil
Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings. In this paper we revisit properties of an LP relaxation for $k$-cut proposed by Naor and Rabani, and analyzed by Chekuri, Guha and Naor. We show that the dual of the LP yields a tree packing, that when combined with an uppe…
One of the most fundamental problems in wireless networks is to achieve high throughput. Fractional Connected Dominating Set (FCDS) Packings can achieve a throughput of $Θ(k/\log n)$ messages for networks with node connectivity $k$, which is optimal regarding routing-based message transmission. FCDS were proposed by Censor-Hillel \emph{et al.} [SODA'14,PODC'14] and are a natural generalization to Connected Dominating Sets (CDS), allowing each node to participate with a fraction of its weight in multiple FCDS. Thus, $Ω(k)$ co-existing transmission backbones are established, taking full advantag…
We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.
We study graph connectivity problem in MPC model. On an undirected graph with $n$ nodes and $m$ edges, $O(\log n)$ round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a $O(\log D \log\log_{m/n} n)$ time connectivity algorithm for diameter-$D$ graphs, using $Θ(m)$ total memory. If our algorithm can use more memory, it can terminate in…
