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19th Workshop on Circuits and Systems

作者： Taoka, Satoshi Takafuji, Daisuke Iguchi, Takashi Watanabe, Toshimasa Hiroshima Univ Grad Sch Engn Higashihiroshima 7398527 Japan

An edge-weighted directed graph is referred to as a network in this paper, and an edge operation is an operation that increases or decreases an edge weight. Decreasing an edge weight from the infinite to a finite value or increasing any edge weight from a finite one to the infinite corresponds to addition or deletion of this edge, respectively. The dynamic shortest path problem (DSPP for short) is defined by "Given any network with a specified vertex (denoted as s), and any sequence of edge operations, construct a shortest path tree of each network obtained by executing those edge operations one by one in the order of the sequence." As an application, fast routing for an interior network using link state protocols, such as OSPF and IS-IS, requires solving DSPP efficiently. In this paper, among as many existing algorithms as possible, including those which execute several edge operations simultaneously, fundamental and/or important algorithms are implemented and their capability is evaluated based on the results of computational experiments.

关键词： networks shortest paths dynamic algorithms static algorithms computational experiments

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Simple Deterministic algorithms for Fully dynamic Maximal Matching 13

Simple Deterministic Algorithms for Fully Dynamic Maximal Ma...

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45th Annual ACM Symposium on the Theory of Computing (STOC)

作者： Neiman, Ofer Solomon, Shay Ben Gurion Univ Negev POB 653 IL-84105 Beer Sheva Israel Weizmann Inst Sci IL-76100 Rehovot Israel

ISBN: (纸本)9781450320290

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n-vertex graphs using a trivial deterministic algorithm with a worst case update time of 0(n). No deterministic algorithm that outperforms the na ve 0(n) one was reported up to this date. The only progress in this direction is due to Ivkovie and Lloyd [14], who in 1993 devised a deterministic algorithm with an amortized update time of 0((n+m) / 2), where m is the number of edges. In this paper we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O(N',1/7). Moreover, our algorithm maintains a matching which is in fact a 3/2 -approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 0 -approximate MCM improving upon the na ve 0(n) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sub -logarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of 0 (log n/ log log n). This result addresses an open question of Onak and Rubinfeld [19]. We also show a deterministic algorithm with optimal space usage of 0(n + m), that for arbitrary graphs maintains a maximal matching with amortized update time of OHI/n,).

关键词： dynamic algorithms maximal matching maximum matching

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Point enclosure problem for homothetic polygons

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THEORETICAL COMPUTER SCIENCE 2025年 1030卷

作者： Akram, Waseem Saxena, Sanjeev Indian Inst Technol Dept Comp Sci & Engn Kanpur 208016 India

In this paper, we investigate the following problem: "given a set S of n homothetic polygons, preprocess S to efficiently report all the polygons of S containing a query point." A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling and translation operations. The problem is a counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (1987) [9]. We show that after preprocessing a set of homothetic polygons with a constant number of vertices, queries can be answered in O (log n + k) optimal time, where k is the output size. The preprocessing takes O(n log n) space and time. We also study the problem in a dynamic setting where insertion and deletion operations are allowed. The results we obtain also hold for c-oriented triangles, where c is a fixed constant.

关键词： Geometric intersection algorithms Data structures dynamic algorithms

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Robust Online algorithms for Certain dynamic Packing Problems 1

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17th International Workshop on Approximation and Online algorithms (WAOA)

作者： Berndt, Sebastian Dreismann, Valentin Grage, Kilian Jansen, Klaus Knof, Ingmar Univ Kiel Dept Comp Sci Kiel Germany Univ Warwick Coventry W Midlands England

ISBN: (数字)9783030394790

ISBN: (纸本)9783030394790;9783030394783

Online algorithms that allow a small amount of migration or recourse have been intensively studied in the last years. They are essential in the design of competitive algorithms for dynamic problems, where objects can also depart from the instance. In this work, we give a general framework to obtain so called robust online algorithms for a variety of dynamic problems: these online algorithms achieve an asymptotic competitive ratio of gamma + epsilon with migration O(1/epsilon), where gamma is the best known offline asymptotic approximation ratio. For our framework, we require only two ingredients: (i) the existence of an online algorithm for the static case (without departures) that provides a provably good solution compared to the total volume of the instance and (ii) that the optimal solution always exceeds this total volume. If these criteria are met, we can complement the online algorithm with any offline algorithm. While these criteria are naturally fulfilled by many dynamic problems, they are especially suited for packing problems. In order to show the usefulness of our approach in this area, we improve upon the best known robust algorithms for the dynamic versions of generalizations of Strip Packing and Bin Packing, including the first robust algorithms for general d-dimensional Bin Packing and Vector Packing.

