We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2/logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn/logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by Kapron et al. To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms.

Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

We consider the classical Minimum Balanced Cut problem: given a graph $G$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $n$-vertex $m$-edge graph $G$ and any parameter $1\leq r\leq O(\log n)$, computes a $(\log m)^{r^2}$-approximation for Minimum Balanced Cut on $G$, in time $O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$. In particular, we obtain a $(\log m)^{1/\epsilon}$-approximation in time $m^{1+O(1/\sqrt{\epsilon})}$ for any constant $\epsilon$, and a $(\log m)^{f(m)}$-approximation in time $m^{1+o(1)}$, for any slowly growing function $m$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. 
Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $G$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $n$-vertex graph is $n^{o(1)}$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $n$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs.

A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

Computing the Strongly-Connected Components (SCCs) in a graph G=(V,E) is known to take only O(m+n) time using an algorithm by Tarjan from 1972[SICOMP 72] where m = |E|, n=|V|. For fully-dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for decremental graphs, i.e. graphs that undergo edge deletions. In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time O(m), thus only a polylogarithmic factor from the optimal running time. Previously such a result was only known for the special case of planar graphs [Italiano et al, STOC 17]. Our result should be compared to the formerly best algorithm for general graphs achieving O(m√n) total update time by Chechik et.al. [FOCS 16] which improved upon a breakthrough result leading to O(mn0.9 + o(1)) total update time by Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]; these results in turn improved upon the longstanding bound of O(mn) by Roditty and Zwick [STOC 04]. All of the above results also apply to the decremental Single-Source Reachability (SSR) problem, which can be reduced to decrementally maintaining SCCs. A bound of O(mn) total update time for decremental SSR was established already in 1981 by Even and Shiloach [JACM 81].

Decremental strongly-connected components and single-source reachability in near-linear time

In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The best sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(log2 n) amortized time per edge insertion or deletion, and O(log n) time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where Δ is the average batch size of all deletions, our algorithm achieves O(log n log(1+n / Δ)) expected amortized work per edge insertion and deletion and O(log3 n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k log(1 + n/k)) expected work and O(log n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

https://dl.acm.org/doi/pdf/10.1145/3323165.3323196

Parallel Batch-Dynamic Graph Connectivity

In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The most well known sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves $O(\log^2 n)$ amortized time per edge insertion or deletion, and $O(\log n / \log\log n)$ time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where $\Delta$ is the average batch size of all deletions, our algorithm achieves $O(\log n \log(1 + n / \Delta))$ expected amortized work per edge insertion and deletion and $O(\log^3 n)$ depth w.h.p. Our algorithm answers a batch of $k$ connectivity queries in $O(k \log(1 + n/k))$ expected work and $O(\log n)$ depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a new randomized dynamic connectivity structure with O(log n(log log n)2) amortized expected update time and O(log n/log log log n) query time, which comes within an O((log log n)2) factor of a lower bound due to Pǎtrascu and Demaine. The new structure is based on a dynamic connectivity algorithm proposed by Thorup in an extended abstract at STOC 2000, which left out some important details.

Fully dynamic connectivity in O(log n(log log n)2) amortized expected time

In this paper we explore fundamental problems in randomized communication complexity such as computing Set Intersection on sets of size k and Equality Testing between vectors of length k. Brody et al. [BCK+16] and Saglam and Tardos [ST13] showed that for these types of problems, one can achieve optimal communication volume of O(k) bits, with a randomized protocol that takes O(log* k) rounds. They also proved [BCK+16, ST13] that this is one point along the optimal round-communication tradeoff curve. Aside from rounds and communication volume, there is a third parameter of interest, namely the error probability perr. It is straightforward to show that protocols for Set Intersection or Equality Testing need to send [MATH HERE] bits. Is it possible to simultaneously achieve optimality in all three parameters, namely [MATH HERE] communication and O(log* k) rounds? In this paper we prove that there is no universally optimal algorithm, and complement the existing round-communication tradeoffs [BCK+16, ST13] with a new tradeoff between rounds, communication, and probability of error. In particular: • Any protocol for solving Multiple Equality Testing in r rounds with failure probability perr = 2−E has communication volume Ω(Ek1/r). • There exists a protocol for solving Multiple Equality Testing in r + log* (k/E) rounds with O(k + rEk1/r) communication, thereby essentially matching our lower bound and that of [BCK+16, ST13]. • Lower bounds on Equality Testing extend to Set Intersection, for every r, k, and perr (which is trivial); in the reverse direction, upper bounds on Equality Testing for r, k, perr imply similar upper bounds on Set Intersection with parameters r + 1, k, and perr. Our original motivation for considering perr as an independent parameter came from the problem of enumerating triangles in distributed (CONGEST) networks having maximum degree Δ. We prove that this problem can be solved in O(Δ/log n + log log Δ) time with high probability 1 − 1/poly(n). This beats the trivial (deterministic) O(Δ)-time algorithm and is superior to the O(n1/3) algorithm of [CPZ19, CS19] when Δ = O(n1/3).

