Problem Solving of graph correspondence using Genetics Algorithm and ACO Algorithm

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1 Proble Solving of graph correspondence using Genetics Algorith and ACO Algorith Alireza Rezaee, 1, Azizeh Ajalli 2 Assistant professor,departent of Mechatronics Engineering, Faculty of New Sciences and Technologies, University of Tehran,, P.O.Box ,Tehran, Iran 1 Meber of Young Researchers Club, Abhar Branch, Islaic Azad University, Abhar, Iran 2 Abstract: In this paper, new genetics and Ant colony optiization algorith for solving the proble of graph correspondence is presented. When using the genetics technique for the proble of graph correspondence, it is not easy to define the crossover operator. our attept will be to present a definition holding the integration of the population graph in a one-to-one correspondence. we present new and suitable definitions for the target function and a function giving score to a solution at the end of any cycle. We copare both algoriths and try to find their advantages and their shortcoings. Keywords: Graph Correspondence, Genetics Algorith, AOC Algorith I. INTRODUCTION The proble of graph correspondence has been noticed in a wide range of various fields and has becoe an iportant instruent for odeling and solving different probles. This proble is a NP-Coplete one and the tie required for solving it increases exponentially in ters of the nuber of the input graphs. Recently, any researches are conducted for iproveent of the efficiency of graph correspondence algoriths, both in tie and in eory consuption. Soe of these algoriths reduce teporal coplexities through applying liitations over the structure of the input graphs. Another ethod for reducing the coplexity of the process of correspondence is to present a special display for seeking and deleting useless routes of the search space. In these ethods, no liitation is applied on the structure of the input graphs. So they can be used in a larger doain of applications. One of the first articles in this context [1] suggests an algorith applying soe distortion on the input graphs and builds an interediate display on which the action of correspondence is done uch rapider. But [2] showed that the assuptions used by this ethod are not always true and this liits the doain of using this algorith. Another ethod reducing the search space considerably is Ullann algorith that uses backwards technique [3] and is still widely used. Messer in [4] copared this algorith with other algoriths and showed its high efficiency in a one-to-one correspondence. Another algorith using backwards techniques is Schidt& Druffel algorith stated in [5]. This algorith uses the inforation existing in the distance atrix. Another iportant algorith in graphical correspondence is Mckay's Nauty algorith [6]. This algorith is based on a series of transforations. These transforations transfor the graphs to a focused for and correspondence test is uch rapider in this focused for. Another algorith for the proble of graphical correspondence is presented in [7]. In this algorith, instead of trying to reduce teporal coplexity of the algorith, it is tried to reduce the coplexity in a case there is a constant odel graph and the graphs are constantly passed in front of this odel graph and the saples of the odel graph are extracted fro any target graphs. Since the odel graph is deeed constant in this algorith, so a table of inforation related to the odel graph and the various states of having it in the target graph can be built in a preprocessing anner and it can be used in the process of correspondence. Using techniques inspired fro the nature in the proble of graph correspondence is exained in several articles. [8] suggested a ethod for solving the proble of graph correspondence using the genetics algorith. In his suggestion, Copyright to IJIRSET

