Source Vertex Research Articles

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

We infer upper and lower bounds on the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several $n$-vertex heteropolygonal Archimedean lattices $\Lambda$. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.

Indoor Navigation With Virtual Graph Representation: Exploiting Peak Intensities of Unmodulated Luminaries

The ubiquitous luminaries provide a new dimension for indoor navigation, as they are often well-structured and the visible light is reliable for its multipath-free nature. However, existing visible light-based technologies, which are generally frequency-based, require the modulation on light sources, modification to the device, or mounting extra devices. The combination of the cost-extensive floor map and the localization system with constraints on customized hardwares for capturing the flashing frequencies, no doubt, hinders the deployment of indoor navigation systems at scale in, nowadays, smart cities. In this paper, we provide a new perspective of indoor navigation on top of the virtual graph representation. The main idea of our proposed navigation system, named PILOT, stems from exploiting the peak intensities of ubiquitous unmodulated luminaries. In PILOT, the pedestrian paths with enriched sensory data are organically integrated to derive a meaningful graph, where each vertex corresponds to a light source and pairwise adjacent vertices (or light sources) form an edge with a computed length and direction. The graph, then, serves as a global reference frame for indoor navigation while avoiding the usage of pre-deployed floor maps, localization systems, or additional hardwares. We have implemented a prototype of PILOT on the Android platform, and extensive experiments in typical indoor environments demonstrate its effectiveness and efficiency.

Top-k Critical Vertices Query on Shortest Path

Shortest path query is one of the most fundamental and classic problems in graph analytics, which returns the complete shortest path between any two vertices. However, in many real-life scenarios, only critical vertices on the shortest path are desirable and it is unnecessary to search for the complete path. This paper investigates the shortest path sketch by defining a top- $k$ critical vertices ( $k$ CV) query on the shortest path. Given a source vertex $s$ and target vertex $t$ in a graph, $k$ CV query can return the top- $k$ significant vertices on the shortest path $SP(s,t)$ . The significance of the vertices can be predefined. The key strategy for seeking the sketch is to apply off-line preprocessed distance oracle to accelerate on-line real-time queries. This allows us to omit unnecessary vertices and obtain the most representative sketch of the shortest path directly. We further explore a series of methods and optimizations to answer $k$ CV query on both centralized and distributed platforms, using exact and approximate approaches, respectively. We evaluate our methods in terms of time, space complexity and approximation quality. Experiments on large-scale real-world networks validate that our algorithms are of high efficiency and accuracy.

UniWalk: Unidirectional Random Walk Based Scalable SimRank Computation over Large Graph

SimRank is an important measure of vertex-pair similarity according to the structure of graphs. Although progress has been achieved, existing methods still face challenges to handle large graphs. Besides huge index construction and maintenance cost, existing methods may require considerable search space and time overheads in the online SimRank query. In this paper, we design a Monte Carlo based method, UniWalk, to enable the fast top- $k$ SimRank computation over large undirected graphs. UniWalk directly locates the top- $k$ similar vertices for any single source vertex $u$ via $R$ sampling paths originating from $u$ , which avoids selecting candidate vertex set $\mathcal{C}$ and the following $O(|\mathcal{C}|R)$ bidirectional sampling paths. We also devise a path enumeration strategy to improve the SimRank precision by using path probabilities instead of path frequencies when sampling, a space-efficient method to reduce intermediate results, and a path-sharing strategy to lower the redundant path sampling cost for multiple source vertices. Furthermore, we extend UniWalk to existing distributed graph processing frameworks to improve its scalability. We conduct extensive experiments to illustrate that UniWalk has high scalability, and outperforms the state-of-the-art methods by orders of magnitude.

