Showing papers in "Algorithmica in 2006"

Deterministic Rendezvous in Graphs

Abstract: Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Optimal Coding and Sampling of Triangulations

Abstract: We present a bijection between the set of plane triangulations (aka maximal planar graphs) and a simple subset of the set of plane trees with two leaves adjacent to each node. The construction takes advantage of Schnyder tree decompositions of plane triangulations. This bijection yields an interpretation of the formula for the number of plane triangulations with n vertices. Moreover, the construction is simple enough to induce a linear random sampling algorithm, and an explicit information theory optimal encoding. Finally, we extend our bijection approach to triangulations of a polygon with k sides with m inner vertices, and develop in passing new results about Schnyder tree decompositions for these objects.

Scalable Parallel Algorithms for FPT Problems

Abstract: Algorithmic methods based on the theory of fixed-parameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NP-hard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and various forms of parallel algorithms are devised, implemented, and compared. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. Target problems need only be amenable to reduction and decomposition. Applications in high throughput computational biology are also discussed.

Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs

Abstract: Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of planar straight-line representation has not been solved completely. In this paper we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

Abstract: We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio α of the level and the logarithm of tree size lies in [0,e). Convergence of all moments is shown to hold only for α ∈ [0,1] (with only convergence of finite moments when α ∈ (1,e)). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for α = 0 and a "quicksort type" limit law for α = 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on the contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

Maximizing throughput in multi-queue switches

Abstract: We study a basic problem in Multi-Queue switches. A switch connectsm input ports to a single output port. Each input port is equipped with an incoming FIFO queue with bounded capacityB. A switch serves its input queues by transmitting packets arriving at these queues, one packet per time unit. Since the arrival rate can be higher than the transmission rate and each queue has limited capacity, packet loss may occur as a result of insufficient queue space. The goal is to maximize the number of transmitted packets. This general scenario models most current networks (e.g. IP networks) which only support a “best effort” service in which all packet streams are treated equally. A 2-competitive algorithm for this problem was designed in [5] for arbitraryB. Recently, a (17/9 ≈ 1.89)-competitive algorithm was presented forB>1 in [3]. Our main result in this paper shows that forB which is not too small our algorithm can do better than 1.89, and approach a competitive ratio ofe/(e − 1) ≈ 1.58.

Maximum matchings in planar graphs via gaussian elimination

Abstract: We present a randomized algorithm for finding maximum matchings in planar graphs in timeO(n ω/2), whereω is the exponent of the best known matrix multiplication algorithm. Sinceω<2.38, this algorithm breaks through theO(n 1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an algorithm for generating perfect matchings in planar graphs uniformly at random usingO(n ω/2) arithmetic operations. Our algorithms are based on the Gaussian elimination approach to maximum matchings introduced in [16].

Simultaneous Optimization via Approximate Majorization for Concave Profits or Convex Costs

Abstract: For multicriteria problems and problems with a poorly characterized objective, it is often desirable to approximate simultaneously the optimum solution for a large class of objective functions. We consider two such classes: (1) Maximizing all symmetric concave functions. (2) Minimizing all symmetric convex functions. The first class corresponds to maximizing profit for a resource allocation problem (such as allocation of bandwidths in a computer network). The concavity requirement corresponds to the law of diminishing returns in economics. The second class corresponds to minimizing cost or congestion in a load balancing problem, where the congestion/cost is some convex function of the loads. Informally, a simultaneous α-approximation for either class is a feasible solution that is within a factor α of the optimum for all functions in that class. Clearly, the structure of the feasible set has a significant impact on the best possible α and the computational complexity of finding a solution that achieves (or nearly achieves) this α. We develop a framework and a set of techniques to perform simultaneous optimization for a wide variety of problems. We first relate simultaneous α-approximation for both classes to α-approximate majorization. Then we prove that α-approximately majorized solutions exist for logarithmic values of α for the concave profits case. For both classes, we present a polynomial-time algorithm to find the best α if the set of constraints is a polynomial-sized linear program and discuss several non-trivial applications. These applications include finding a (log n)-majorized solution for multicommodity flow, and finding approximately best α for various forms of load balancing problems. Our techniques can also be applied to produce approximately fair versions of the facility location and bi-criteria network design problems. In addition, we demonstrate interesting connections between distributional load balancing (where the sizes of jobs are drawn from known probability distributions but the actual size is not known at the time of placement) and approximate majorization.