2 the suggested crossover function ay not work correctly for a one-to-one correspondence. The exaple he used for testing the algorith is the exaination of the reseblance between the solar syste and an ato. In his work, no coparison between the suggested algorith and other ethods has been done. In [9], using an ACO technique, a ethod for solving the proble of graph correspondence is suggested. In this article, definitions of the target function and also a function giving scores to a solution found at the end of any cycle are stated. [10] has also proposed an algorith using ACO. At first, we try to iprove genetics and ACO algoriths suggested for the proble of graph correspondence. For genetics algorith, we present a new definition of crossover operator. [11]For ACO algorith, we try to present exact definitions fitting the proble of graph correspondence for the target function and a function giving scores to a solution at the end of any cycle. Finally, we do a coparison between the two proposed ethods. [12] The structure of this article is as follows: first, we propose the genetics algorith suggested for solving the proble of correspondence. [13,14]Then, we describe the proposed ethod using ACO technique. In the next section, the results are exained and both algoriths are copared. Finally, conclusions are derived fro the works. [15,16] II. GENETICS ALGORITHM Genetics algorith is an optiizing algorith based on using evolutionary ethods. The ain idea of this algorith is that instead of a solution, a population of solutions is held for a proble of optiization and doing operations over this population, the new population is generated. Any solution is coposed of a set of genes and any gene shows one or ore properties fitting that solution. Genes can be of any type of data, but they are often considered as binary ones. Any solution fors a vector of binary values although except the vector, other data structures can be used (as it is done for solving the proble of graph correspondence and instead of using a vector, we use a two diensional atrix). Any eber of this population can pair based on the closeness to a target function and produce soe offspring. These offspring can be a part of the new population of the solutions. After all reproductions are perfored, individuals (solutions) being ore beneficial to the target function than other solutions are returned as the final solution of the proble. [17] Genetics algorith has good capabilities in solving probles of discrete optiization. Naturally, genetics algoriths are suitable for solving those probles having any optial points and in addition, having this property that sub-solutions can be locally cobined and a better general solution is obtained. This is done using the crossover function. [18] In the proble of graph correspondence, a correspondence function is defined identifying the relation between nodes of the target graph and those of the odel graph. This correspondence function is shown as a two diensional binary atrix M when used in Genetics algorith. Any row of this atrix shows a node of the target graph ( t i ) and any of its coluns shows a node of the odel graph ( b j ). If the eleent M ij of this atrix is one, then f ( ti ) will be equal to b. Only an eleent of a row can have a non-zero value and other eleents ust be zero, because any node of j the target graph can only correspond to a node of the odel graph. Now we need to define utation and crossover operators in this representation of the proble of graph correspondence. These definitions ust perfor such that the integration of the atrix holds, i.e. after applying the; any node of the target graph only corresponds to a node of the odel graph. For utation, we use the shift of two rows of the atrix, i.e. in utation of a solution (a atrix), we replace nodes of the odel corresponded to two target nodes. [19] Definition of crossover function is not as easy as that of utation. Our definition fro crossover function is that first, we deterine a position between one to the nuber of the nodes of the target graph (the nuber of rows of the atrix). This position divides any of the two atrixes and also the atrix ust be divided to two parts. The new atrix takes any parts fro one of the two available atrixes. A proble we face is that the integration of the resulting atrix ay not hold and nodes fro the target graph correspond to soe nodes fro the odel graph. In this case, the crossover function replaces soe non-zero values of the colun having ore than one non-zero value with zero values of the coluns not having non-zero values. [20] Copyright to IJIRSET

3 III. ACO ALGORITHM One of the other algoriths inspired fro the nature we have selected for solving the proble of graphical correspondence is Ant Colony Optiization (ACO) algorith. [21] In any cycle of this algorith, any ant fors a coplete correspondence and the effect of pheroone eerges. This algorith uses a construction graph in solving the proble of graph correspondence. The construction graph is a graph the artificial ants leave their pheroone traces on its sides. The vertexes of this graph are various parts of the solution that every vertex can be selected by an ant and added to the solution constructed by then (vertexes and sides selected by then). [22] Suppose two graphs their reseblance ust be deterined are as G ( V, r V, re ) and G ( V, r V, r E ). A construction graph is a nondirected coplete graph in which there is a vertex for any )V V. a. Pheroone As we know, pheroone is the eans of counication aong ants leaved on the edges of construction graph by the. The aount of pheroone on an edge ), v ) is denoted by T ), v ) and shows the utility of the correspondence u with and v with v. When selecting a new vertex by an ant, a side lead to the selection of that vertex takes a value that shows the content of the pheroone leaved by the ant on that side. When we construct a correspondence graph, those vertexes corresponded to each other in the correspondence graph are also ore likely to correspond to each other in the new correspondence graph, of course if the new correspondence graph has taken vertexes fro the previous correspondence graph. In other words, the ore correspondent vertexes the new correspondence graph has, the higher the utility and the probability of the selection of the reaining vertexes of the previous correspondence graph that correspond to each other but have not appeared in the new correspondence graph. b. Construction of a correspondence by an ant In any cycle, any ant constructs a correspondence and this correspondence states that which vertexes correspond to which vertexes of the second graph.[23] Any ant starts fro a null correspondence and adds pairs iteratively to it. These pairs are selected fro the set { ) V V }, where is the set of the pairs selected before. The probability of selecting the pair ) cand is as follows: [ T ( v, v ') cand Where: u' [ T. [ h v' u'.[ h v' (1) T ) is the pheroone factor and is equal to the set of all traces of pheroone that are between the new candidate (, u ) selected before and added to, i.e: T u') u and any pair v ) v') T u'), v') (2) If is equal to 0, i.e. when we want to select the first pair, we ignore the effect of pheroone factor. For this, we put T ) equal to 1. In this state, the probability is only related to the heuristic factor. ) is the heuristic factor that causes those pairs to be selected that increase the score function ore. h and are two paraeters deterining the iportance of any above paraeters. Ants stop selection and adding new pairs to the correspondence when no pair increases the score function directly or when the score function is not increased in the three last iterations. [24,25] Copyright to IJIRSET