The communication complexity of set intersection and multiple equality testing

Despite decades of research on approximate query processing (AQP), our understanding of sample-based joins has remained limited and, to some extent, even superficial. The common belief in the community is that joining random samples is futile. This belief is largely based on an early result showing that the join of two uniform samples is not an independent sample of the original join, and that it leads to quadratically fewer output tuples. However, unfortunately, this result has little applicability to the key questions practitioners face. For example, the success metric is often the final approximation's accuracy, rather than output cardinality. Moreover, there are many non-uniform sampling strategies that one can employ. Is sampling for joins still futile in all of these settings? If not, what is the best sampling strategy in each case? To the best of our knowledge, there is no formal study answering these questions. 
This paper aims to improve our understanding of sample-based joins and offer a guideline for practitioners building and using real-world AQP systems. We study limitations of offline samples in approximating join queries: given an offline sampling budget, how well can one approximate the join of two tables? We answer this question for two success metrics: output size and estimator variance. We show that maximizing output size is easy, while there is an information-theoretical lower bound on the lowest variance achievable by any sampling strategy. We then define a hybrid sampling scheme that captures all combinations of stratified, universe, and Bernoulli sampling, and show that this scheme with our optimal parameters achieves the theoretical lower bound within a constant factor. Since computing these optimal parameters requires shuffling statistics across the network, we also propose a decentralized variant where each node acts autonomously using minimal statistics.

Joins on Samples: A Theoretical Guide for Practitioners

Despite decades of research on AQP (approximate query processing), our understanding of sample-based joins has remained limited and, to some extent, even superficial. The common belief in the community is that joining random samples is futile. This belief is largely based on an early result showing that the join of two uniform samples is not an independent sample of the original join, and that it leads to quadratically fewer output tuples. Unfortunately, this early result has little applicability to the key questions practitioners face. For example, the success metric is often the final approximation's accuracy, rather than output cardinality. Moreover, there are many non-uniform sampling strategies that one can employ. Is sampling for joins still futile in all of these settings? If not, what is the best sampling strategy in each case? To the best of our knowledge, there is no formal study answering these questions.This paper aims to improve our understanding of sample-based joins and offer a guideline for practitioners building and using real-world AQP systems. We study limitations of offline samples in approximating join queries: given an offline sampling budget, how well can one approximate the join of two tables? We answer this question for two success metrics: output size and estimator variance. We show that maximizing output size is easy, while there is an information-theoretical lower bound on the lowest variance achievable by any sampling strategy. We then define a hybrid sampling scheme that captures all combinations of stratified, universe, and Bernoulli sampling, and show that this scheme with our optimal parameters achieves the theoretical lower bound within a constant factor. Since computing these optimal parameters requires shuffling statistics across the network, we also propose a decentralized variant in which each node acts autonomously using minimal statistics. We also empirically validate our findings on popular SQL and AQP engines.

/pdf/joins-on-samples-a-theoretical-guide-for-practitioners-26cuysj9bo.pdf

Joins on samples: a theoretical guide for practitioners

In this paper we present linear time approximation schemes for several generalized matching problems on nonbipartite graphs. Our results include $O_\epsilon(m)$-time algorithms for $(1-\epsilon)$-maximum weight $f$-factor and $(1+\epsilon)$-approximate minimum weight $f$-edge cover. As a byproduct, we also obtain direct algorithms for the exact cardinality versions of these problems running in $O(m\sqrt{f(V)})$ time. 
The technical contributions of this work include an efficient method for maintaining {\em relaxed complementary slackness} in generalized matching problems and approximation-preserving reductions between the $f$-factor and $f$-edge cover problems.

Dawei Huang

Papers

Fully dynamic connectivity in O(log n(log log n)2) amortized expected time

The communication complexity of set intersection and multiple equality testing

Joins on Samples: A Theoretical Guide for Practitioners

Joins on samples: a theoretical guide for practitioners

Approximate Generalized Matching: $f$-Factors and $f$-Edge Covers.