An Efficient Ant Colony Optimization Algorithm for Multiple Graph Alignment Tran Ngoc Ha Do Duc Dong, Hoang Xuan Huan Thai Nguyen University of Education, hatn84@gmail.com Vietnam National University - Hanoi, dongdoduc@vnu.edu.vn , huanhx@vnu.edu.vn Abstract - The Multiple Graph Alignment (MGA) is a new method to analyze the structure of biological molecules This method allows detect functional similarities in the structure of biological systems This article introduces an ant colony optimization algorithm combined with local search for optimal align multi-graph analysis of protein structures Experiment results showed that the new algorithm outperformed the other heuristic approach and existing evolutionary computing evolutionary algorithm called GAVEO Experiments show that it is more efficient than the greedy algorithm For NP-hard problems, there were many natural simulation approaches to find approximate solutions In particular, the experiments showed that the ant colony optimization (ACO) method is better than evolutionary algorithms in many typical problems [3, 4, 7] This article introduces an ant colony optimization algorithm incorporating local search to aligning the multi-graph called ACO-MGA The simulation results show that ACOMGA algorithm is more outstanding effective than the GAVEO and Greedy algorithms Keyword -Multiple Graph Alignment, label, Ant Colony Optimization, Local Search, Pheromone update rule I INTRODUCTION The multiple graph alignment techniques [14] are a useful tool to analyze the similarity of DNA sequences or proteins, thereby we can detect the similarity of different molecules based on genetics However, the functional similarities among the genes and proteins are closely related to the structure rather than sequential features [5.13] so it is necessary to develop new research approaches The rest of this article is organized as follows: Section mathematic defines the MGA problem and introduces the schema of ACO method New algorithm is introduced in Section 3, the experiment results which comparing the new algorithm with the GAVEO and Greedy algorithm are presented in Section The conclusions are presented in the last section There have been different proposed approaches to explore the structure similarities (see [2, 8-13, 16-18]), that mainly due to correct graphs matching technique and get the meaningful results when studying the functional evolution of heterogeneous molecules However, these methods are difficult to discover biological meaningful patterns that are stored approximately II MULTIPLE GRAPH ALIGNMENT PROBLEM AND RELATED WORKS A Multiple graph alignment problem Weskamp et al [15] proposed using the MGA problem to study protein characteristics, where graphs are used to approximately describe the binding pockets This approach is extended to analysis structure of biological molecules which include chemical compounds and protein binding sites by Fober et al [5] Mathematical definition of MGA problem is as follows (more details see [5]) Weskamp et al [15] firstly introduced the concept of multigraph alignment (MGA) in 2007; they used it to analyze protein active sites, and proposed a heuristic algorithm to find greed-based solutions In this approach, each binding pocket is modeled by a connected graph G(V, E) and the MGA problem is defined as follows Given a set of connected graphs G = {G1(V1, E1), , Gn(Vn, En)}, each vertex is labeled in a given label set and the weighted edges; in each graph, there are four operations: deleting a node, inserting a node, changing a label of a node and changing the weight of an edge Task of the MGA problem is aligning the nodes of the graphs in the set G to optimize a predefined objective function Multigraph Multigraph is a set of graphs G = {G1(V1,E1),…,Gn(Vn,En)}, where the graphs Gi(Vi,Ei) are connected graphs, node is labeled under a given set L, the weighted edges represent the distance between the vertices In the model of protein binding sites, the labels of the nodes can be: hydrogen-bond donor, acceptor, mixed donor/acceptor, hydrophobic aliphatic and aromatic In each graphs, there are edit operations which is mathematic defined as follow: MGA is the NP-hard problem (see [5.15]), the heuristic algorithms is only suitable for small size problems, so it is not suitable for real applications Fober et Definition On the graph G(V, E) of multigraph G there are edit operations: al [5] have extended the use of this problem for the structural analysis of biomolecules and have proposed an 978-1-4673-2088-7/13/$31.00 ©2013 IEEE 386 i) ii) iii) Insertion or deletion of a node: A node v ∈ V and all relationships with it (edges) can be deleted or inserted Change of the label of a node: The label ݈ሺݒሻ of a node ܸ ∈ ݒcan be changed by another label in set L Change of the weight of an edge: The weight w(e) of