Efficient Information Flow Maximization in Probabilistic Graphs

Reliable propagation of information through large networks, e.g., communication networks, social networks or sensor networks is very important in many applications concerning marketing, social networks, and wireless sensor networks. However, social ties of friendship may be obsolete, and communication links may fail, inducing the notion of uncertainty in such networks. In this paper, we address the problem of optimizing information propagation in uncertain networks given a constrained budget of edges. We show that this problem requires to solve two NP-hard subproblems: the computation of expected information flow, and the optimal choice of edges. To compute the expected information flow to a source vertex, we propose the F-tree as a specialized data structure, that identifies independent components of the graph for which the information flow can either be computed analytically and efficiently, or for which traditional Monte-Carlo sampling can be applied independently of the remaining network. For the problem of finding the optimal edges, we propose a series of heuristics that exploit properties of this data structure. Our evaluation shows that these heuristics lead to high quality solutions, thus yielding high information flow, while maintaining low running time.

Oriented bipartite graphs with minimal trace norm

The trace norm of a digraph D is the trace norm of its adjacency matrix A (i.e. the sum of the singular values of A). In particular, when G is a graph then its trace norm is the well-known energy of G introduced by I. Gutman in 1978. In this paper, we address the following problem: among all orientations of a given bipartite graph, which has minimal trace norm? We show that the minimal trace norm is attained in sink-source orientations (i.e. orientations in which every vertex is a sink vertex or a source vertex). Consequently, if is a set of connected bipartite graphs with minimal energy G, then the sink-source orientations of G have minimal trace norm among all orientations of graphs in . This is a consequence of our main result which relates the energy of a graph with the trace norm of its orientations. As another application we find the extremal values of the trace norm among all orientations of unicyclic graphs.

Superbubbles, Ultrabubbles, and Cacti.

A superbubble is a type of directed acyclic subgraph with single distinct source and sink vertices. In genome assembly and genetics, the possible paths through a superbubble can be considered to represent the set of possible sequences at a location in a genome. Bidirected and biedged graphs are a generalization of digraphs that are increasingly being used to more fully represent genome assembly and variation problems. In this study, we define snarls and ultrabubbles, generalizations of superbubbles for bidirected and biedged graphs, and give an efficient algorithm for the detection of these more general structures. Key to this algorithm is the cactus graph, which, we show, encodes the nested decomposition of a graph into snarls and ultrabubbles within its structure. We propose and demonstrate empirically that this decomposition on bidirected and biedged graphs solves a fundamental problem by defining genetic sites for any collection of genomic variations, including complex structural variations, without need for any single reference genome coordinate system. Further, the nesting of the decomposition gives a natural way to describe and model variations contained within large variations, a case not currently dealt with by existing formats [e.g., variant cell format (VCF)].

Path-Value Functions for Which Dijkstra’s Algorithm Returns Optimal Mapping

Dijkstra’s algorithm (DA) is one of the most useful and efficient graph-search algorithms, which can be modified to solve many different problems. It is usually presented as a tool for finding a mapping which, for every vertex v, returns a shortest-length path to v from a fixed single source vertex. However, it is well known that DA returns also a correct optimal mapping when multiple sources are considered and for path-value functions more general than the standard path-length. The use of DA in such general setting can reduce many image processing operations to the computation of an optimum-path forest with path-cost function defined in terms of local image attributes. In this paper, we describe the general properties of a path-value function defined on an arbitrary finite graph which, provably, ensure that Dijkstra’s algorithm indeed returns an optimal mapping. We also provide the examples showing that the properties presented in a 2004 TPAMI paper on the image foresting transform, which were supposed to imply proper behavior of DA, are actually insufficient. Finally, we describe the properties of the path-value function of a graph that are provably necessary for the algorithm to return an optimal mapping.

Modelling simultaneous broadcasting by level-disjoint partitions

Simultaneous broadcasting of multiple messages from the same source vertex in synchronous networks is considered under restrictions that each vertex receives at most one message in a unit time step, every received message can be sent out only in the next time step, no message is sent to already informed vertices. The number of outgoing messages is unrestricted, messages have unit length, and we assume full-duplex mode. In Gregor et al. (2015), we developed a concept of level-disjoint partitions to study simultaneous broadcasting under this model. In this work, we consider the optimal number of level-disjoint partitions. We also provide a necessary condition in terms of eccentricity and girth on existence of k v-rooted level-disjoint partitions of optimal height. In particular, we provide a structural characterization of graphs admitting two level-disjoint partitions with the same root.

