Source Vertex Research Articles

STEP: Sequence of time-aligned edge plots

We present sequence of time-aligned edge plots (STEP): a sequence- and edge-scalable visualization of dynamic networks and, more broadly, graph ensembles. We construct the graph sequence by ordering the individual graphs based on specific criteria, such as time for dynamic networks. To achieve scalability with respect to long sequences, we partition the sequence into equal-sized subsequences. Each subsequence is represented by a horizontal axis placed between two vertical axes. The horizontal axis depicts the order within the subsequence, while the two vertical axes depict the source and destination vertices. Edges within each subsequence are depicted as segmented lines extending from the source vertices on the left to the destination vertices on the right throughout the entire subsequence, and only the segments corresponding to the sequence members where the edges occur are drawn. By partitioning the sequence, STEP provides an overview of the graphs’ structural changes and avoids aspect ratio distortion. We showcase the utility of STEP for two realistic datasets. Additionally, we evaluate our approach by qualitatively comparing it against three state-of-the-art techniques using synthetic graphs with varying complexities. Furthermore, we evaluate the generalizability of STEP by applying it to a graph ensemble dataset from the architecture domain.

Distributed Single-Source Shortest Path with Only Local Relaxation

Finding the shortest path from a source vertex to any other vertices on a graph (single-source shortest path, SSSP) is used in a wide range of applications. With the rapid expansion of graph data volume, graphs are too large to be stored and processed in a standalone machine. Therefore, performing SSSP distributively in the computer clusters becomes an inevitable way. We found that the performance of existing distributed SSSP algorithms is limited by the communication cost between workers, which is caused by global relaxation. To eliminate the expensive communication cost, we propose an efficient distributed SSSP algorithm LR-SSSP that replaces global relaxation with local relaxation. Furthermore, we propose two optimizations, i.e., lazy synchronization and forward relaxation, to reduce invalid synchronization and communication. Our results show that LR-SSSP can achieve up to 6–20× speedup over the state-of-the-art Δ-stepping++ algorithm.

I/O Efficient Label-Constrained Reachability Queries in Large Graphs

Computing the reachability between two vertices in a graph is a fundamental problem in graph data analysis. Most of the existing works assume that the edges in the graph have no labels, but in many real application scenarios, edges naturally come with edge-labels, and label constraints may be placed on the edges appearing on a valid path between two query vertices. Therefore, we study the label-constrained reachability (LCR) queries in this paper, where we are given a source vertex s , a target vertex t , a label set Δ, and the goal is to check whether there exists any path from s to t such that all the labels of edges on the path belong to Δ. A plethora of methods have been proposed in the literature to support the LCR queries. All these methods take the assumption that the graph is resident in the main memory of a machine. Nevertheless, the graphs in many real application scenarios are generally big and may not reside in memory. In these cases, existing methods suffer from serious scalability problem, i.e., result in huge I/O costs. Motivated by this, in this paper, we study the I/O efficient LCR query problem and aim to efficiently answer the LCR queries when the graph cannot fit in the main memory. To achieve this goal, we propose a reduction-based indexing approach. We introduce two elegant graph reduction operators which aims to reduce the size of the graph loaded in memory while preserving the LCR information among the remaining vertices. With these two operators, we devise an index named LCR-Index and propose algorithms to adaptively construct the index based on the available memory. Equipped with LCR-Index, we can answer a LCR query by only scanning the LCR-Index sequentially. Experiments demonstrate our query processing algorithm can handle graphs with billions of edges.

The Rabin numbers of enhanced hypercubes

The ω-Rabin number rω(G) and strong ω-Rabin number rω⁎(G) are two effective parameters to assess transmission latency and fault tolerance of an interconnection network G. As determining the Rabin number of a general graph is NP-complete, we consider the Rabin number of the enhanced hypercube Qn,k which is a variant of the hypercube Qn. For n≥k≥5, we prove that rω(Qn,k)=rω⁎(Qn,k)=d(Qn,k) for 1≤ω<n−⌊k2⌋; rω(Qn,k)=rω⁎(Qn,k)=d(Qn,k)+1 for n−⌊k2⌋≤ω≤n+1, where d(Qn,k) is the diameter of Qn,k. In addition, we present algorithms to construct internally disjoint paths of length at most rω⁎(Qn,k) from a source vertex to other ω (1≤ω≤n+1) destination vertices (not necessarily distinct) in Qn,k.

Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology

In the Multiagent Path Finding (MAPF for short) problem, we focus on efficiently finding non-colliding paths for a set of k agents on a given graph G, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule l, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our Fixed-Parameter Tractability (FPT) results. We show that MAPF is W[1]-hard with respect to k (even if k is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan l are fixed constants. On the positive side, we show an FPT algorithm for k+l. As we continue, the structure of G comes into play. We give an FPT algorithm for parameter k plus the diameter of the graph G. The MAPF problem is W[1]-hard for cliquewidth of G plus l while it is FPT for treewidth of G plus l.

Parameterized complexity of broadcasting in graphs

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge set, or by the size k of a vertex cover, or by k=n−t, where t is the input deadline for the broadcast protocol to complete.

Fault-tolerant unicast paths constructive algorithms in a family of recursive networks

ABSTRACT In parallel and distributed systems, interconnection networks and data center networks are two crucial networks. They are also crucial parts of high-performance computing and cloud computing services. With the rapid growth of high-performance computing and cloud computing, both networks are growing in size. At the same time, vertex faults are unavoidable. In computer/communication networks, unicast refers to one-to-one communication between the source vertex and the destination vertex. Fault-tolerant routing of vertices in networks is an important problem that has been widely discussed. In this paper, for a class of recursive networks with less than g -restrict connectivity fault vertices and each fault-free vertex having at least g fault-free neighbors, we provide a fault-tolerant unicast path design algorithm in this study. The algorithm’s correctness is then demonstrated and its time complexity is analyzed. In addition, the results obtained can be used for unknown and known networks, including the dragonfly network, DCell, and generalized DCell.

A method of secure network traffic routing based on specified criterias

Due to the implementation of new network services, the increase amount of data that need to be transmitted, and the use of networks in various sectors with diverse communication requirements, there is a need to develop new approaches to ensure the quality of such communications. Leading network equipment manufacturers and standardization organizations are developing new routing algorithms, resulting in the introduction of new routing protocols or improvements to existing ones. However, all these algorithms cover routing principles for general-purpose networks and do not consider the communication requirements of specialized networks. Therefore, the task arises to research optimization directions for network traffic routing, define optimization criteria, and further develop a method for secure network traffic routing based on the specified criteria. In this work, a routing method is proposed that takes into account the defined requirements when searching for the optimal route. In the case of dynamic routing, each router calculates the shortest routes to all other networks based on the shortest path search algorithm. This work defines a method for calculating metrics based on specified criteria and formally describes the algorithm for finding the shortest path. Quality of communication criteria is introduced, which will enable meeting communication requirements in specialized networks. Calculation methods for these criteria are demonstrated, and data collection methods for the calculation of specified criteria are determined. A formula for calculating metrics is proposed, which includes the possibility of selecting T-values and determining their numerical parameters to prioritize specific criteria. Default values for criteria are defined, and metric calculations are tested by default for different types of interfaces. After calculating metrics, the task reduces to finding the shortest paths in a weighted graph using an algorithm based on Dijkstra's algorithm. The proposed algorithm for finding the shortest path involves identifying the primary (shortest) and backup paths from a given source vertex to all other graph vertices. A formal description of the proposed algorithm is provided.

METHODOLOGY FOR PLANNING THE LOCATION OF SWITCHING NODES OF IP-TELEPHONY NETWORKS

The relevance of the topic is due to the high pace of development and implementation of new systems and processes in information structures and the lack of a methodology for determining the effectiveness of using new infocommunication technologies in them. Currently, there are no unified scientifically based methods for assessing the feasibility and effectiveness of using IP telephony technology. This is due to insufficient experience in the operation of telecommunication systems based on IP telephony technology and their significant difference from traditional telephone systems. It is indicated that for optimal planning of the location of switching nodes of IP-telephony networks, a methodology has been developed, consisting of the development of a mathematical basis for constructing the projected IP-telephony network, algorithmic implementation and estimation of the complexity of the implementation of this algorithm, which can be used in the design of real IP-telephony networks. The algorithmic implementation of the developed technique uses three arrays from among the nodes of the network, i.e. the first array contains labels with two values, the second array contains the distances - the current shortest distances from the specified vertex to the corresponding vertex, and the third array contains the vertex numbers. In turn, the algorithmic implementation depends on how the source vertex is found, how the set of unmarked vertices is stored, and how labels are updated. Keywords: Telecommunications network, IP-telephony technology, packet switching of flows, planning of the location of switching nodes, algorithmic implementation of the planning methodology, subscriber network, parameters of information systems, network parameters.

