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Dynamically maintaining the minimum cut in a graph $G$ under edge insertions and deletions is a fundamental problem in dynamic graph algorithms for which no conditional lower bound on the time per operation exists. In an $n$-node graph the…

Data Structures and Algorithms · Computer Science 2025-01-07 Antoine El-Hayek , Monika Henzinger , Jason Li

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional…

Data Structures and Algorithms · Computer Science 2023-02-16 Aleksander B. G. Christiansen , Jacob Holm , Ivor van der Hoog , Eva Rotenberg , Chris Schwiegelshohn

We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee {\em worst-case update time} and work against an adaptive adversary, meaning that an edge…

Data Structures and Algorithms · Computer Science 2017-04-19 Danupon Nanongkai , Thatchaphol Saranurak

We present a deterministic fully dynamic algorithm with subpolynomial worst-case time per graph update such that after processing each update of the graph, the algorithm outputs a minimum cut of the graph if the graph has a cut of size at…

Data Structures and Algorithms · Computer Science 2024-01-19 Wenyu Jin , Xiaorui Sun , Mikkel Thorup

We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximate directed densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortized time or $\tilde{O}(\log^4(n)/\varepsilon^7)$ worst-case time per edge update (where…

Data Structures and Algorithms · Computer Science 2023-12-19 Richard Li , Kent Quanrud

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than $2$. Specifically, we obtain a…

Data Structures and Algorithms · Computer Science 2023-04-28 Sayan Bhattacharya , Peter Kiss , Thatchaphol Saranurak , David Wajc

While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a…

Data Structures and Algorithms · Computer Science 2015-09-23 Sayan Bhattacharya , Monika Henzinger , Danupon Nanongkai , Charalampos E. Tsourakakis

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…

Data Structures and Algorithms · Computer Science 2016-11-17 Monika Henzinger , Stefan Neumann

Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But amortized data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For…

Data Structures and Algorithms · Computer Science 2021-03-12 Aaron Bernstein , Sebastian Forster , Monika Henzinger

We present an algorithm for updating the Reeb graph under fully dynamic changes of the function values. The basic event is the interchange of two consecutive vertex values. The algorithm updates the Reeb graph in $O(l g{n})$ worst-case…

Computational Geometry · Computer Science 2015-07-20 Salman Parsa

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for…

Data Structures and Algorithms · Computer Science 2023-01-30 Monika Henzinger , Ami Paz , A. R. Sricharan

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a $(2+\epsilon)$-approximate maximum matching in general graphs with $O(\text{poly}(\log n, 1/\epsilon))$ update time. (2) An…

Data Structures and Algorithms · Computer Science 2016-04-21 Sayan Bhattacharya , Monika Henzinger , Danupon Nanongkai

We study the point location problem on dynamic planar subdivisions that allows insertions and deletions of edges. In our problem, the underlying graph of a subdivision is not necessarily connected. We present a data structure of linear size…

Computational Geometry · Computer Science 2018-03-13 Eunjin Oh , Hee-Kap Ahn

We propose a fully dynamic algorithm for maintaining reachability information in directed graphs. The proposed deterministic dynamic algorithm has an update time of $O((ins*n^{2}) + (del * (m+n*log(n))))$ where $m$ is the current number of…

Data Structures and Algorithms · Computer Science 2007-11-22 Venkata Seshu Kumar Kurapati

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in $\tilde{O}((\log n)^2)$ amortized time, and can find a bridge in the component of any given vertex, or a…

Data Structures and Algorithms · Computer Science 2018-08-28 Jacob Holm , Eva Rotenberg , Mikkel Thorup

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…

Data Structures and Algorithms · Computer Science 2018-05-09 Alexandr Andoni , Clifford Stein , Zhao Song , Zhengyu Wang , Peilin Zhong

In edge orientations, the goal is usually to orient (direct) the edges of an undirected $n$-vertex graph $G$ such that all out-degrees are bounded. When the graph $G$ is fully dynamic, i.e., admits edge insertions and deletions, we wish to…

Data Structures and Algorithms · Computer Science 2013-12-06 Tsvi Kopelowitz , Robert Krauthgamer , Ely Porat , Shay Solomon

In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal…

Data Structures and Algorithms · Computer Science 2015-06-16 Andrew McGregor , David Tench , Sofya Vorotnikova , Hoa T. Vu

In this paper we study the fundamental problem of maintaining a dynamic collection of strings under the following operations: concat - concatenates two strings, split - splits a string into two at a given position, compare - finds the…

Data Structures and Algorithms · Computer Science 2016-04-11 Paweł Gawrychowski , Adam Karczmarz , Tomasz Kociumaka , Jakub Łącki , Piotr Sankowski

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…

Data Structures and Algorithms · Computer Science 2019-03-22 Richard Peng , Bryce Sandlund , Daniel D. Sleator