Résultats de recherche pour « Single-source shortest paths »

One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with non-dyadic, higher-order interactions. Nevertheless, the connectivity properties of real-world hypergraphs remain largely understudied. In this work we introduce path size as a measure to characterise higher-order connectivity and quantify the relevance of non-dyadic ties for efficient shortest paths in a diverse set of empirical networks with and without temporal information. By comparing our results with simple randomised null models, our analysis presents a nuanced picture, suggesting that non-dyadic ties are often central and are vital for system connectivity, while dyadic edges remain essential to connect more peripheral nodes, an effect which is particularly pronounced for time-varying systems. Our work contributes to a better understanding of the structural organisation of systems with higher-order interactions.

We summarize the currently best known theoretical results for the single-source shortest paths problem for directed graphs with non-negative edge weights. We also point out that a recent result due to Cherkassky, Goldberg and Silverstein (1996) leads to even better time bounds for this problem than claimed by the authors.

Shortest path problems are a touchstone for evaluating the computing performance and functional range of novel computing substrates. Much has been published in recent years regarding the use of biocomputers to solve minimal path problems such as route optimisation and labyrinth navigation, but their outputs are typically difficult to reproduce and somewhat abstract in nature, suggesting that both experimental design and analysis in the field require standardising. This chapter details laboratory experimental data which probe the path finding process in two single-celled protistic model organisms, Physarum polycephalum and Paramecium caudatum, comprising a shortest path problem and labyrinth navigation, respectively. The results presented illustrate several of the key difficulties that are encountered in categorising biological behaviours in the language of computing, including biological variability, non-halting operations and adverse reactions to experimental stimuli. It is concluded that neither organism examined are able to efficiently or reproducibly solve shortest path problems in the specific experimental conditions that were tested. Data presented are contextualised with biological theory and design principles for maximising the usefulness of experimental biocomputer prototypes.

Finding the shortest paths from a single source to all other vertices is a fundamental method used in a variety of higher-level graph algorithms. We present three parallel friendly and work-efficient methods to solve this Single-Source Shortest Paths (SSSP) problem: Work front Sweep, Near-Far and Bucketing. These methods choose different approaches to balance the trade off between saving work and organizational overhead. In practice, all of these methods do much less work than traditional Bellman-Ford methods, while adding only a modest amount of extra work over serial methods. These methods are designed to have a sufficient parallel workload to fill modern massively-parallel machines, and select reorganizational schemes that map well to these architectures. We show that in general our Near-Far method has the highest performance on modern GPUs, outperforming other parallel methods. We also explore a variety of parallel load-balanced graph traversal strategies and apply them towards our SSSP solver. Our work-saving methods always outperform a traditional GPU Bellman-Ford implementation, achieving rates up to 14x higher on low-degree graphs and 340x higher on scale free graphs. We also see significant speedups (20-60x) when compared against a serial implementation on graphs with adequately high degree.

The algorithms-with-predictions framework has been used extensively to develop online algorithms with improved beyond-worst-case competitive ratios. Recently, there is growing interest in leveraging predictions for designing data structures with improved beyond-worst-case running times. In this paper, we study the fundamental data structure problem of maintaining approximate shortest paths in incremental graphs in the algorithms-with-predictions model. Given a sequence $σ$ of edges that are inserted one at a time, the goal is to maintain approximate shortest paths from the source to each vertex in the graph at each time step. Before any edges arrive, the data structure is given a prediction of the online edge sequence $\hatσ$ which is used to ``warm start'' its state. As our main result, we design a learned algorithm that maintains $(1+ε)$-approximate single-source shortest paths, which runs in $\tilde{O}(m η\log W/ε)$ time, where $W$ is the weight of the heaviest edge and $η$ is the prediction error. We show these techniques immediately extend to the all-pairs shortest-path setting as well. Our algorithms are consistent (performing nearly as fast as the offline algorithm) when predictions are nearly perfect, have a smooth degradation in performance with respect to the prediction error and, in the worst case, match the best offline algorithm up to logarithmic factors. As a building block, we study the offline incremental approximate single-source shortest-paths problem. In this problem, the edge sequence $σ$ is known a priori and the goal is to efficiently return the length of the shortest paths in the intermediate graph $G_t$ consisting of the first $t$ edges, for all $t$. Note that the offline incremental problem is defined in the worst-case setting (without predictions) and is of independent interest.