Connectivity algorithm with depth first search (DFS) on simple graphs

This paper discusses an algorithm to detect connectivity of a simple graph using Depth First Search (DFS). The DFS implementation in this paper differs than other research, that is, on counting the number of visited vertices. The algorithm obtains s from the number of vertices and visits source vertex, following by its adjacent vertices until the last vertex adjacent to the previous source vertex. Any simple graph is connected if s equals 0 and disconnected if s is greater than 0. The complexity of the algorithm is O(n2).

IBAS: Index Based A-Star

The A-star algorithm is an efficient classical algorithm for solving the shortest path problem. The efficiency of the algorithm depends on the evaluation function, which is used to estimate the heuristic value of the shortest path from the current vertex to the target. When the vertex coordinates are known, the heuristic value of the shortest path is usually generated by the distance. In this paper, we present an Index Based A-Star algorithm (IBAS), which aims to solve the shortest path problem in a weighted directed acyclic graph with unknown vertex coordinates. This paper constructs three indexes for each vertex, i.e., the earliest arrival index, reverse earliest arrival index, and latest arrival index. We can compute the lower bound and the upper bound of the shortest distance from the source vertex to the target based on the three indexes and prune the intermediate vertices which are not in shortest path according to the lower and upper bounds. The IBAS algorithm not only makes use of the earliest arrival index to construct the evaluation function of the A-star algorithm but also utilizes the three indexes to prune useless vertices, so as to improve the performance of the algorithm. Compared with the A-star algorithm, the additional time complexity and space complexity of the IBAS algorithm are $O(|V|+|E|)$ and $O(|V|)$ , respectively. A real road network and benchmark datasets with large-scale network are selected to verify the performance of IBAS. Experimental results verify the effectiveness of the proposed algorithm.

Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal

Let $G$ be a directed graph with $n$ vertices, $m$ edges, and a designated source vertex $s$. We address the problem of single-source reachability (SSR) from $s$ in the presence of failures of vertices/edges. We show that for every $k\geq1$, there is a subgraph $H$ of $G$ with at most $2^kn$ edges that preserves the reachability from $s$ even after the failure of any $k$ edges. Formally, given a set $F$ of $k$ edges, a vertex $v\in V(G)$ is reachable from $s$ in $G\setminus F$ if and only if $v$ is reachable from $s$ in $H\setminus F$. We call $H$ a $k$-fault tolerant reachability subgraph ($\textsc{$k$-FTRS}$). We also prove a matching lower bound of $\Omega(2^kn)$ edges for such subgraphs that holds for all $n,k$ with $2^k\leq n$. Our results extend to vertex failures without any extra overhead. The construction of ${$k$-FTRS}$ is interesting from several different perspectives. From the Graph theory perspective it reveals a separation between SSR and single-source shortest paths (SSSP) in directed grap...

Dynamic SPT Increment Algorithm for Aircraft Booking Based on Hadoop

In the civil aviation industry, the number of flights and the number of airports have been increasing. Airlines recommendation service which to provide customers with more efficient and shorter routes has become one of the effective measures for improving the service quality of aviation. Actually, Airports and air-lines can be formed a weighted directed graph (for example, we use plane fares to represent the weights on the edges of the graph, and represent the vertices in the graph by the airports), and Airlines recommendation is to help user to find out the shortest path from the source vertex to the target vertices of the weighted directed graph. The shortest path search can be done by using the Dijkstra incremental optimization algorithm[1]. However, as the incremental part become larger and larger, the time spent on shortest path computation will be growing rapidly. In order to shorten the computing time, this paper proposes a dynamic SPT increment algorithm based on Hadoop platform. By the Hadoop efficient distributed platform[2], using MapReduce to take a dynamic search for the incremental part of the SPT. Finally, regenerates the new shortest path tree to improve the algorithm efficiency and performance.