关键词： Online algorithms dynamic algorithms Competitive ratio Recourse Packing problems

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Semidynamic algorithms for maintaining single-source shortest path trees

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ALGORITHMICA 1998年第3期22卷 250-274页

作者： Frigioni, D Marchetti-Spaccamela, A Nanni, U Univ Aquila Dipartimento Matemat Pura & Appl I-67010 Coppito AQ Italy Univ Rome La Sapienza Dipartimento Informat & Sistemist I-00198 Rome Italy

We consider the problem of updating a single-source shortest path tree in either a directed or an undirected graph, with positive real edge weights. Our algorithms for the incremental problem (handling edge insertions and cost decrements) work for any graph;they have optimal space requirements and query time, but their performances depend on the class of the considered graph. The cost of updates is computed in terms of amortized complexity and depends on the size of the output modifications. In the case of graphs with bounded genus (including planar graphs), graphs with bounded arboricity (including bounded degree graphs), and graphs with bounded treewidth, the incremental algorithms require O(log rr) amortized time per vertex update: where a vertex is considered updated ii it reduces its distance from the source. For general graphs with n vertices and m edges our incremental solution requires O(root mlog n) amortized time perverter update. We also consider the decremental problem for planar graphs, providing algorithms and data structures with analogous performances. The algorithms, based on Dijkstra's technique [6]. require simple datastructures that are really suitable for a practical and straightforward implementation.

关键词： single-source shortest path dynamic algorithms amortized complexity output complexity

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Simple Deterministic algorithms for Fully dynamic Maximal Matching

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ACM TRANSACTIONS ON algorithms 2016年第1期12卷 7-7页

作者： Neiman, Ofer Solomon, Shay Ben Gurion Univ Negev Dept Comp Sci POB 653 IL-84105 Beer Sheva Israel Weizmann Inst Sci IL-76100 Rehovot Israel Tel Aviv Univ Sch Comp Sci IL-69978 Tel Aviv Israel

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n vertex graphs using a trivial deterministic"algorithm with a worst-case update time of O(n). No deterministic algorithm that outperforms the naive 0(n) one was reported up to this date. The only progress in this direction is due to Ivkovie and Lloyd, who in 1993 devised a deterministic algorithm with an amortized update time of 0((n + m)boolean AND 5/2), where m is the number of edges. In this article, we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O(m). Moreover, our algorithm maintains a matching, which in fact is a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 - epsilon)-approximate MCM improving upon the naive 0(n) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sublogarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of 0(logn/ log log n). This result addresses an open question of Onak and Rubinfeld [2010]. We also show a deterministic algorithm with optimal space usage, which for arbitrary graphs maintains a maximal matching in amortized O(m) time and uses only O(n + m) space.

关键词： dynamic algorithms maximal matching maximum matching

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Fully dynamic recognition algorithm and certificate for directed cographs

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DISCRETE APPLIED MATHEMATICS 2006年第12期154卷 1722-1741页

作者： Crespelle, C. Paul, C. LIRMM CNRS Dept Informat Montpellier France

This paper presents an optimal fully dynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) time where d is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular decomposition tree is updated;otherwise, a certificate is returned within the same complexity. (c) 2006 Elsevier B.V. All rights reserved.

关键词： dynamic algorithms directed cographs certifying algorithms

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dynamic Kernels for Hitting Sets and Set Packing

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ALGORITHMICA 2022年第11期84卷 3459-3488页

作者： Bannach, Max Heinrich, Zacharias Reischuk, Ruediger Tantau, Till Univ Lubeck Inst Theoret Informat Ratzeburger Allee 160 D-23562 Lubeck Germany

Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size k(d) (which is a polynomial as d is a constant), and they do so in time m . 2(d) poly(d) for a small polynomial poly(d) (which is linear in the hypergraph size for d fixed). We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-k(d) kernel. This paper presents a deterministic solution with worst-case time 3(d) poly(d) for updating the kernel upon inserts and time 5(d) poly(d) for updates upon deletions. These bounds nearly match the time 2(d) poly(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times O(c(k)) where c = d - 1 + O(1/d) equals the best base known for the static setting.

关键词： Kernelization dynamic algorithms Hitting Set Set Packings

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dynamic matrix rank

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THEORETICAL COMPUTER SCIENCE 2009年第41期410卷 4085-4093页

作者： Frandsen, Gudmund Skovbjerg Frandsen, Peter Frands Univ Aarhus Dept Comp Sci DK-8200 Aarhus N Denmark Ramboll Management AS DK-8200 Aarhus N Denmark

We consider maintaining information about the rank of a matrix under changes of the entries. For n x n matrices, we show an upper bound of O(n(1.575)) arithmetic operations and a lower bound of Omega(n) arithmetic operations per element change. The upper bound is valid when changing up to O(n(0.575)) entries in a single Column of the matrix. We also give an algorithm that maintains the rank using O(n(2)) arithmetic operations per rank one update. These bounds appear to be the first nontrivial bounds for the problem. The upper bounds are valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound for element updates uses fast rectangular matrix Multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic Computation of rational functions. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Matrix rank dynamic algorithms Lower bounds

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Fully dynamic representations of interval graphs

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THEORETICAL COMPUTER SCIENCE 2019年 759卷 14-49页

作者： Crespelle, Christophe Univ Lyon UCB Lyon 1 ENS Lyon CNRSInriaLIP UMR 5668 15 Parvis Rene Descartes F-69342 Lyon France

We present a fully dynamic algorithm that maintains three different representations of an interval graph: a minimal interval model of the graph, the P Q-tree of its maximal cliques, and its modular decomposition. After each vertex or edge modification (insertion or deletion), the algorithm determines whether the new graph is an interval graph in O (n) time, and, in the positive, updates the three representations within the same complexity. Furthermore, we state that the modular decomposition tree of an interval graph and the P Q-tree of its maximal cliques are algorithmically equivalent and we completely explicit the relationship between the two structures. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Interval graphs dynamic algorithms PQ-tree Modular decomposition

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