4 c. The stage of updating pheroone For siulating vaporization of pheroone, any trace of pheroone where is the rate of vaporization and is in the range of 0 and 1. T ), v ) is ultiplied by (1- ), The best ant of the cycle leaves its pheroone. Suppose k is the best correspondence copared to the score function that is constructed during this cycle. If there are soe correspondences with identical scores, one of the is randoly selected. Suppose k is the correspondence of the current cycle and best is the best correspondence constructed fro the beginning of the progra to now (consider also the current cycle). The content of the pheroone leaved has an inverse relation with the difference between the score k and the score best, i.e. it is equal to: 1 (1 score( best) score( k )) This content of pheroone is leaved on any edge ( ), v)) linking two different pairs ) and v ) of k. IV. RESULTS ACO algorith highly depends on its paraeters and aong the paraeters, pheroone factor ( ) and the factor of the rate of vaporization ( ) have the highest effects. In this algorith, when or increase, the ants reach a score of correspondence. For exaple, consider the following figures: correspondence ore quickly although this decreases the The nuber of cycles Figure 1: we have changed the values of, where is constant. The result is that increasing, correspondence is obtained ore quickly, but with a lowers score. Copyright to IJIRSET

5 The nuber of cycles Figure 2: we have changes the values of, where is constant. The result is that increasing, the correspondence is obtained ore quickly, but with a lower score. In coparison between genetics algorith and ACO algorith, the output results highly depend on the paraeters of ACO algorith. For exaple, consider that =0.1 and =1. In these conditions, ACO algorith finds the answer uch quicker than genetics algorith. The following figure shows this result. Genetics Algorith The nuber of nodes Figure 3: Given =0.1 and =1, ACO algorith works quicker than genetics algorith On the other hand, for exaple suppose that =0.05 and =0.5. In these conditions, Genetics algorith works quicker than ACO algorith. The following figure shows this result. Copyright to IJIRSET

6 Genetics Algorith ACO Algorith Figure 4: The nuber of nodes given =0.05 and =0.5, genetics algorith works quicker than ACO algorith. Another proble is the nuber of the nodes existing in the graph. If the size of the graphs increase (in our experients, ore than 150 nodes), then regardless of the paraeters of ACO algorith, genetics algorith works better than ACO algorith. For exaple, consider the above case ( =0.05 and =0.5). in this case, when the nuber of nodes is low, ACO algorith works quicker (as it is shown in figure 4), but when the nuber of nodes is high, genetics algorith finds the answer quicker. Genetics Algorith ACO Algorith The nuber of nodes Figure 5: Efficiency of both algoriths with increasing the nuber of nodes. The vertical axis shows the operation tie. First, ACO algorith works better, but gradually increasing the nuber of nodes (ore than 150 nodes), the operation tie of genetics algorith becoes less than that of ACO algorith. V. CONCLUSIONS In this paper, we first presented a genetics algorith and an ACO algorith for the proble of graph correspondence. We stated a new definition fro crossover operator in genetics algorith that holds the integration of the population graph. In ACO algorith, we presented new definitions for the target function and a function giving scores to a solution at the end of any cycle. In the next step, we copared both algoriths presented. ACO algorith is very sensitive to the paraeters and Copyright to IJIRSET

