Storage Model of Graph Based on Variable Collection

Transcription

1 Advanced Materials Research Online: ISSN: , Vols , pp doi: / Trans Tech Publications, Switzerland Storage Model of Graph Based on Variable Collection CHEN Zheng-sheng 1,a, LU Zhi-ping 1,b, LI Chang-gui 2,c 1 Institute of Surveying and Mapping, Information Engineering University, Zhengzhou , China, 2 Unit 73603, Nanjing , China a czsgeo@126.com, b ssscenter@126.com, c li_changgui@sina.com Keywords: variable collection; storage model; adjacent matrix; adjacent links; graph Abstract. As the traditional implements of graph are complicated in data structure and hard to maintain, or short in storage utilization and low computational efficiency, this paper designs one storage model of graph based on variable collection according to the object oriented method, and implements it with the variable collection data type that high level programing languages process. Comparing with adjacency matrix and list, analysis and cases show that this model is comprehensible and extensible with high calculation efficiency. Introduction In recent years, Graph has been widely used in many areas of computer, telecommunications engineering, chemistry etc. Due to the complicated data structure of graph, where any two nodes may be relevant, there are no physical locations in the storage area to represent the relationships of the nodes, namely it has no sequential storage structure. The commonly used method is by means of a two-dimension array (adjacency matrix) or an adjacency list data structure to represent graph [1-5]. Graph represented by adjacency matrix has simple, intuitive features, but the storage space is large, and the implementation of the efficiency is not high; the adjacency list representation use pointers contact the nodes with edges. This method is more complicated than an array. In order to obtain better computing and storage efficiency, different graph needs different adjacency list, therefore this method is less versatile. The difficulty to represent a graph is that the degree of each node is different, namely the node degree is variable. Currently the mainstream programming languages, such as C++, C #, Java, provide variable collection data structure, which is a proper way to represent graph. Common Storage Structure of Graph Graph. A graph [6-8] is an object consisting of two sets called its node (vertex) set and its edge set, commonly expressed as Eq. 1. G = ( V, E) (1) G represents a graph, V is node set of G, and E is the edge set of G. If the edge between node v i and v j has no direction, this edge is called undirected edge, which can be represent with non-order node pair (v i, v j ) and the graph is called undirected graph, else the edge called direction edge represented with ordered pair <v i, v j >, v i is called edge head, and v j is edge tail, and the graph is called directed graph or net whose edge has weight, as shown in Fig. 1. The weight of edge is a numerical value, which stand for actual meanings, such as road length in traffic route net of a road net. Fig. 1 Directed graph Fig. 3 The adjacency list table of directed graph All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-12/05/16,07:44:49)

2 Advanced Materials Research Vols Storage Structure of Graph. Graph is a non-linear data structure, whose nodes have a many to many relationship, and the degree of each node may be different. The difference between the maximum degree and the minimum degree may be very large. The commonly data structure of graph is adjacency matrix and adjacency list. 1) Adjacency Matrix Set the number of nodes of graph as n, each node in turn recorded as: v 0, v 1,, v n-1, the matrix of graph is a n by n dimension mathematical matrix. Its i row include edges whose starting point is v i, while its column j include edges whose end node is v j. The following Eq.2 is the definition of a graph matrix: w ij,exist<v i, v j > Matrix[ i][ j] = +, not exist <v i, v j > (2) For a common graph, the value of w ij is 1; for a net, the value of w ij is its weight. Each element of matrix will stand a storage space, so the space complexity is O(n 2 ). 2) Adjacent List An adjacency list[9] representation for a graph associates each node with its neighboring nodes by establishing a single linked list, where the i-th node is composed with related edges. For a edge of node v i, it only need store one other node of the edge, as shown in Fig. 2. The adjvex is the adjacent node of v i, and the weight is a numeric value, next represent next pointer of a node. Fig. 2 The node structure of links The following Fig. 3 is an adjacent list, which represents the directed graph shown in Fig. 1. In the adjacency list of undirected graph, the degree of node v i equals node count in the i-th link; while in the directed graph, the node count of the i-th link is only the out-degree of v i. In order to get the in-degree, you must loop over all the link tables. To get the in-degree expediently, you can also create an inverse adjacency list instead. Crossing chain list is another commonly used storage method. It can be regard as a combination of an adjacent and inverse adjacent list, where each edge has one node, and each node has one edge, too, as shown is Fig. 4. Fig. 4 Crossing chain of directed graph The in-degree and out-degree can easily calculated in the cross chain, and its time complexity equals the adjacent links. The Variable Collection Storage Mode In accordance with the idea of object-oriented modeling, the relations between nodes can be defined as edge object. A node of directed graph have an out-edge collection and an in-edge collection, which is a one-to-many relationship with edge, and the edge has one-to-two relationships with node, one stand for in-node, other is out-node, as shown in Fig. 5. Also the node object of undirected graph will use only one edge set, and it will be much simpler. The storage model of graph based on variable collection naturally expressed the relationship of nodes, which is much easier to understand. Because the edge count of each node in the same graph are not always equal, to implement this storage model need the program language has the ability to support such variable data structure or develop one by the user themselves. Currently the main