an edge e ∈ E can be changed depending on the different forms n s ( A) = ∑ ns ( a i ) + i =1 ∑ es( a i , a j ) (1) 1≤ i < j ≤ n Where ns is the assessment score of the suitability of the corresponding column and calculated by the expression (2): nsm a nsmm ns M = ∑ a i 1≤ j < k ≤ m nsdummy m nsdummy i Multiple Graph Alignments Give multigraph G ={G1(V1,E1),…,Gn(Vn,En)}, for each vertex sets Vi , we add to it a dummy node (denoted ⊥) that is not connected to the other nodes, an alignment of G is defined as follows l(a ij )=l(aki ) l(a ij ) ≠ l(aki ) (2) a ij = ⊥ , aki ≠⊥ a ij ≠⊥ , aki =⊥ and es evaluate the compatibility of the edge length and is calculated by the expression (3): Definition (Multigraph Alignment) Set { ⊆ܣV1 ∪ ሼ⊥ሽ} × … × {Vm ∪ ሼ⊥ሽ} is an alignment of multigraph G if and only if it verifies two conditions: esmm (aki ,akj ) ∈ Ek , (ali ,alj ) ∉ El a1i a1j (aki ,akj ) ∉ Ek , (ali ,alj ) ∈ El esmm es M , M = ∑ d klij ≤ ε (3) a j 1≤ k ε mm kl For all i=1,…,n and for ܸ ∈ ݒ , there exists exactly one a = (a1,…,an) ∈ ܣsuch that ݒൌ ܽ For each a = (a1,…,an) ∈ ܣ, there exists at least one ≤ i ≤ n such that ܽ ് ⊥ In the expression (3) ݀ ൌ หݓ൫ܽ ൯ െ ݓ൫ܽ ൯ห Five parameters (nsm, nsmm, nsdummy, esm, esmm) are reused as [15]: nsm = 1.0; nsmm = -5.0; nsdummy = -2.5; esm = 0.2; esmm =-0.1 Fig shows an alignment of four graphs, where dummy nodes are presented by square, labeled nodes are presented by circular Noting that in each graph, there is only one dummy node, but for ease of visualization, in the first and the fourth graph there are two dummy nodes, that means the nodes in the corresponding row are aligned with dummy nodes in these graphs The solution of MGA problem is an alignment that maximizing scoring function ݏሺܣሻ This problem is NP-hard (see [5.15]), the complexity of algorithms is very large, for example, if you use an exhaustive method, the complexity is O(ሺܸ݉ܽݔሻ! ) with Vmax is the number of vertices of the graph that there is maximum node and m is the number of graphs Weskamp et al [15] introduced the greedy algorithm; it transforms the comparing of multiple graphs become the problem of comparing two graphs to find a solution that is good enough to solve the problem in a short time Fober et al [5] proposed genetic algorithms called GAVEO significantly improve performance compared with the greedy algorithm, although it runs in a longer time Fig 1: A multiple graph alignment of the four graphs, the node labels are indicated by the letters assigned to the nodes (presented by circles) and dummy nodes are indicated by squares B Ant Colony Optimization method To assess the quality of an alignment, we use the scoring function for the edit distance This function is defined based on the set of edit operations mentioned above to match the pairwise graphs followed the selected alignment ACO method had been proposed by Dorigo in 1991 (see [4]) Until now, it had been developed into many variations to solve hard combinatorial optimization problems In these algorithms, the under-examined problem is transformed into the path finding problem on a construction graph G= (V,E,Ω, η,T), where V is a set of vertices, E is a set of edges, Ω is a set of constraints for solution building, η and T is the vectors that denotes heuristic information and reinforcement learning information for solution finding (their elements can be on the vertices or on the edges) For ease of presentation, in the rest of the article we keep the notation convention G ={G1(V1,E1),…,Gn(Vn,En)} to refer to the multigraph in which the graph Gi has additional dummy node Vi for all i=1,…,n The scoring function for alignment quality Define 3(Scoring function) In each iteration, each ant in the m ant colony will build the solution on the Construction graph from a starting set C0 and randomly sequential develop based on reinforcement learning information at pheromone trail and For each alignment matrix A of multigraph G, the scoring function s(A) is defined as (1): 387 heuristic information follow random walk procedure satisfy the constraints Ω Then, those solutions are evaluated and used for updating the pheromone trails as reinforcement learning information that helps ant colony constructs solutions in the next loops, more details see [4] This procedure is specified in Fig specially, the dummy nodes allow many lines passed through it The set of these paths can be seen as an only path as the concept of the common ACO algorithm with indicates that this line starts from a node of G1, passes through the next graphs, when reaching to the first or the last layer, "walking" to the other node on the same layer and return back until through every node exactly once time Procedure of ACO algorithms; Begin Initialize; // initialize pheromone trail matrix and u ants Repeat Construct solutions; // each ant constructs its own