7 , and its operation tie and also the quality of the solution it finds (the score it gains) depends on these paraeters. This relation is presented in the section, epirical results and the coparison of both algoriths, and also in figures 1, 2, 3 and 4. If the size of these graphs increases (for exaple ore than 150 nodes), regardless of the paraeters of ACO algorith, genetics algorith works better than ACO algorith (it obtains correspondence in less tie). REFERENCES [1] D.G. Corneil, C.C. Gotlieb, An efficient algorith for graph isoorphis, Journal of the Association for Coputing Machinery, 17, pp , [2] R. Mathon, Saple graphs for isoorphis testing, Congressus Nuerantiu, 21, pp , [3] M. Dorigo and G. Di Caro. The Ant Colony Optiization Meta-heuristic. In D. Corne, M. Dorigo, and F. Glover, editors, New Ideas in Optiization. McGraw Hill, London, UK, pages 11-32, [4] B. T. Messer, Efficient Graph Matching Algoriths for Preprocessed Model Graphs, Ph.D. Thesis, Inst. of Cop. Science and Appl. Matheatics, University of Bern, [5] T. St utzle and H.H. Hoos. MAX MIN Ant Syste. Journal of Future Generation Coputer Systes, 16: ,2000. [6] B.D. McKay, Practical Graph Isoorphis, Congressus Nuerantiu, 30, pp , [7] H. Bunke, B.T. Messer, Efficient Attributed Graph Matching and its Application to Iage Analysis, in Iage Analysis and Processing. (C. Braccini, L. De Floriani, G. Vernazza, eds), Springer, Berlin Heidelberg, pp , [8] Tesse, Bjørnar, Genetic Algoriths for Analogical Mapping. In Proc. of Intl. Conf. on Evolutionary Coputation (ICEC `98), pp , Anchorage, Alaska. IEEE [9] Olfa Saoud, Sébastien Sorlin, Christine Solnon, and Khaled Ghédira. Ant Algorith for the Graph Matching Proble. 5th European Conference on Evolutionary Coputation in Cobinatorial Optiization. April [10] Olfa Saoud, Sébastien Sorlin, Christine Solnon, and Khaled Ghédira. A oparative Study of Ant Colony Optiization and Reactive Search for Graph Matching Probles. 6th European Conference on Evolutionary Coputation in Cobinatorial Optiization [11] H. Kälviäinen E. Oja, Coparisons of Attributed Graph Matching Algoriths for Coputer Vision, Lappenraantra University of Technology, Departent of Inforation Technology, Research Report 18, [12] T. Miyazaki, The coplexity of McKay's canonical labeling algorith, in Groups and Coputation, II (L. Finkelstein and W.M. Kantor, eds.), Aer. Math. Soc., Providence, RI, pp , [13] X. Jiang, H. Bunke, Including geoetry in graph representations: a quadratic tie isoorphis algorith and its applications, in Perner, P., Wang, P., Rosenfeld, A. (eds.): Advances in Structural and Syntactic Pattern Recognition, Springer Verlag, Lecture Notes in Coputer Science 1121, 1996, [14] S. Sorlin and C. Solnon. Reactive Tabu Search for Measuring Graph Siilarity. to appear in 5th IAPR Workshop on Graph-based Representations in Pattern Recognition (GbR 2005), LNCS, Springer Verlag, [15] R. Battiti and M. Protasi. Reactive Local Search for the Maxiu Clique Proble. Algorithica, Springer-Verlag, (29), , [16] P. Foggia, C. Sansone, M.Vento, A Database of Graphs for Isoorphis and Sub-Graph Isoorphis Bencharking, Proc. of the 3rd IAPR TC-15 International Workshop on Graph-based Representations, Italy, [17] H. Bunke, M.Vento, Bencharking of Graph Matching Algoriths Proc. 2nd IAPRTC15 Workshop on Graph-based Representations, Handhorf, [18] L.P. Cordella, P. Foggia, C. Sansone, M. Vento, Evaluating Perforance of the VF Graph Matching Algorith, Proc. of the 10th International Conference on Iage Analysis and Processing, IEEE Coputer Society Press, pp , [19] P. Foggia C. Sansone, M.Vento, An Iproved Algorith for Matching Large Graphs, Proc. of the 3rd IAPR-TC-15 International Workshop on Graph-based Representations, Italy, [20] J.R. Ullann, An Algorith for Subgraph Isoorphis, Journal of the Association for Coputing Machinery, vol. 23, pp , [21] D. Conte, P. Foggia, C. Sansone, and M. Vento. Thirty years of graph atching in pattern recognition. International Journal of Pattern Recognition and Artificial Intelligence, 18(3): , [22] P. Chapin and C. Solnon. Measuring the siilarity of labeled graphs. 5th International Conference on Case-Based Reasoning (ICCBR). Lecture Notes in Coputer Science - Springer Verlag, [23] H. Bunke and X. Jiang. Graph atching and siilarity. Volue Teodorescu, H-N, Mlynek, D.Kandel, A. Zierann, H-J. (ds.): Intelligent Systes and Interfaces, chapter 1, [24] M. Boeres, C. Ribeiro, and I. Bloch. A Randoized Heuristic for Scene Recognition by Graph Matching. WEA 2004, , [25] R. Abauen, S. Fischer, and H. Bunke. Graph Edit Distance with Node Splitting and Merging, and Its Application to Diato Identification. IAPR-TC15 Wksp on Graph-based Representation in Pattern Recognition, LNCS, Springer Verlag, , Copyright to IJIRSET

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