MITra: A Framework for Multi-Instance Graph Traversal

This paper presents MITra, a framework for composing multi-instance graph algorithms that traverse from multiple source vertices simultaneously over a single thread. Underlying MITra is a model of multi-instance traversal that uniformly captures traversal sharing across instances. Based on this, MITra provides a programming model that allows users to express traversals by declaring vertex ranks and specify computation logic via an edge function. It synthesizes multi-instance traversal algorithms from declared vertex ranks and edge functions adopted from classic single-instance algorithms, automatically sharing computation across instances and benefiting from SIMD. We show that MITra can generate multi-instance algorithms provably better than existing ones, while being more expressive than traditional frameworks. In addition to the ease of programming, we experimentally verify that MITra is on average an order of magnitude faster than approaches based on existing frameworks for common graph algorithms, and is comparable to the state-of-the-art highly optimized one-off algorithms.

Shortest Journeys in Directed Temporal Graphs

Consider a directed temporal graph [Formula: see text] with time ranges on the edges. There can be more than one range on an edge, and each range carries a positive traversal time. Let [Formula: see text] and let [Formula: see text] be the total number of time ranges in [Formula: see text]. We assume that [Formula: see text]. We study the problem of computing shortest journeys that start from a fixed source vertex [Formula: see text] within a given time interval [Formula: see text], where the cost of a journey is equal to the sum of traversal times of the edges on it at the times of crossing those edges. We can construct in [Formula: see text] time a data structure of size [Formula: see text] such that for any vertex [Formula: see text] and any time [Formula: see text], we can report in [Formula: see text] time the cost of the shortest journey that starts from [Formula: see text] within [Formula: see text] and arrives at [Formula: see text] no later than [Formula: see text]. The journey achieving the reported cost can be produced in time linear in its complexity.

METHOD OF ROUTING A GROUP OF MOBILE ROBOTS IN A FIXED NETWORK FOR SEARCHING THE MISSING OBJECTS IN A TECHNOLOGICAL DISASTER ZONE

Context. The relevance of the article is determined by the need for further development of models of collective behavior of systems with multi-agent structure construction endowed with intelligence that ensures synchronization of the joint efforts of various agents while achieving the goals set for the system. The method proposed in the article solves the problem of competition between different agents of a multi-agent system, which is important while performing search, rescue, and monitoring tasks in crisis areas of various origins.&#x0D; Objective is to develop a method for determining the sufficient population of a multi-agent system and the optimal routes of movement of its individual elements in a stationary network for the most complete examination of a technological disaster zone (any given zone based on a certain transport network).&#x0D; Method. We implemented the concept of a dynamic programming to search for all possible edge-simple longest paths connecting the directed subsets of vertices-sources and vertices-sinks in the structure of the model weighted directed graph. To this end, the modified Dijkstra method was applied. The modification comprises representing the weights of the arcs of the modeling directed graph with the negative values, which are further used in calculations according to the Dijkstra method. After finding the next edgesimple longest path, the arcs that make up it are fixed in the memory of the computer system (in the route plan) and removed from the graph structure, and the process is iteratively repeated. The search for paths takes place as long as the transitive closure between the vertices that are part of the specified subsets of source vertices and sink vertices is preserved. The developed method makes it possible to find such a set of traffic routes for the elements of the multi-agent system, which maximizes the area examined by them in a technological disaster zone (or the number of checked objects on the traffic routes) in one “wave” of the search and distributes the elements of a multi-agent system by routes that do not have common areas. A derivative of the application of the developed method is the determination of a sufficient population of a multi-agent system for effective search activities within the defined zone.&#x0D; Results. 1) A method of routing a group of mobile robots in a stationary network for searching the missing objects in a technological disaster zone has been developed. 2) The working expression of the Dijkstra method for searching in the structure of a network object (in the structure of a model graph) for the longest paths has been formalized. 3) We have suggested a set of indicators for a comprehensive evaluation of route plans of a multi-agent system. 4) The method has been verified on test problems.&#x0D; Conclusions. Theoretical studies and several experiments confirm the efficiency of the developed method. The solutions made using the developed method are accurate, which allows recommending it for practical use in determining in an automated mode route plans for multi-agent systems, as well as the required number of agents in such systems to perform the required amount of search tasks in a particular crisis area.

