Data Structures for Node Connectivity Queries

Let $\kappa(s,t)$ denote the maximum number of internally disjoint paths in an undirected graph $G$. We consider designing a data structure that includes a list of cuts, and answers the following query: given $s,t \in V$, determine whether $\kappa(s,t) \leq k$, and if so, return a pointer to an $st$-cut of size $\leq k$ (or to a minimum $st$-cut) in the list. A trivial data structure that includes a list of $n(n-1)/2$ cuts and requires $\Theta(kn^2)$ space can answer each query in $O(1)$ time. We obtain the following results. In the case when $G$ is $k$-connected, we show that $n$ cuts suffice, and that these cuts can be partitioned into $(2k+1)$ laminar families. Thus using space $O(kn)$ we can answers each min-cut query in $O(1)$ time, slightly improving and substantially simplifying a recent result of Pettie and Yin. We then extend this data structure to subset $k$-connectivity. In the general case we show that $(2k+1)n$ cuts suffice to return an $st$-cut of size $\leq k$,and a list of size $k(k+2)n$ contains a minimum $st$-cut for every $s,t \in V$. Combining our subset $k$-connectivity data structure with the data structure of Hsu and Lu for checking $k$-connectivity, we give an $O(k^2 n)$ space data structure that returns an $st$-cut of size $\leq k$ in $O(\log k)$ time, while $O(k^3 n)$ space enables to return a minimum $st$-cut

(No) Quantum space-time tradeoff for USTCON

Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T = Oe(n2/S) for any S such that S = Ω(log(n)) and S = O(n2/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time Oe(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and Oe(n1.5) time, or Oe(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk

Equivalence Classes and Conditional Hardness in Massively Parallel Computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model

Parallel Graph Connectivity in Log Diameter Rounds

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 $\Theta(m)$ total memory. If our algorithm can use more memory, it can terminate in fewer rounds, and there is no lower bound on the memory per processor. We extend our results to related graph problems such as spanning forest, finding a DFS sequence, exact/approximate minimum spanning forest, and bottleneck spanning forest. We also show that achieving similar bounds for reachability in directed graphs would imply faster boolean matrix multiplication algorithms. We introduce several new algorithmic ideas. We describe a general technique called double exponential speed problem size reduction which roughly means that if we can use total memory $N$ to reduce a problem from size $n$ to $n/k$, for $k=(N/n)^{\Theta(1)}$ in one phase, then we can solve the problem in $O(\log\log_{N/n} n)$ phases. In order to achieve this fast reduction for graph connectivity, we use a multistep algorithm. One key step is a carefully constructed truncated broadcasting scheme where each node broadcasts neighbor sets to its neighbors in a way that limits the size of the resulting neighbor sets. Another key step is random leader contraction, where we choose a smaller set of leaders than many previous works do