Large Deviations for the Weighted Height of an Extended Class of Trees

Abstract: We use large deviations to prove a general theorem on the asymptotic edge-weighted height Hn* of a large class of random trees for which Hn* ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees, random recursive trees and plane oriented trees for instance. New applications include the heights of some random lopsided trees and of the intersection of random trees.

Error Compensation in Leaf Power Problems

Abstract: The k-Leaf Power recognition problem is a particular case of graph power problems: For a given graph it asks whether there exists an unrooted tree—the k-leaf root—with leaves one-to-one labeled by the graph vertices and where the leaves have distance at most k iff their corresponding vertices in the graph are connected by an edge. Here we study "error correction" versions of k-Leaf Power recognition—that is, adding or deleting at most l edges to generate a graph that has a k-leaf root. We provide several NP-completeness results in this context, and we show that the NP-complete Closest 3-Leaf Power problem (the error correction version of 3-Leaf Power) is fixed-parameter tractable with respect to the number of edge modifications or vertex deletions in the given graph. Thus, we provide the seemingly first nontrivial positive algorithmic results in the field of error compensation for leaf power problems with k > 2. To this end, as a result of independent interest, we develop a forbidden subgraph characterization of graphs with 3-leaf roots.

Hardness and approximation results for packing steiner trees

Abstract: We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for four terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee ofO (√n logn), wheren denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of Θ(m 1/3/−ɛ) and give an approximation guarantee ofO(m 1/2/+ɛ), wherem denotes the number of edges. We have similar results for packing Steiner-node-disjoint priority Steiner trees of undirected graphs.

MOVE: A Distributed Framework for Materialized Ontology View Extraction

Abstract: The use of ontologies lies at the very heart of the newly emerging era of semantic web. Ontologies provide a shared conceptualization of some domain that may be communicated between people and application systems. As information on the web increases significantly in size, web ontologies also tend to grow bigger, to such an extent that they become too large to be used in their entirety by any single application. Moreover, because of the size of the original ontology, the process of repeatedly iterating the millions of nodes and relationships to form an optimized sub-ontology becomes very computationally extensive. Therefore, it is imperative that parallel and distributed computing techniques be utilized to implement the extraction process. These problems have stimulated our work in the area of sub-ontology extraction where each user may extract optimized sub-ontologies from an existing base ontology. The extraction process consists of a number of independent optimization schemes that cover various aspects of the optimization process, such as ensuring consistency of the user-specified requirements for the sub-ontology, ensuring semantic completeness of the sub-ontology, etc. Sub-ontologies are valid independent ontologies, known as materialized ontologies, that are specifically extracted to meet certain needs. Our proposed and implemented framework for the extraction process, referred to as Materialized Ontology View Extractor (MOVE), has addressed this problem by proposing a distributed architecture for the extraction/optimization of a sub-ontology from a large-scale base ontology. We utilize coarse-grained data-level parallelism inherent in the problem domain. Such an architecture serves two purposes: (a) facilitates the utilization of a cluster environment typical in business organizations, which is in line with our envisaged application of the proposed system, and (b) enhances the performance of the computationally extensive extraction process when dealing with massively sized realistic ontologies. As ontologies are currently widely used, our proposed approach for distributed ontology extraction will play an important role in improving the efficiency of ontology-based information retrieval.

