Fast Dynamic Graph Algorithms for Parameterized Problems

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are $o(|G|)$, that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT). We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size $k$. These dynamic graphs achieve almost the best possible update time $O(\mathrm{poly}(k)\log n)$ and the query time $O(f(\mathrm{poly}(k),k))$, where $f(n,k)$ is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem. We also give a dynamic graph for Chromatic Number parameterized by the solution size of Cluster Vertex Deletion, and a dynamic graph for bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming the parameter is a constant, each dynamic graph can be updated in $O(\log n)$ time and can compute a solution in $O(1)$ time. These results are obtained by another approach.Comment: SWAT 2014 to appea

Faster Fully Dynamic Transitive Closure in Practice

The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been thoroughly investigated in theory and many specialized algorithms for solving it have been proposed in the last decades. In two large studies [Frigioni ea, 2001; Krommidas and Zaroliagis, 2008], a number of these algorithms have been evaluated experimentally against simple, static algorithms for graph traversal, showing the competitiveness and even superiority of the simple algorithms in practice, except for very dense random graphs or very high ratios of queries. A major drawback of those studies is that only small and mostly randomly generated graphs are considered. In this paper, we engineer new algorithms to maintain all-pairs reachability information which are simple and space-efficient. Moreover, we perform an extensive experimental evaluation on both generated and real-world instances that are several orders of magnitude larger than those in the previous studies. Our results indicate that our new algorithms outperform all state-of-the-art algorithms on all types of input considerably in practice

Algorithm for interactive sorting

Sekä ihmisille että koneille on vaikeaa järjestää aineistoa, jossa järjestys määräytyy käyttäjän subjektiivisen kokemuksen kautta. Tällaisia aineistoja ovat esimerkiksi listaukset, joihin on valittu kymmenen parasta elokuvaa tai kirjaa. Koska käyttäjän kokemuksella kirjasta ei ole numeerista arvoa, tietokone ei pysty itse järjestämään aineistoa, vaan tarvitsee tähän käyttäjän syötettä. Tässä työssä kehitimme yleiskäyttöisen interaktiivisen algoritmin, joka toistuvasti kysyy käyttäjältä tietoa järjestettävien tietoalkioiden välisestä suhteellisesta suuruusjärjestyksestä ja täydentää suuruusjärjestystä mallintavaa suunnattua verkkoa. Käyttämällä topologista järjestämistä, algoritmi järjestää tietoalkiot tuloslistaksi. Algoritmi on suunniteltu virhesietoiseksi ja itseään korjaavaksi käyttäjän virheiden varalta. Se ei myöskään ota kantaa järjestettävän aineiston tyyppiin. Algoritmi osoittautui tehokkaaksi lyhyiden listojen järjestämisessä ja käyttäjän virheistä huolimatta pystyy tuottamaan enimmäkseen oikean lopputuloksen. Tämän työn pohjalta on mahdollista jatkokehittää algoritmia erityisempiä käyttötarkoituksia varten, kuten asiakaspalautteen ja tutkimusmateriaalin keräämiseen tai muiden sovellusten osana.Both humans and computers struggle with sorting datasets where the ordering is defined by a user's subjective preference or experience. Datasets like this include for example "top 10" lists of books or movies. Since a user's preference does not have a numerical value, the computer can not sort the dataset by itself, but it needs input from the user. In this thesis, we solve this problem by introducing a new algorithm, which iteratively requests the user to input a relative order for a couple of items in the unsorted set. Using these partial orderings iteratively, the algorithm updates a directed graph of "greater than" relations and uses a topological sort algorithm to sort the items into a result list. The algorithm is designed to be self-repairing and tolerant of mistakes when the user makes errors in input. Our algorithm proves to be efficient with small datasets and despite user error provides a mostly correct result. Based on this work it is possible to further develop the algorithm for more specific use cases in marketing and research or as a part of an external application which can benefit from free-form sorting

Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs

We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with O?(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in O?(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs. Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in O?(n?m) worst-case time and queries in O(?m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008]

