We present a deterministic fully dynamic algorithm to answer $c$-edge connectivity queries on pairs of vertices in $n^{o(1)}$ worst case update and query time for any positive integer $c = (\log n)^{o(1)}$ for a graph with $n$ vertices. Previously, only polylogarithmic and $O(\sqrt{n})$ worst case update time fully dynamic algorithms were known for answering $1$, $2$ and $3$-edge connectivity queries respectively [Henzinger and King 1995, Frederikson 1997, Galil and Italiano 1991].
Our result extends the $c$-edge connectivity vertex sparsifier [Chalermsook et al. 2021] to a multi-level sparsification framework. As our main technical contribution, we present a novel update algorithm for the multi-level $c$-edge connectivity vertex sparsifier with subpolynomial update time.
We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min\{λ^{\frac{r-3}{r-1}} n^2, n^r/λ^{r/(r-1)}\})$ (the $\hat O_r(\cdot)$ hides the terms subpolynomial in the main parameter and terms that depend only on $r$) where $p$ is the size, $n$ is the number of vertices, and $r$ is the rank of the hypergraph. Our algorithm is faster than existing algorithms when the the rank is constant and the connectivity $λ$ is $ω(1)$. At the heart of our algorithm is a structural result regarding min-cuts in simple hypergraphs. We show a trade-off between the number…
We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are $ u $ and $ v $ strongly connected in the graph $ G \setminus w $?", for any triple of vertices $ u, v, w $, while $ G $ undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in $O(m n \log{n})$ total time and …
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 $…
Clustering is an important task with applications in many fields of computer science. We study the fully dynamic setting in which we want to maintain good clusters efficiently when input points (from a metric space) can be inserted and deleted. Many clustering problems are $\mathsf{APX}$-hard but admit polynomial time $O(1)$-approximation algorithms. Thus, it is a natural question whether we can maintain $O(1)$-approximate solutions for them in subpolynomial update time, against adaptive and oblivious adversaries. Only a few results are known that give partial answers to this question. There a…
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size $d$; after the update we have to answer queries. In this paper, we consider the dynamic subgraph connectivity problem with sensitivity $d$: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to d vertices are changed. Then we get a sequence of connectivity queries in …
We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a planar graph, subject to connectivity and $2$-vertex-connectivity (biconnectivity) queries between pairs of vertices. Whenever a query pair is connected and not biconnected, we find the first and last cutvertex separating them. Additionally, we allow local changes to the embedding by flipping the embedding of a subgraph that is connected by at most two vertices …
We give offline algorithms for processing a sequence of $2$ and $3$ edge and vertex connectivity queries in a fully-dynamic undirected graph. While the current best fully-dynamic online data structures for $3$-edge and $3$-vertex connectivity require $O(n^{2/3})$ and $O(n)$ time per update, respectively, our per-operation cost is only $O(\log n)$, optimal due to the dynamic connectivity lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer scheme that transforms a graph into smaller equivalents that preserve connectivity information. This construction of equivalents i…
Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst-case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity…
We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in $O(\log^2n/\log\log n)$ amortized time and connectivity queries in $O(\log n/\log\log n)$ worst-case time, where $n$ is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have $O(\log^2n)$ amortized update time and $O(\log n/\log\log n)$ worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer m…
This paper considers fully dynamic graph algorithms with both faster worst case update time and sublinear space. The fully dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes, process an online sequence of edge insertions, edge deletions, and queries of the form "Is there a path between nodes a and b?" In 2013, the first data structure was presented with worst case time per operation which was polylogarithmic in n. In this paper, we shave off a factor of log n from that time, to O(log^4 n) per update. For sequences which are polynomial in length, our al…
Classic dynamic data structure problems maintain a data structure subject to a sequence S of updates and they answer queries using the latest version of the data structure, i.e., the data structure after processing the whole sequence. To handle operations that change the sequence S of updates, Demaine et al. (TALG 2007) introduced retroactive data structures. A retroactive operation modifies the update sequence S in a given position t, called time, and either creates or cancels an update in S at time t. A partially retroactive data structure restricts queries to be executed exclusively in the …
We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(log n) time, where n is the number of vertices of the graph, while vertex operations cost O(log n + d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in…
Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+ε)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be fo…
Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We address the problem of maintaining a DFS tree when the graph is undergoing {\em updates} (insertion and deletion of vertices or edges). We present the following results for this problem. (a) Fault tolerant DFS tree: There exists a data structure of size ${O}(m ~polylog~ n)$ such that given any set ${\cal F}$ of failed vertices or edges, a DFS tree of the gra…
Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages. This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge property or total monotonicity. In this paper, we consider a new condition which assumes (among some other technical assumptions) that the rows of the DP table are monotone. Under this assumption, we introduce a nove…
Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems. Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by "on" vertices. (For instance, this is closer to applications in networks of routers, where node faults …
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…
Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data --- data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM …
Connectivity Oracles for Predictable Vertex Failures
Max Planck Institute of Informatics（马克斯·普朗克信息学研究所）；University of Ioannina（约阿尼纳大学）；UC San Diego（加州大学圣迭戈分校）；ISTA（ISTA（奥地利科学技术研究所））；Bocconi University（博科尼大学）