Book Embeddability of Series–Parallel Digraphs

Abstract: In this paper we deal with the problem of computing upward two-page book embeddings of Two Terminal Series-Parallel (TTSP) digraphs, which are a subclass of series-parallel digraphs. An optimal O(n) time and space algorithm to compute an upward two-page book embedding of a TTSP-digraph with n vertices is presented. A previous algorithm of Alzohairi and Rival [1] runs in O(n3) time and assumes that the input series-parallel digraph does not have transitive edges. An application of this result to a computational geometry problem is also discussed. More precisely, upward two-page book embeddings are used to deal with the upward point-set embeddability problem, i.e., the problem of mapping planar digraphs onto a given set of points in the plane so that all edges are monotonically increasing in a common direction. The equivalence between upward two-page book embeddability and upward point-set embeddability with at most one bend per edge on any given set of points is proved. An O(n log n)-time algorithm for computing an upward point-set embedding with at most one bend per edge for TTSP-digraphs is presented.

Adaptive Spatial Partitioning for Multidimensional Data Streams

Abstract: We propose a space-efficient scheme for summarizing multidimensional data streams. Our sketch can be used to solve spatial versions of several classical data stream queries efficiently. For instance, we can track e-hot spots, which are congruent boxes containing at least an e fraction of the stream, and maintain hierarchical heavy hitters in d dimensions. Our sketch can also be viewed as a multidimensional generalization of the e-approximate quantile summary. The space complexity of our scheme is O((1/e) log R) if the points lie in the domain [0, R]d, where d is assumed to be a constant. The scheme extends to the sliding window model with a log (e n) factor increase in space, where n is the size of the sliding window. Our sketch can also be used to answer e-approximate rectangular range queries over a stream of d-dimensional points.

Combinatorial algorithms for the unsplittable flow problem

Abstract: We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.

Asymptotics of the Moments of Extreme-Value Related Distribution Functions

Abstract: The asymptotic cost of many algorithms and combinatorial structures is related to the extreme-value Gumbel distribution exp(-exp(-x)). The following list is not exhaustive: Trie, Digital Search Tree, Leader Election, Adaptive Sampling, Counting Algorithms, trees related to the Register Function, Composition of Integers, some structures represented by Markov chains (Column-Convex Polyominoes, Carlitz Compositions), Runs and number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Sometimes we can start from an exact (discrete) probability distribution, sometimes from an asymptotic analysis of the discrete objects (e.g., urn models) before establishing the relationship with the Gumbel distribution function. Also some Markov chains are either exactly and directly given by the structure itself, or as a limiting Markov process. The main motivation of the paper is to compute the asymptotic distribution and the moments of the random variables in question. The moments are usually given by a dominant part and a small fluctuating part. We use Laplace and Mellin transforms and singularity analysis and aim for a unified treatment in all cases. Furthermore, our goal is a purely mechanical computation of dominant and fluctuating components, with the help of a computer algebra system. We provide each time the first three moments, but the treatment is (almost) completely automatic. We need some real analysis for the approximations and apart from that only easy complex analysis; simple poles and a few special functions.

Recognizing Hole-Free 4-Map Graphs in Cubic Time

Abstract: We present a cubic-time algorithm for the following problem: Given a simple graph, decide whether it is realized by adjacencies of countries in a map without holes, in which at most four countries meet at any point.

FastLSA: A Fast, Linear-Space, Parallel and Sequential Algorithm for Sequence Alignment

Abstract: Sequence alignment is a fundamental operation for homology search in bioinformatics. For two DNA or protein sequences of length m and n, full-matrix (FM), dynamic programming alignment algorithms such as Needleman-Wunsch and Smith-Waterman take O(m × n) time and use a possibly prohibitive O(m × n) space. Hirschberg's algorithm reduces the space requirements to O(min(m, n)), but requires approximately twice the number of operations required by the FM algorithms. The Fast Linear-Space Alignment (FastLSA) algorithm adapts to the amount of space available by trading space for operations. FastLSA can effectively adapt to use either linear or quadratic space, depending on the specific machine. Our experiments show that, in practice, due to memory caching effects, FastLSA is always as fast or faster than the Hirschberg and FM algorithms. To improve the performance of FastLSA further, we have parallelized it using a simple but effective form of wavefront parallelism. Our experimental results show that Parallel FastLSA exhibits good speedups, almost linear for eight processors or less, and also that the efficiency of Parallel FastLSA increases with the size of the sequences that are aligned. Consequently, parallel and sequential FastLSA can be flexibly and effectively used with high performance in situations where space and the number of parallel processors can vary greatly.