3 1458 Advanced Information and Computer Technology in Engineering and Manufacturing, Environmental Engineering program languages provided such data type, C++(Vector), Java(Set), C#(ICollection), JavaScript(Array), etc. They can automatically manage the element count and allocate or collect the storage of element. The following Fig. 6 shows an implementation of graph storage model based on variable collection use UML, where the variable collection data type implemented by List. Fig. 5 Directed graph and its variable collection storage model Fig. 6 The UML implementation of graph storage model based on variable collection The object of DirectedGraph have two collection attributes, which are Edges and Nodes; Each DirectedEdge object have two attribtes of DirectedNode object, which are InNode and OutNode; Each DirectedNode have two DirectedEdge object set, where one is InEdges, the other is OutEdges. Comparison and Application Comparison. Compare with Fig. 4 and Fig. 5, you can find that the variable collection and the adjacent list model are very similar with each other. They all link nodes with edges directly. The difference is that the storage management of the graph elements. The adjacent list model manages the storage itself, while the variable collection model hands this troublesome stuff to the software running environment. The matrix storage model uses a two-dimension array to store graph. Adjacent array is a n n matrix, whose space complexity is O(n 2 ), while the space complexity of adjacent list and the variable collection depend both on the number of nodes and edges, whose space complexity is O(n+ e). Because most of the applications of graph are sparse, in such cases, the adjacent array has a higher space complexity than the later ones. Application. Graph theory is wildly used many areas, not only it can reflect the two-dimension net systems of the reality, but also can show its topological structure and do quantitative analysis, for examples, finding shortest path on a city road network, topological sorting, grouping, coloring etc. The following is an example of path finding algorithm using different storage models of graph. Dijkstra's algorithm[10], conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree, which is widely recognized as a classical algorithm for solving the shortest path problem. We execute this algorithm in the same graph with different storage models, and compare their exculpating time. The operation system of this test is Windows XP SP3, with a 2.0GHz CPU, and the memory size of the test computer is 2 GB. Programing language is C#. Respectively, the number of nodes of graph is from 193 to 5194; the number of edges is from 306 to The test results show in Table 1 and Fig. 7.

4 Advanced Materials Research Vols Table 1 The elapsed time of three storage model of graph number nodes edges matrix(ms) adjacency list (ms) variable collection(ms) Fig. 7 The time elapses of three storage model of graph The test result shows that the computation time of the adjacency matrix increases exponentially as the road network expanded, while the consuming time of adjacency list and the variable collection increase linearly. And the later one is a little shorter and the former one. Summary Adjacency matrix is one value mapping method with graph, which the node and edge of graph are located by its relative location or index of matrix. So its performance is limited. And it provides storage cells for all possible edges, thus caused storage wasting especially for sparse graphs. And its calculation time is much longer than the other two models. Adjacency list and variable collection perform better than adjacency matrix both in storage space and execution time efficiency. Adjacency list construct the two-dimension graph with one-dimension data structure, whose advantage is the storage space can be accurately controlled, but its data structure is too complex to maintenance, and to achieve best performances, different graph needs different list. While the variable collection model use object-oriented modeling method to express graph in a natural, intuitive way, and it retain the structural and relationship characteristics of the graph, meanwhile its performances is no bad. Acknowledgement This work is Supported by National Science Foundation of China ( ) and National 863 Project of China (2013AA122501). References [1] Gross, Jonathan L., & Yellen, Jay, ed. Handbook of Graph Theory. CRC Press (2003). [2] Bang-Jensen, J.; Gutin, G. Digraphs: Theory, Algorithms and Applications. Springer (2000). [3] Harary, Frank. Graph Theory. Addison Wesley Publishing Company (1995). [4] Biggs, Norman. Algebraic Graph Theory (2nd ed.). Cambridge University Press (1993). [5] Duane A. Bailey. Java TM Structures: Data Structures in Java TM for the Principled Programmer [M]. Boston: WCB/McGraw-Hill, 1999: [6] Trudeau, Richard J. Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. Retrieved 8 (2012), p. 19.

5 1460 Advanced Information and Computer Technology in Engineering and Manufacturing, Environmental Engineering [7] Gross, Jonathan L.; Yellen, Jay. Graph Theory and Its Applications. CRC Press(1998). [8] Biggs, Norman. Algebraic Graph Theory (2nd ed.). Cambridge University Press. (1993). [9] Michael T. Goodrich and Roberto Tamassia. Algorithm Design: Foundations, Analysis, and Internet Examples. John Wiley & Sons (2002). [10] Dijkstra, Edsger; Thomas J. Misa, Editor. An Interview with Edsger W. Dijkstra. Communications of the ACM, 2010, 53 (8):

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