Scalable parallel graph algorithms with matrix–vector multiplication evaluated with queries

Graph problems are significantly harder to solve with large graphs residing on disk compared to main memory only. In this work, we study how to solve four important graph problems: reachability from a source vertex, single source shortest path, weakly connected components, and PageRank. It is well known that the aforementioned algorithms can be expressed as an iteration of matrix–vector multiplications under different semi-rings. Based on this mathematical foundation, we show how to express the computation with standard relational queries and then we study how to efficiently evaluate them in parallel in a shared-nothing architecture. We identify a common algorithmic pattern that unifies the four graph algorithms, considering a common mathematical foundation based on sparse matrix–vector multiplication. The net gain is that our SQL-based approach enables solving “big data” graph problems on parallel database systems, debunking common wisdom that they are cumbersome and slow. Using large social networks and hyper-link real data sets, we present performance comparisons between a columnar DBMS, an open-source array DBMS, and Spark’s GraphX.

Tractability, hardness, and kernelization lower bound for and/or graph solution

And/or graphs are well-known data structures with several applications in many fields of computer science, such as Artificial Intelligence, Distributed Systems, Software Engineering, and Operations Research. An and/or graph is an acyclic digraph G containing a single source vertex s, where every vertex v∈V(G) has a label f(v)∈{ and,or }. In an and/or graph, (weighted) edges represent dependency relations between vertices: a vertex labeled and depends on all of its out-neighbors, while a vertex labeled or depends on only one of its out-neighbors. A solution subgraph H of an and/or graph G is a subdigraph of G containing its source vertex and such that if an and-vertex (resp. or-vertex) is included in H then all (resp. one) of its out-edges must also be included in H. In general, solution subgraphs represent feasible solutions of problems modeled by and/or graphs. The optimization problem associated with an and/or graph G consists of finding a minimum weight solution subgraphH of G, where the weight of a solution subgraph is the sum of the weights of its edges. Because of its wide applicability, in this work we develop a multivariate investigation of this optimization problem. In a previous paper (Souza et al., 2013) we have analyzed the complexity of such a problem under various aspects, including parameterized versions of it. However, the main open question has remained open: Is the problem of finding a solution subgraph of weight at most k (where k is the parameter) in FPT? In this paper we answer negatively to this question, proving the W[1]-hardness of the problem, and its W[P]-completeness when zero-weight edges are allowed. We also show that the problem is fixed-parameter tractable when parameterized by the tree-width, but it is W[SAT]-hard with respect to the clique-width and k as aggregated parameters. In addition, we show that when the out-edges of each or-vertex have all the same weight (a condition very common in practice), the problem becomes fixed-parameter tractable by the clique-width. Finally, using a framework developed by Bodlaender et al. (2009) and Fortnow and Santhanam (2011), based upon the notion of compositionality, we show that the tractable cases do not admit a polynomial kernel unless NP⊆coNP∕poly, even restricted to instances without or-vertices with out-degree greater than two.

Scalable Single Source Shortest Path Algorithms for Massively Parallel Systems

We consider the single-source shortest path (SSSP) problem: given an undirected graph with integer edge weights and a source vertex $v$ , find the shortest paths from $v$ to all other vertices. In this paper, we introduce a novel parallel algorithm, derived from the Bellman-Ford and Delta-stepping algorithms. We employ various pruning techniques, such as edge classification and direction-optimization, to dramatically reduce inter-node communication traffic, and we propose load balancing strategies to handle higher-degree vertices. These techniques are particularly effective on power-law graphs, as demonstrated by our extensive performance analysis. In the largest tested configuration, an R-MAT graph with $2^{38}$ vertices and $2^{42}$ edges on 32,768 Blue Gene/Q nodes, we have achieved a processing rate of three Trillion Edges Per Second (TTEPS), a four orders of magnitude improvement over the best published results.