Multicriteria Global Minimum Cuts

Abstract: We consider two multicriteria versions of the global minimum cut problem in undirected graphs. In the k-criteria setting, each edge of the input graph has k non-negative costs associated with it. These costs are measured in separate, non-interchangeable, units. In the AND-version of the problem, purchasing an edge requires the payment of _emphiall_/emph the k costs associated with it. In the OR-version, an edge can be purchased by paying erbany oneerb of the k costs associated with it. Given k bounds b1,b2,. . . ,bk, the basic multicriteria decision problem is whether there exists a cut C of the graph that can be purchased using a budget of bi units of the ith criterion, for 1 ≤ i ≤ k. We show that the AND-version of the multicriteria global minimum cut problem is polynomial for any fixed number k of criteria. The OR-version of the problem, on the other hand, is NP-hard even for k = 2, but can be solved in pseudo-polynomial time for any fixed number k of criteria. It also admits an FPTAS. Further extensions, some applications, and multicriteria versions of two other optimization problems are also discussed.

The Geometric Dilation of Finite Point Sets

Abstract: Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that π/2 = 1.570... is sometimes necessary in order to accommodate a finite set of points.

Efficient Algorithms for k Maximum Sums

Abstract: We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers $\left\langle x_{1},x_{2},\ldots ,x_{n}\right\rangle $ and an integer parameter k, $1\leq k\leq \frac{1}{2}n(n-1),$ the problem involves finding the k largest values of $\sum_{\ell =i}^{j}x_{\ell }$ for $1\leq i\leq j\leq n.$ The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature and is linear-time solvable. Recently, Bae and Takaoka presented a $\Theta(nk)$ -time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the above problem in $O( \min \{k+n\log^{2}n,n\sqrt{k}\}) $ time in the worst case. Our algorithm is optimal for $k = \Omega(n \log^2 n)$ and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.

Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

Abstract: We present some fundamental structural properties for minimum length networks (known as Steiner minimum trees) interconnecting a given set of points in an environment in which edge segments are restricted to λ uniformly oriented directions. We show that the edge segments of any full component of such a tree contain a total of at most four directions if λ is not a multiple of 3, or six directions if λ is a multiple of 3. This result allows us to develop useful canonical forms for these full components. The structural properties of these Steiner minimum trees are then used to resolve an important open problem in the area: does there exist a polynomial time algorithm for constructing a Steiner minimum tree if the topology of the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner topology.

Parallelizing Feature Selection

Abstract: Classification is a key problem in machine learning/data mining. Algorithms for classification have the ability to predict the class of a new instance after having been trained on data representing past experience in classifying instances. However, the presence of a large number of features in training data can hurt the classification capacity of a machine learning algorithm. The Feature Selection problem involves discovering a subset of features such that a classifier built only with this subset would attain predictive accuracy no worse than a classifier built from the entire set of features. Several algorithms have been proposed to solve this problem. In this paper we discuss how parallelism can be used to improve the performance of feature selection algorithms. In particular, we present, discuss and evaluate a coarse-grained parallel version of the feature selection algorithm FortalFS. This algorithm performs well compared with other solutions and it has certain characteristics that makes it a good candidate for parallelization. Our parallel design is based on the master--slave design pattern. Promising results show that this approach is able to achieve near optimum speedups in the context of Amdahl's Law.

