Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity

Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions

Time Versus Cost Tradeoffs for Deterministic Rendezvous in Networks

Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as $\mathit{rendezvous}$. Agents move in synchronous rounds. Each agent has a distinct integer label from the set $\{1,\dots,L\}$. Two main efficiency measures of rendezvous are its $\mathit{time}$ (the number of rounds until the meeting) and its $\mathit{cost}$ (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound $E$ on the time of exploration (where $E$ and a corresponding exploration procedure are known to both agents) and of the size $L$ of the label space. We present two natural rendezvous algorithms. Algorithm $\mathtt{Cheap}$ has cost $O(E)$ (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly $E$) and time $O(EL)$. Algorithm $\mathtt{Fast}$ has both time and cost $O(E\log L)$. Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically $E$ (i.e., of cost $E+o(E)$) must have time $\Omega(EL)$. On the other hand, we show that any deterministic rendezvous algorithm with time complexity $O(E\log L)$ must have cost $\Omega (E\log L)$

Implicit Decomposition for Write-Efficient Connectivity Algorithms

The future of main memory appears to lie in the direction of new technologies that provide strong capacity-to-performance ratios, but have write operations that are much more expensive than reads in terms of latency, bandwidth, and energy. Motivated by this trend, we propose sequential and parallel algorithms to solve graph connectivity problems using significantly fewer writes than conventional algorithms. Our primary algorithmic tool is the construction of an $o(n)$-sized "implicit decomposition" of a bounded-degree graph $G$ on $n$ nodes, which combined with read-only access to $G$ enables fast answers to connectivity and biconnectivity queries on $G$. The construction breaks the linear-write "barrier", resulting in costs that are asymptotically lower than conventional algorithms while adding only a modest cost to querying time. For general non-sparse graphs on $m$ edges, we also provide the first $o(m)$ writes and $O(m)$ operations parallel algorithms for connectivity and biconnectivity. These algorithms provide insight into how applications can efficiently process computations on large graphs in systems with read-write asymmetry

The Random-Query Model and the Memory-Bounded Coupon Collector

We study a new model of space-bounded computation, the random-query model. The model is based on a branching-program over input variables x_1,…,x_n. In each time step, the branching program gets as an input a random index i ∈ {1,…,n}, together with the input variable x_i (rather than querying an input variable of its choice, as in the case of a standard (oblivious) branching program). We motivate the new model in various ways and study time-space tradeoff lower bounds in this model. Our main technical result is a quadratic time-space lower bound for zero-error computations in the random-query model, for XOR, Majority and many other functions. More precisely, a zero-error computation is a computation that stops with high probability and such that conditioning on the event that the computation stopped, the output is correct with probability 1. We prove that for any Boolean function f: {0,1}^n → {0,1}, with sensitivity k, any zero-error computation with time T and space S, satisfies T ⋅ (S+log n) ≥ Ω(n⋅k). We note that the best time-space lower bounds for standard oblivious branching programs are only slightly super linear and improving these bounds is an important long-standing open problem. To prove our results, we study a memory-bounded variant of the coupon-collector problem that seems to us of independent interest and to the best of our knowledge has not been studied before. We consider a zero-error version of the coupon-collector problem. In this problem, the coupon-collector could explicitly choose to stop when he/she is sure with zero-error that all coupons have already been collected. We prove that any zero-error coupon-collector that stops with high probability in time T, and uses space S, satisfies T⋅(S+log n) ≥ Ω(n^2), where n is the number of different coupons

Scalable Peer-to-Peer Indexing with Constant State

We present a distributed indexing scheme for peer to peer networks. Past work on distributed indexing traded off fast search times with non-constant degree topologies or network-unfriendly behavior such as flooding. In contrast, the scheme we present optimizes all three of these performance measures. That is, we provide logarithmic round searches while maintaining connections to a fixed number of peers and avoiding network flooding. In comparison to the well known scheme Chord, we provide competitive constant factors. Finally, we observe that arbitrary linear speedups are possible and discuss both a general brute force approach and specific economical optimizations