The single-source shortest path problem arises in many applications, such as roads, social applications, and computer networks. Finding the shortest path is challenging, especially for graphs that contain a large number of vertices and edges. In this work, we propose a novel hybrid method that first sparsifies a given graph by removing most edges that cannot form the shortest path tree and then applies a classical shortest path algorithm to the sparser graph. Removing all the edges that cannot form the shortest path tree would be expensive since it is equivalent to solving the original problem. Therefore, we propose an iterative bioinspired algorithm, namely the Physarum algorithm, as the first stage to sparsify the graph. We prove that the resulting sparser graph always contains the shortest path tree of the original graph. Next, a state-of-the-art algorithm such as Dijkstra's is applied to find the single-source shortest path on the resulting graph. The proposed method is therefore a two-stage hybrid algorithm and it computes the single-source shortest path exactly. We compare the accuracy and solution time of the proposed hybrid method against state-of-the-art implementation of Dijkstra's algorithm and the BFS algorithm on directed weighted and unweighted graphs, respectively, as a baseline. The results show that the proposed hybrid method achieves a significant speed improvement compared to the baseline.

Cellular Automata (CAs) are computational models that can capture the essential features of systems in which global behavior emerges from the collective effect of simple components, which interact locally. During the last decades, CAs have been extensively used for mimicking several natural processes and systems to find fine solutions in many complex hard to solve computer science and engineering problems. Among them, the shortest path problem is one of the most pronounced and highly studied problems that scientists have been trying to tackle by using a plethora of methodologies and even unconventional approaches. The proposed solutions are mainly justified by their ability to provide a correct solution in a better time complexity than the renowned Dijkstra's algorithm. Although there is a wide variety regarding the algorithmic complexity of the algorithms suggested, spanning from simplistic graph traversal algorithms to complex nature inspired and bio-mimicking algorithms, in this chapter we focus on the successful application of CAs to shortest path problem as found in various diverse disciplines like computer science, swarm robotics, computer networks, decision science and biomimicking of biological organisms' behaviour. In particular, an introduction on the first CA-based algorithm tackling the shortest path problem is provided in detail. After the short presentation of shortest path algorithms arriving from the relaxization of the CAs principles, the application of the CA-based shortest path definition on the coordinated motion of swarm robotics is also introduced. Moreover, the CA based application of shortest path finding in computer networks is presented in brief. Finally, a CA that models exactly the behavior of a biological organism, namely the Physarum's behavior, finding the minimum-length path between two points in a labyrinth is given.

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of Ω (n + D) rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where n is the number of nodes in the network and D is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs Õ(n^2/3 D^1/3 + n^5/6) rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing Õ(√n D) rounds and the second performing Õ(√n D^1/4 + n^3/5 + D) rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a (1+ε)-approximation Õ((√n D^1/4+D)/ε)-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.

Computing shortest paths is a fundamental primitive for several social network applications including socially-sensitive ranking, location-aware search, social auctions and social network privacy. Since these applications compute paths in response to a user query, the goal is to minimize latency while maintaining feasible memory requirements. We present ASAP, a system that achieves this goal by exploiting the structure of social networks. ASAP preprocesses a given network to compute and store a partial shortest path tree (PSPT) for each node. The PSPTs have the property that for any two nodes, each edge along the shortest path is with high probability contained in the PSPT of at least one of the nodes. We show that the structure of social networks enable the PSPT of each node to be an extremely small fraction of the entire network; hence, PSPTs can be stored efficiently and each shortest path can be computed extremely quickly. For a real network with 5 million nodes and 69 million edges, ASAP computes a shortest path for most node pairs in less than 49 microseconds per pair. ASAP, unlike any previous technique, also computes hundreds of paths (along with corresponding distances) between any node pair in less than 100 microseconds. Finally, ASAP admits efficient implementation on distributed programming frameworks like MapReduce.

In this paper we propose the first experimental study of the fully dynamic single-source shortest-paths problem on directed graphs with positive real edge weights. In particular, we perform an experimental analysis of three different algorithms: Dijkstra's algorithm, and the two output bounded algorithms proposed by Ramalingam and Reps in [30] and by Frigioni, Marchetti-Spaccamela and Nanni in [18], respectively. The main goal of this paper is to provide a first experimental evidence for: (a) the effectiveness of dynamic algorithms for shortest paths with respect to a traditional static approach to this problem; (b) the validity of the theoretical model of output boundedness to analyze dynamic graph algorithms. Beside random generated graphs, useful to capture the "asymptotic" behavior of the algorithms, we also developed experiments by considering a widely used graph from the real world, i.e., the Internet graph.