The Floyd‐Warshall all‐pairs shortest paths algorithm for disconnected and very sparse graphs

AbstractThe Floyd‐Warshall algorithm is the most popular algorithm for determining the shortest paths between all vertex pairs in a graph. It is a very simple and an elegant algorithm. However, for graphs without any negative weighted edges, using Dijkstra's shortest path algorithm for every vertex as a source vertex to produce all‐pairs shortest paths works significantly better than the Floyd‐Warshall algorithm, especially for large graphs. Furthermore, for graphs with negative weighted edges, with no negative cycle, in general Johnson's algorithm also performs better than the Floyd‐Warshall algorithm for large graphs. Johnson's algorithm first transforms the graph into a non‐negative one by using the Bellman‐Ford algorithm, then applies the Dijkstra's algorithm to the transformed graph. Thus, mainly, the Floyd‐Warshall algorithm is quite inefficient, especially for large graphs. In this paper, we show a simple improvement on the Floyd‐Warshall algorithm that will increases its efficiency, especially for very sparse graphs (i.e., the number of its edges is less than the number of its vertices), so it can be used instead of more complicated alternatives. We also show that our approach is also very effective for denser disconnected graphs. Since the new algorithm modifies the original Floyd‐Warshall algorithm, it is mainly aimed for directed graphs without negative cycles. Most programmers prefer to implement the Floyd‐Warshall algorithm over more complicated but more efficient alternatives for solving all‐pairs shortest path problems. In this work, we show that without the addition of any complicated data structures, the performance of the Floyd‐Warshall algorithm can be improved very easily. Our practical approach works even better than its alternatives for large sparse graphs.

A Robust Theory of Series Parallel Graphs

Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded µ-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called synchronized series-parallel graphs (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs. SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of counting complexity that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with n states and k counting complexity to a deterministic automaton with 2 n 2 states and kn counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.

Cache-Efficient Approach for Index-Free Personalized PageRank

Personalized PageRank (PPR) measures the importance of vertices with respect to a source vertex. Since real-world graphs are evolving rapidly, PPR computation methods need to be index-free and fast. Unfortunately, existing index-free methods suffer from cache misses. They follow the state-of-the-art algorithm that first performs the Forward Push (FP) phase and subsequently runs the random walk Monte-Carlo simulation (MC) phase. Although existing methods succeed in reducing cache misses in the FP phase, an inefficient data layout limits their performance improvement. Besides, existing methods have overlooked the importance of reducing cache misses in the MC phase. In this paper, we propose a cache-efficient approach that accelerates both FP and MC phases. In the FP phase, we first reorder the data layout with low overheads. Specifically, we utilize the Breadth First Search result so that vertices near the source vertex are co-located on the reordered data layout. We subsequently perform optimized FP, namely Distance-Extension Forward Push (DEFP). By preferentially proceeding FP around the source vertex, DEFP improves memory access locality. In the MC phase, we perform optimized MC, namely Vertex-Centric Random Walk (VCRW). VCRW aggregates random walks at each vertex to eliminate redundant memory access for repeatedly obtaining neighbor vertices. We prove that most of the random walks can be aggregated while maintaining accuracy guarantees. Experimental results show that the proposed method is up to 4.7x faster than existing index-free methods and outperforms the state-of-the-art index-oriented method under rigorous accuracy guarantees.

ECONOMICALLY EFFICIENT CACHE FRIENDLY ADJACENCY LIST FOR GRAPH PROCESSING ALGORITHMS

Graphs are a fundamental part of computer science and various scientific fields. They are used in numerous applications, from social networks to electronic design automation. Efficient graph representation is important for modern algorithms, especially in economical contexts, where resource-efficient implementations play a significant role by reducing the resource consumption. This is essential in environments where computing resources are a high priority. There are many representations of graphs as data structure. One of these types is adjacency list: it is collection of linked lists which shows the connection of source and destination vertices. Adjacency lists are more memory-efficient than adjacency matrices because they only allocate memory for vertices with at least one edge in the graph. However, traditional implementations of adjacency lists do not fully use the features provided by modern CPUs, such as the cache line. Our design of Adjacency List is cache friendly and is keeping the main structure of traditional implementation. The main idea is derived from C#’s dictionary implementation for hash tables, which uses one array combining all data that should be in linked lists. Our results show that cache friendly implementation of Adjacency list outperforms it in lookup and insertion operations.