Struktury danych i algorytmy dynamiczne dla grafów planarnych

Obtaining provably efficient algorithms for the most basic graph problems like finding (shortest) paths or computing maximum matchings, fast enough to handle real-world-scale graphs (i.e., consisting of millions of vertices and edges), is a very challenging task. For example, in a very general regime of strongly-polynomial algorithms (see, e.g., [65]), we still do not know how to compute shortest paths in a real-weighted sparse directed graph significantly faster than in quadratic time, using the classical, but somewhat simple-minded, Bellman-Ford method. One way to circumvent this problem is to consider more restricted computation models for graph algorithms. If, for example, we restrict ourselves to graphs with integral edge weights, we can improve upon the Bellman-Ford algorithm [14, 31]. Although these results are very deep algorithmically, their theoretical efficiency is still very far from the only known trivial linear lower bound on the actual time complexity of the negatively-weighted shortest path problem. Another approach is to develop algorithms specialized for certain graph classes that appear in practice. Planar graphs constitute one of the most important and well-studied such classes. Many of the real-world networks can be drawn on a plane with no or few edge crossings. The examples include not very complex road networks and graphs considered in the domain of VLSI design. Complex road networks, although far from being planar, share with planar graphs some useful properties, like the existence of small separators [20]. Special cases of planar graphs, such as grids, appear often in the area of image processing (e.g., [7]). And indeed, if we restrict ourselves to planar graphs, many of the classical polynomial-time graph problems, in particular computing shortest paths [35, 58] and maximum flows [4, 5, 21] in real-weighted graphs, can be solved either optimally or in nearly-linear time. The very rich combinatorial structure of planar graphs often allows breaking barriers that appear in the respective problems for general graphs by using techniques from computational geometry (e.g., [27]), or by applying sophisticated data structures, such as dynamic trees [4, 10, 21, 66]. In this thesis, we focus on the data-structural aspect of planar graph algorithmics. By this, we mean that rather than concentrating on particular planar graph problems, we study more abstract, “low-level” problems. Efficient algorithms for these problems can be used in a blackbox manner to design algorithms for multiple specific problems at once. Such an approach allows us to improve upon many known complexity upper bounds for different planar graph problems simultaneously, without going into the specifics of these problems. We also study dynamic algorithms for planar graphs, i.e., algorithms that maintain certain information about a dynamically changing graph (such as “is the graph connected?”) much more efficiently than by recomputing this information from scratch after each update. We consider the edge-update model where the input graph can be modified only by adding or removing 1 single edges. A graph algorithm is called fully-dynamic if it supports both edge insertions and edge deletions, and partially dynamic if it supports either only edge insertions (then we call it incremental) or only edge deletions (then it is called decremental). When designing dynamic graph algorithms, we care about the update time, i.e., the time needed by the algorithm to adapt to an elementary change of the graph, and query time, i.e., the time needed by the algorithm to recompute the requested portion of the maintained information. Sometimes, especially in partially dynamic settings, it is more convenient to measure the total update time, i.e., the total time needed by the algorithm to process any possible sequence of updates. For some dynamic problems, it is worth focusing on a more restricted explicit maintenance model where the entire maintained information is explicitly updated (so that the user is notified about the update) after each change. In this model the query procedure is trivial and thus we only care about the update time. Note that there is actually no clear distinction between dynamic graph algorithms and graph data structures, since dynamic algorithms are often used as black-boxes to obtain efficient static algorithms (e.g., [26]). For example, the incremental connectivity problem, where one needs to process queries about the existence of a path between given vertices, while the input undirected graph undergoes edge insertions, is actually equivalent to the disjoint-set data structure problem, also called the union-find data structure problem (see, e.g., [15]). We concentrate mostly on the decremental model and obtain very efficient decremental algorithms for problems on unweighted planar graphs related to reachability and connectivity. We also apply our dynamic algorithms to static problems, thus confirming once again the datastructural character of these results. In the following, let G = (V, E) denote the input planar graph with n vertices. For clarity of this summary, assume G is a simple graph. Then, by planarity, it has O(n) edges. When we talk about general graphs, we denote by m the number of edges of the graph. 2 Contracting a Planar Graph The first part of the thesis is devoted to the data-structural aspect of contracting edges in planar graphs. Edge contraction is one of the fundamental graph operations. Given an undirected graph and its edge e, contracting the edge e consists in removing it from the graph and merging its endpoints. The notion of contraction has been used to describe a number of prominent graph algorithms, including Edmonds’ algorithm for computing maximum matchings [19], or Karger’s minimum cut algorithm [44]. Edge contractions are of particular interest in planar graphs, as a number of planar graph properties can be described using contractions. For example, it is well-known that a graph is planar precisely when it cannot be transformed into K5 or K3,3 by contracting edges, or removing vertices or edges (see e.g., [17]). Moreover, contracting an edge preserves planarity. We would like to have at our disposal a data structure that performs contractions on the input planar graph and still provides access to the most basic information about our graph, such as the sizes of neighbors sets of individual vertices and the adjacency relation. While contraction operation is conceptually very simple, its efficient implementation is challenging. This is because it is not clear how to represent individual vertices’ adjacency lists so that adjacency list merges, adjacency queries, and neighborhood size queries are all efficient. By using standard data structures (e.g., balanced binary search trees), one can maintain adjacency lists of a graph subject to contractions in polylogarithmic amortized time. However, in many planar graph algorithms this becomes a bottleneck. As an example, consider the problem of computing a 5-coloring of a planar graph. There exists a very simple algorithm based on contractions [53] that only relies on a folklore fact that 2 a planar graph has a vertex of degree no more than 5. However, linear-time algorithms solving this problem use some more involved planar graph properties [23, 53, 60]. For example, the algorithm by Matula et al. [53] uses the fact that every planar graph has either a vertex of degree at most 4 or