High Performance Graph Processing with Locality Oriented Design

Many distributed graph processing frameworks have emerged for helping doing large scale data analysis for many applications including social network and data mining. The existing frameworks usually focus on the system scalability without consideration of local computing performance. We have observed two locality issues which greatly influence the local computing performance in existing systems. One is the locality of the data associated with each vertex/edge. The data are often considered as a logical undividable unit and put into continuous memory. However, it is quite common that for some computing steps, only some portions of data (called as some properties) are needed. The current data layout incurs large amount of interleaved memory access. The other issue is their execution engine applies computation at a granularity of vertex. Making optimization for the locality of source vertex of each edge will often hurt the locality of target vertex or vice versa. We have built a distributed graph processing framework called Photon to address the above issues. Photon employs Property View to store the same type of property for all vertices and edges together. This will improve the locality while doing computation with a portion of properties. Photon also employs an edge-centric execution engine with Hilbert-Order that improve the locality during computation. We have evaluated Photon with five graph applications using five real-world graphs and compared it with four existing systems. The results show that Property View and edge-centric execution design improve graph processing by 2.4X.

Source-wise round-trip spanners

In this paper, we study a new type of graph spanners, called source-wise round-trip spanners. Given any source vertex set S⊆V in a directed graph G(V,E), such a spanner (with stretch k) has the property that the round-trip shortest path distance from any vertex s∈S to any vertex v∈V is at most k times of their round-trip distance in G. We present an algorithm to construct the source-wise round-trip spanners with stretch (2k+ϵ) and size O((k2/ϵ)ns1/klog⁡(nw)) where n=|V|,s=|S| and w is the maximum edge weight. The result out-performs the state-of-the-art traditional round-trip spanners when the source vertex set S has small cardinality, and at the same time, it matches the traditional spanners when S is the whole vertex set V.

Development learning algorithm of mobile robot to detect obstacles in confined space

XXI century – the age of high technology. Now our life is surrounded by information technology: everyone has a cell phone, computer and other appliances (sometimes called gadgets). Themselves without noticing, we increasingly use them in their lives. Already it reached the point that we begin to use the robots for their own purposes. In less than ten years, each will have their own personal mobile robot. Properly programmed robot can perform many tasks. One of the common problems encountered in the design of robots, is the problem of developing an effective obstacle avoidance algorithm. Robot during their movement should not get stuck and stop until they have accomplished the task. To successfully perform this task in robot memory card should be stored workspace, which marked the places where there are obstacles to the robot. It is necessary to develop an algorithm that allows the robot to build a map of the area for the final (minimal) time. The map is a surface enclosed space that is divided into squares. The area of each square equal to the area of the robot base or square robot step length. To study the space used by an algorithm based on the algorithms of “Spiral” movement of the robot and the “movement along the wall”. There are many obstacles crawling algorithms. The most effective is Dijkstra's algorithm. This algorithm applies only to graphs with non-negative weights. Dijkstra's algorithm finds the shortest way to the top of the graph in order to remove them from a given source vertex. Central to Dijkstra's algorithm is that it is enough to compare the length of such paths

Computing Maximum Cardinality Matchings in Parallel on Bipartite Graphs via Tree-Grafting

It is difficult to obtain high performance when computing matchings on parallel processors because matching algorithms explicitly or implicitly search for paths in the graph, and when these paths become long, there is little concurrency. In spite of this limitation, we present a new algorithm and its shared-memory parallelization that achieves good performance and scalability in computing maximum cardinality matchings in bipartite graphs. Our algorithm searches for augmenting paths via specialized breadth-first searches (BFS) from multiple source vertices, hence creating more parallelism than single source algorithms. Algorithms that employ multiple-source searches cannot discard a search tree once no augmenting path is discovered from the tree, unlike algorithms that rely on single-source searches. We describe a novel tree-grafting method that eliminates most of the redundant edge traversals resulting from this property of multiple-source searches. We also employ the recent direction-optimizing BFS algorithm as a subroutine to discover augmenting paths faster. Our algorithm compares favorably with the current best algorithms in terms of the number of edges traversed, the average augmenting path length, and the number of iterations. We provide a proof of correctness for our algorithm. Our NUMA-aware implementation is scalable to 80 threads of an Intel multiprocessor and to 240 threads on an Intel Knights Corner coprocessor. On average, our parallel algorithm runs an order of magnitude faster than the fastest algorithms available. The performance improvement is more significant on graphs with small matching number.