Finding Minimum Production Time and Optimal Procedure for a Project When Step Times are Given by Constructive Real Numbers

In the production process, some jobs can only be done sequentially, while others can be done synchronously. Given the time needed to complete each job, an oriented graph with a source vertex and a sink vertex can be generated. Each route starting from the source vertex and ending at the sink vertex represents a way of completing the project. It is possible to find the minimum production time and the optimal production path when time for each job is given by a rational number. However, this paper proves that there does not exist a computer program that can always generate the minimum time and optimal path when the time needed for each job is a constructive real number.

Geodesic Tracks: Computing Discrete Geodesics With Track-Based Steiner Point Propagation.

This article presents a simple yet effective method for computing geodesic distances on triangle meshes. Unlike the popular window propagation methods that partition mesh edges into intervals of varying lengths, our method places evenly-spaced, source-independent Steiner points on edges. Given a source vertex, our method constructs a Steiner-point graph that partitions the surface into mutually exclusive tracks, called geodesic tracks. Inside each triangle, the tracks form sub-regions in which the change of distance field is approximately linear. Our method does not require any pre-computation, and can effectively balance speed and accuracy. Experimental results show that with 5 Steiner points on each edge, the mean relative error is less than 0.3 % for common 3D models used in the graphics community. We propose a set of effective filtering rules to eliminate a large amount of useless broadcast events. For a 1000K-face model, our method runs 10 times faster than the conventional Steiner point method that examines a complete graph of Steiner points in each triangle. We also observe that using more Steiner points increases the accuracy at only a small extra computational cost. Our method works well for meshes with poor triangulation and non-manifold configuration, which often poses challenges to the existing PDE methods. We show that geodesic tracks, as a new data structure that encodes rich information of discrete geodesics, support accurate geodesic path and isoline tracing, and efficient distance query. Our method can be easily extended to meshes with non-constant density functions and/or anisotropic metrics.

An Efficient Index-Based Approach to Distributed Set Reachability on Small-World Graphs

Set reachability query in directed graphs has a plethora of graph-based applications such as dependency analysis and graph centrality calculation. Given two sets <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> of source and target vertices, set reachability query needs to acquire all pairs <inline-formula><tex-math notation="LaTeX">$(s,t)$</tex-math></inline-formula> where <inline-formula><tex-math notation="LaTeX">$s{\in }S$</tex-math></inline-formula> , <inline-formula><tex-math notation="LaTeX">$t{\in }T$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> can reach <inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula> . The state-of-the-art approach distributed set reachability (DSR) investigates the set reachability query in a distributed environment and adopts a static graph-based index to enhance the query efficiency. Nevertheless, DSR needs to store the graph-based index in all partitions, which causes a huge space overhead. Furthermore, it cannot efficiently solve the negative query <inline-formula><tex-math notation="LaTeX">$(s,t)$</tex-math></inline-formula> where <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> cannot reach <inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula> , since DSR needs to traverse the whole reachable paths and becomes unable to efficiently reduce the computations. To alleviate these issues, we propose a novel multi-level 2-hop (ML2hop) index for the set reachability query in a distributed environment. Based on ML2hop, we further present a bi-directional query algorithm, called MLQA, to achieve efficient support for both positive and negative queries in Pregel-like systems. Generally, MLQA is equipped with the following three significant properties: (1) Low computation costs. It reduces redundant local computations in each partition by controlling the rounds of path traversals. (2) Low communication costs. It restricts the message exchange among different partitions within one single round with guaranteed accuracy of query results. (3) High parallelism. It adopts a bi-directional query technique for message propagation, achieving the better query efficiency than the forward-traversal query strategy utilized in DSR. Experimental results over several real-world graphs demonstrate that MLQA significantly outperforms the state-of-the-art algorithm by up to two orders of magnitude speedup.

Structural parameterizations of Tracking Paths problem

Given a graph G with source and destination vertices s,t∈V(G) respectively, Tracking Paths asks for a minimum set of vertices T⊆V(G), such that the sequence of vertices restricted to T encountered in each simple path from s to t is unique. The problem was proven NP-hard [3] and was found to admit a quadratic kernel when parameterized by the size of the desired solution [8]. Following recent trends, for the first time, we study Tracking Paths with respect to structural parameters of the input graph, parameters that measure how far the input graph is, from an easy instance. We prove that Tracking Paths admits fixed-parameter tractable (FPT) algorithms when parameterized by the following parameters: (i) size of a 2-dominating set, (ii) size of cluster vertex deletion set, and (iii) size of split deletion set. En route we also look at Vertex Cover parameterized by the size of edge clique cover and show it fixed-parameter tractable.