a vertex of degree 5 adjacent to at least four vertices, each having degree at most 11. Similarly, although there exists a very simple algorithm for computing a minimum spanning tree of a planar graph based on edge contractions, various different methods have been used to implement it efficiently [23, 51, 52]. The problem of maintaining a planar graph under contractions has been studied before. In their book, Klein and Mozes [46] showed that there exists a (a bit more general) data structure maintaining a planar graph under edge contractions and deletions, and answering adjacency queries in O(1) worst-case time. The update time is O(log n). This result is based on the work of Brodal and Fagerberg [8], who showed how to maintain a bounded-outdegree orientation of a dynamic planar graph so that the edge set updates are supported in O(log n) amortized time. Gustedt [32] showed an optimal solution to the union-find problem in the case when at any time the actual subsets form disjoint and connected subgraphs of a given planar graph G. In other words, in this problem the allowed unions correspond to the edges of a planar graph and the execution of a union operation can be seen as a contraction of the respective edge. Our Results We show a data structure that can efficiently maintain a planar graph subject to edge contractions in linear total time, assuming the standard word-RAM model with word size Ω(log n). It can report groups of parallel edges and self-loops that emerge. It also supports constant-time adjacency queries and maintains the neighbor lists and degrees explicitly. The data structure can be used as a black-box to implement planar graph algorithms that use contractions. As an example, our data structure can be used to give clean and conceptually simple lineartime implementations of algorithms for computing 5-coloring or minimum spanning tree. More importantly, by using our data structure, we give improved algorithms for a few problems in planar graphs. In particular, we obtain optimal algorithms for decremental 2-edgeconnectivity (see, e.g., [30]), finding a unique perfect matching [26], and computing maximal 3-edge-connected subgraphs [12]. In order to obtain our result, we first partition the graph into small pieces of roughly logarithmic size (using so-called r-divisions [24]). Then we solve our problem recursively for each of the pieces, and separately using a simple-minded approach for the subgraph induced by o(n) vertices contained in multiple pieces (the so-called boundary vertices). Such an approach proved successful in obtaining optimal data structures for the planar union-find problem [32] and decremental connectivity [50]. In fact, our data-structural problem can be seen as a generalization of the former problem. However, maintaining the status of each edge e of the initial graph G (i.e., whether e has become a self-loop or a parallel edge) subject to edge contractions, and supporting constant-time adjacency queries without resorting to randomization, turn out to be serious technical challenges. Overcoming these difficulties is our main contribution of this part of the thesis. 3 Decremental Reachability The second part of this thesis is devoted to dynamic reachability problems in planar graphs. In the dynamic reachability problem we are given a (directed) graph G subject to edge updates and the goal is to design a data structure that would allow answering queries about the existence of a path between a pair of query vertices u, v ∈ V . 3 Two variants of dynamic reachability are studied most often. In the all-pairs variant, our data structure has to support queries between arbitrary pairs of vertices. This variant is also called the dynamic transitive closure problem, since a path u → v exists in G if uv is an edge of the transitive closure of G. In the single-source reachability problem, a source vertex s ∈ V is fixed from the very beginning and the only allowed queries are about the existence of a path s → v, where v ∈ V . If we work with undirected graphs, the dynamic reachability problem is called the dynamic connectivity problem. Note that in the undirected case a path u → v exists in G if and only if a path v → u exists in G. State of the Art Dynamic reachability in general directed graphs turns out to be a very challenging problem. First of all, it is computationally much more demanding than its undirected counterpart. For undirected graphs, fully-dynamic all-pairs algorithms with polylogarithmic amortized update and query bounds are known [36, 38, 71]. For directed graphs, on the other hand, in most settings (either single-source or all-pairs, either incremental, decremental or fully-dynamic) the best known algorithm has either polynomial update time or polynomial query time. The only exception is the incremental single-source reachability problem, for which a trivial extension of depth-first search [68] achieves O(1) amortized update time. One of the possible reasons behind such a big gap between the undirected and directed settings is that one needs only linear time to compute the connected components of an undirected graph, and thus there exists a O(n)-space static data structure that can answer connectivity queries in undirected graphs in O(1) time. On the other hand, the best known algorithm for computing the transitive closure runs in Oe(min(n ω , nm)) = Oe(n 2 ) 1 time [11, 59]. So far, the best known bounds for fully-dynamic reachability are as follows. For dynamic transitive closure, there exist a number of algorithms with O(n 2 ) update time and O(1) query time [16, 61, 64]. These algorithms, in fact, maintain the transitive closure explicitly. There also exist a few fully-dynamic algorithms that are better for sparse graphs, each of which has Ω(n) amortized update time and query time which is o(n) but still polynomial in n [62, 63, 64]. For the single-source variant, the only known non-trivial (i.e., other than recompute-from-scratch) algorithm has O(n 1.53) update time and O(1) query time [64]. Algorithms with O(nm) total update time are known for both incremental [39] and decremental [48, 62] transitive closure. Note that for sparse graphs this bound is only poly-logarithmic factors away from the best known static transitive closure upper bound [11]. All the known partially-dynamic single-source reachability algorithms work in the explicit maintenance model. As mentioned before, for incremental single-source reachability, an optimal (in the amortized sense) algorithm is known. Interestingly, the first algorithms with O(mn1− ) total update time (where > 0) have been obtained only recently [33, 34]. The best known algorithm to date has Oe(m √ n) total update time and is due to Chechik et al. [13]. Dynamic reachability has also been previously studied for planar graphs. Diks and Sankowski [18] showed a fully-dynamic transitive closure algorithm with Oe( √ n) update and query times, which works under the assumption that the graph is plane embedded and the inserted edges can only connect vertices sharing some adjacent face. Łącki [48] showed that one can maintain the strongly connected components of a planar graph under edge deletions in O(n √ n) total time. By known reductions, it follows that there exists a decremental single-source reachability algorithm for planar graphs with O(n √ n) total update time. Note that this bound matches the recent best known bound for general graphs [13] up to polylogarithmic factors. 1We denote by Oe(f(n)) the order O(f(n) polylog n)

