Faster Shortest Paths in Dense Distance Graphs, with Applications

We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $\Theta(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). The DDG has $\Theta(n)$ arcs, such that distances in the DDG are equal to the distances in $G$. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in $O(n\log(n) r^{-1/2} \log r)$ time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the $\log n$ dependency in the running time of the shortest-path algorithm, making it $O(n r^{-1/2} \log^2r)$. This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in $O(n \log p)$ time, where $p$ is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with $r$ vertices and $k$ boundary vertices in $O(r \log k)$ time, which is faster than has been previously known for small values of $k$

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)

Propuesta metodológica para el cálculo de las penalidades por giro en modelos de accesibilidad

En esta tesis de maestría se busca desarrollar una metodología para el cálculo de las penalidades por giro a utilizar en los modelos de accesibilidad y en general en los modelos de transportes dada la utilización de algoritmos de caminos mínimos en el cálculo de los tiempos de viaje en la red vial que incluyen penalizaciones y restricciones por giro, entre estos la accesibilidad media global, utilizada en diversos temas como la planificación urbana y de transportes en Manizales (Colombia) y diferentes ciudades alrededor del mundo. En esta ciudad se han utilizado penalidades y restricciones por giro determinadas de manera subjetiva por lo que no se tiene un valor calculado a partir de un método científico. Por lo tanto, se calcularán las penalidades y restricciones por giro para la ciudad de Manizales realizando una cuantificación de los tiempos de giro de los vehículos en diversas intersecciones viales, escogidas a partir de un análisis de priorización y registrando un video en cada una. Con estos datos se podrá obtener el promedio de giro a izquierda y derecha, es decir, las penalidades por giro para Manizales a utilizar en los modelos de accesibilidad calculados en la ciudad o en general para los modelos de transportes. Las penalidades calculadas mediante está metodología serán comparadas con las penalidades utilizadas en investigaciones previas a través del gradiente de ahorro, el cual nos permite cuantificar las diferencias generadas por este dato y su importancia en los modelos de transportes, entre ellos la accesibilidadAbstract: In this Master’s degree thesis seeks develop a methodology for the calculation of turn penalties to use in accessibility models and in general for transport models given in the recent use of algorithms of shortest paths for the calculation of travel times in the road network that includes turn penalties and restrictions, among then the global mean accessibility, used in some issues such as urban and transport planning in Manizales (Colombia) and different cities around the world. At Manizales, turn penalties and restrictions used in accessibility models are determined by a subjective way, so there are not calculated from a scientific method. Therefore, turn and restrictions penalties for Manizales will be calculated, making a quantification of the turn times of the vehicles in different road intersections, chosen from a priorization analysis and recording a video in each one. With this data we can obtain the average time to turn to left and right, that is, the turn penalties for Manizales to be used in the accessibility models calculated in the city or in general in the transport models. The penalties calculated using this methodology will be compared with the penalties used in previous investigations through the saving gradient, which allows us to quantify the differences generated by this data and its importance in transport models, including accessibilityMaestrí

Aportes a los estudios de conflicto y construcción de paz desde Colombia. Tomo 2

Los estudios de paz y conflicto se dedican a investigar las causas de los conflictos violentos, las condiciones que hacen posible su resolución pacífica y la promoción de órdenes de paz estables en los ámbitos local, nacional o internacional. Este campo de investigación académica —aún joven— es interdisciplinario y tiene una clara pretensión normativa: la generación de conocimiento para la contención de la violencia y el fomento de la paz. Por su condición, Colombia se ha convertido en el epicentro de este tipo de investigaciones, y los resultados son seguidos de cerca por actores de la política internacional, activistas de derechos humanos y la academia global. Así, con el presente libro, la Universidad Nacional de Colombia se suma a los esfuerzos por promover la investigación en paz y conflicto, dando a conocer en el ámbito internacional los análisis que académicos, funcionarios y activistas de todo el país hacen de su experiencia y su diario trabajo en la transformación del conflicto violento colombiano.Introducción Perspectivas de análisis del conflicto armado colombiano Stefan Peters La voz de las víctimas. Trayectorias de victimización y agenciamiento político en Colombia Mary Luz Alzate Zuluaga Potencialidades del acompañamiento psicosocial y psicojurídico frente a la participación de víctimas en escenarios transicionales Luisa Galindo Juliette Vargas Juliana Galindo Las reparaciones emancipatorias en contextos transicionales excluyentes Laura Clérico Diana P. Quintero Tania Bolaños Carol Palau Antagonismo mnemónico en Colombia y su impacto en el proceso de reconciliación social Tatiana Fernández-Maya Mateo Orrego Entre lo doméstico y lo productivo. Reorganizando las relaciones de género en clave de paz: aprendizajes del sur del Tolima, Colombia John Jairo Uribe Sarmiento Nohora Isabel Barros Navarro María del Pilar Salamanca Santos Devenires de la paz en las escuelas rurales: entre fragilidades y posibilidades Elida Giraldo Gil Epistemologías decoloniales y de los pueblos hacia la construcción de paz en Colombia María Cárdenas Edwin Alexander Henao Conde Ariel Rosebel Palacios Angulo Esfera pública y conflicto armado en Colombia Mario Fernando Guerrero-Gutiérrez Accesibilidad geográfica y conflicto armado. ¿Cómo construir paz por medio de la infraestructura vial? Diego Alexander Escobar García Jorge Alberto Montoya Santiago Cardona Urrea Deforestación y ampliación de la frontera agropecuaria durante las etapas de negociación y construcción de paz: reflexiones a partir de lo ocurrido en la Amazonia colombiana, en San José del Guaviare Francisco López Loffsner Catalina Riveros Gómez Los autore