Swap and mismatch edit distance

Abstract: There is no known algorithm that solves the general case of theapproximate string matching problem with the extended edit distance, where the edit operations are: insertion, deletion, mismatch and swap, in timeo(nm), wheren is the length of the text andm is the length of the pattern. In an effort to study this problem, the edit operations were analysed independently. It turns out that the approximate matching problem with only the mismatch operation can be solved in timeO(n √m logm). If the only edit operation allowed is swap, then the problem can be solved in timeO(n logm logσ), whereσ=min(m, |Σ|). In this paper we show that theapproximate string matching problem withswap andmismatch as the edit operations, can be computed in timeO(n √m logm).

Left and Right Pathlengths in Random Binary Trees

Abstract: We study the difference between the left and right total pathlengths in a random binary tree. The results include exact and asymptotic formulas for moments and an asymptotic distribution that can be expressed in terms of either the Brownian snake or ISE. The proofs are based on computing expectations for a subcritical binary Galton-Watson tree, and studying asymptotics as the Galton-Watson process approaches a critical one.

Computing All Immobilizing Grasps of a Simple Polygon with Few Contacts

Abstract: We study the output-sensitive computation of all the combinations of edges and concave vertices of a simple polygon that allow an immobilizing grasp with less than four frictionless point contacts. More specifically, if n is the number of edges, and m is the number of concave vertices of the polygon, we show how to compute: in O(m4/3 log1/3 m + K) time, all K combinations that allow a form-closure grasp with two contacts; in O(n2 log4 m + K) time, all K combinations that allow a form-closure grasp with three contacts; in O(n log4 m + (nm)2/3 log2+e m + K) time (for any constant e > 0), all K combinations of one concave vertex and one edge that allow a grasp with one contact on the vertex and one contact on the interior of the edge, satisfying Czyzowicz's weaker conditions for immobilization; in O(n2 log3 n + K) time, all K combinations of three edges that allow a grasp with one contact on the interior of each edge, satisfying Czyzowicz's weaker conditions for immobilization.

Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis

Abstract: In this paper we prove the optimality and other properties of the ź-adic nonadjacent form: this expansion has been introduced in order to compute scalar multiplications on Koblitz curves efficiently. We also refine and extend results about double expansions of scalars introduced by Avanzi, Ciet and Sica in order to improve scalar multiplications further. Our double expansions are optimal and their properties are carefully analysed. In particular, we provide first- and second-order terms for the expected weight, determine the variance and prove a central limit theorem. Transducers for all the involved expansions are provided, as well as automata accepting all expansions of minimal weight.

The Random Multisection Problem, Travelling Waves and the Distribution of the Height of m-Ary Search Trees

Abstract: The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of m-ary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way (travelling wave). The crucial property for the proof is a so-called intersection property that transfers inequalities between two distribution functions (resp. of their Laplace transforms) from one level to the next. It is conjectured that such intersection properties hold in a much more general context. If this property is verified convergence to a travelling wave follows almost automatically.

A Coarse-Grained Parallel Algorithm for the All-Substrings Longest Common Subsequence Problem

Abstract: Given two strings A and B of lengths na and nb, respectively, the All-substrings Longest Common Subsequence (ALCS) problem obtains, for any substring B' of B, the length of the longest string that is a subsequence of both A and B'. The sequential algorithm for this problem takes O(na nb) time and O(nb) space. We present a parallel algorithm for the ALCS problem on the Coarse-Grained Multicomputer (BSP/CGM) model with p < √na processors, that takes O(na nb/p) time, O(log p) communication rounds and O(nb √na) space per processor. The proposed algorithm also solves the basic Longest Common Subsequence (LCS) problem that finds the longest string (and not only its length) that is a subsequence of both A and B. To our knowledge, this is the best BSP/CGM algorithm in the literature for the LCS and ALCS problems.

A Slightly Improved Sub-Cubic Algorithm for the All Pairs Shortest Paths Problem with Real Edge Lengths

Abstract: We present an $O(n^3\sqrt{\log\log n}/\!\log n)$-time algorithm for the All Pairs Shortest Paths (APSP) problem for directed graphs with real edge lengths. This slightly improves previous algorithms for the problem obtained by Fredman, Dobosiewicz, Han, and Takaoka.