Deterministic Fully Dynamic SSSP and More

We present the first non-trivial fully dynamic algorithm maintaining exact single-source distances in unweighted graphs. This resolves an open problem stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019]. Previous fully dynamic single-source distances data structures were all approximate, but so far, non-trivial dynamic algorithms for the exact setting could only be ruled out for polynomially weighted graphs (Abboud and Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main case for which neither a subquadratic dynamic algorithm nor a quadratic lower bound was known. Our dynamic algorithm works on directed graphs, is deterministic, and can report a single-source shortest paths tree in subquadratic time as well. Thus we also obtain the first deterministic fully dynamic data structure for reachability (transitive closure) with subquadratic update and query time. This answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019]. Finally, using the same framework we obtain the first fully dynamic data structure maintaining all-pairs $(1+\epsilon)$-approximate distances within non-trivial sub-$n^\omega$ worst-case update time while supporting optimal-time approximate shortest path reporting at the same time. This data structure is also deterministic and therefore implies the first known non-trivial deterministic worst-case bound for recomputing the transitive closure of a digraph.Comment: Extended abstract to appear in FOCS 202

Recent Advances in Fully Dynamic Graph Algorithms

In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms

MULTIHIERARCHICAL DOCUMENTS AND FINE-GRAINED ACCESS CONTROL

This work presents new models and algorithms for creating, modifying, and controlling access to complex text. The digitization of texts opens new opportunities for preservation, access, and analysis, but at the same time raises questions regarding how to represent and collaboratively edit such texts. Two issues of particular interest are modelling the relationships of markup (annotations) in complex texts, and controlling the creation and modification of those texts. This work addresses and connects these issues, with emphasis on data modelling, algorithms, and computational complexity; and contributes new results in these areas of research. Although hierarchical models of text and markup are common, complex texts often exhibit layers of overlapping structure that are best described by multihierarchical markup. We develop a new model of multihierarchical markup, the globally ordered GODDAG, that combines features of both graph- and range-based models of markup, allowing documents to be unambiguously serialized. We describe extensions to the XPath query language to support globally ordered GODDAGs, provide semantics for a set of update operations on this structure, and provide algorithms for converting between two different representations of the globally ordered GODDAG. Managing the collaborative editing of documents can require restricting the types of changes different editors may make, while not altogether restricting their access to the document. Fine-grained access control allows precisely these kinds of restrictions on the operations that a user is or is not permitted to perform on a document. We describe a rule-based model of fine-grained access control for updates of hierarchical documents, and in this context analyze the document generation problem: determining whether a document could have been created without violating a particular access control policy. We show that this problem is undecidable in the general case and provide computational complexity bounds for a number of restricted variants of the problem. Finally, we extend our fine-grained access control model from hierarchical to multihierarchical documents. We provide semantics for fine-grained access control policies that control splice-in, splice-out, and rename operations on globally ordered GODDAGs, and show that the multihierarchical version of the document generation problem remains undecidable