Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees. Our results, in both the insertion-only and the turnstile models, are as follows. Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic. BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs. DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.ISSN:1868-896

Universally Optimal Deterministic Broadcasting in the HYBRID Distributed Model

In theoretical computer science, it is a common practice to show existential lower bounds for problems, meaning there is a family of pathological inputs on which no algorithm can do better. However, most inputs of interest can be solved much more efficiently, giving rise to the notion of universally optimal algorithms, which run as fast as possible on every input. Questions on the existence of universally optimal algorithms were first raised by Garay, Kutten, and Peleg in FOCS '93. This research direction reemerged recently through a series of works, including the influential work of Haeupler, Wajc, and Zuzic in STOC '21, which resolves some of these decades-old questions in the supported CONGEST model. We work in the HYBRID distributed model, which analyzes networks combining both global and local communication. Much attention has recently been devoted to solving distance related problems, such as All-Pairs Shortest Paths (APSP) in HYBRID, culminating in a $\tilde \Theta(n^{1/2})$ round algorithm for exact APSP. However, by definition, every problem in HYBRID is solvable in $D$ (diameter) rounds, showing that it is far from universally optimal. We show the first universally optimal algorithms in HYBRID, by presenting a fundamental tool that solves any broadcasting problem in a universally optimal number of rounds, deterministically. Specifically, we consider the problem in a graph $G$ where a set of $k$ messages $M$ distributed arbitrarily across $G$, requires every node to learn all of $M$. We show a universal lower bound and a matching, deterministic upper bound, for any graph $G$, any value $k$, and any distribution of $M$ across $G$. This broadcasting tool opens a new exciting direction of research into showing universally optimal algorithms in HYBRID. As an example, we use it to obtain algorithms for approximate and exact APSP in general and sparse graphs

Universally-Optimal Distributed Algorithms for Known Topologies

Many distributed optimization algorithms achieve existentially-optimal running times, meaning that there exists some pathological worst-case topology on which no algorithm can do better. Still, most networks of interest allow for exponentially faster algorithms. This motivates two questions: (1) What network topology parameters determine the complexity of distributed optimization? (2) Are there universally-optimal algorithms that are as fast as possible on every topology? We resolve these 25-year-old open problems in the known-topology setting (i.e., supported CONGEST) for a wide class of global network optimization problems including MST, $(1+\varepsilon)$-min cut, various approximate shortest paths problems, sub-graph connectivity, etc. In particular, we provide several (equivalent) graph parameters and show they are tight universal lower bounds for the above problems, fully characterizing their inherent complexity. Our results also imply that algorithms based on the low-congestion shortcut framework match the above lower bound, making them universally optimal if shortcuts are efficiently approximable. We leverage a recent result in hop-constrained oblivious routing to show this is the case if the topology is known -- giving universally-optimal algorithms for all above problems.Comment: Full version of extended abstract in STOC 202

Near-Optimal Distributed DFS in Planar Graphs

We present a randomized distributed algorithm that computes a Depth-First Search (DFS) tree in ~O(D) rounds, in any planar network G=(V,E) with diameter D, with high probability. This is the first sublinear-time distributed DFS algorithm, improving on a three decades-old O(n) algorithm of Awerbuch (1985), which remains the best known for general graphs. Furthermore, this ~O(D) round complexity is nearly-optimal as Omega(D) is a trivial lower bound. A key technical ingredient in our results is the development of a distributed method for (recursively) computing a separator path, which is a path whose removal from the graph leaves connected components that are all a constant factor smaller. We believe that the general method we develop for computing path separators recursively might be of broader interest, and may provide the first step towards solving many other problems.ISSN:1868-896