solution Improve solutions by local search // if it’s necessary Update trail; Until End condition; End; Random Walk Procedure to build an alignment In each iteration, each ant will perform iterative process to buil the vectors a = (a1,…,an) for an alignment A as follows: Ants randomly select a real node on the construction graph and based on the heuristic information and the pheromone trail to randomly walk to build a solution For ease of envisioning, we assume that this real node is in G1 (denoted as a1), ants will randomly walk across the layers to Gn as follows If ants have built vectors (a1,…,ai) where aq is the vertex j of Gi then selected node k in Gi +1 with probability given by Equation (4) Fig Specification of an ACO algorithm To apply ACO method, there are three factors that need to be resolved: 1) the construction graph and sequential developed procedures according to given constrains, 2) heuristic information, 3) pheromone update rule Below, we introduce an ACO algorithm for the MGA problem called as ACO-MGA ఛೕ,ೖ כቂఎೕ,ೖ ሺሻቃ α Pkij = III ACO-MGA ALGORITHM β α ఛ כቂఎೕ,ೞ ሺሻቃ శభ ೕ,ೞ ∑ೞചೃ_ೇ β (4) where R_Vi is the number of remaining un-aligned nodes on Vi included dummy node, ߬, is intensity of pheromone trail of the edge connected vertex j of Gi with vertex k of Gi+1 , and ߟ, ሺܽሻ is heuristic information calculated by Eq (5) Considering the alignment problem for multi-graph G ={G1(V1,E1),…,Gn(Vn,En), after the addition of the dummy node to the vertices set of the graph Gi as mentioned above, the Construction Graph and the solution building procedure as follows ேሺ,ሻ ሺܽሻ ൌ ቊ ߟ, Construction Graph th Construction Graph consists of n layer, the i layer is the graph Gi of G, the vertices of the upper layer connect to all nodes of the lower one Fig shows the construction graph, where the edges of each graph in each layer aren’t showed, the circles are real node and dummy nodes are represented by a square ߟ ݇ ݅݁݀݊ ݈ܽ݁ݎ ܽ ݏ ݇ ݅ݕ݉݉ݑ݀ ݏ (5) where NL(k,a) is the number of vertexs in {a1,…ai} that its label is like the label l(k) of vertex k, ߟ is given enough small value After vector a is developed to a=(a1,…an), the real vertices in a is removed from the construction graph to continue repeating the alignment procedure of ants until every vertex has been aligned The alignment process of ants is illustrated in Fig 4, where the dummy nodes are numbered -1, the other nodes are numbered 0, 1, 2, Noting that if the real node which is original selected is not on the G1, it is on Gm, the above procedures can be divided into two processes aligning from Gm to Gn and aligning backwards from Gm to G1 Fig The construction graph of n graphs alignment where each graph contains or nodes An alignment of the graph in defined above is a path from G1 through all the layers to Gn layer such that each line passes through a node of each layer and each node of construction graph there is exactly one line passed through, 388 IV EXPERIMENT RESULTS Experiments to compare the ACO-MGA with Greedy algorithm [15] and the evolution algorithm called GAVEO [5] on the solution quality and runtime: 1) Run algorithms with the same data sets and a predefined number of loops to compare the effect and runtime 2) Run algorithms with the same data sets with the same predefined time to compare scoring of the alignment The experiments are performed on a computer with: CPU Dual Core 2.2 Ghz, RAM DDR3 3GB running Windows XP SP3 We run each of the three algorithms 10 times and compare the average results The parameters had been set as follows: • • • Fig Ant builds the solution Pheromone Update Rule After the ants have found the solution, the solutions of iteration are evaluated and selected the best solution to perform local search to improve quality then perform pheromone trail updating SMMAS Pheromone Update Rule is applied as in [2] and [6], detail as follow: ߬ ՚ ሺ1 െ ߩሻ߬ ∆ (6) ߩ߬௫ ሺ݅, ݆ሻܾ߳݁݊݅ݐݑ݈ݏ ݐݏ where: ∆ ൌ ൜ (7) ߩ߬ ݁ݏ݅ݓݎ݄݁ݐ τmax and τmin is predefined parameter The number of ants in each loop is 20 ρ=0.6, ߙ ൌ ߚ ൌ τmax = 1.0 τmin = τmax/(n2*Vmax2), where n is the number of graph, Vmax is the number of node of the graph that has the most node Because there is no real data, we use Graph Generator program to generate data as in [5] where each graph has 20 or 50 vertices and the number of graph alternately is 4, 8, 16 and 32 A Effect and Runtime comparisons Table and table below are the results of comparing the method about score and runtime Table is the result of the alignment of the graphs has average 20 vertices and table results of the alignment of graphs with an average of 50 vertices The best score are shown in bold Local search Table Comparison of the score and runtime with the data sets including 4, 8, 16 and 32 graphs, and the average number of the vertices of each set is 20 nodes Method/Number of 16 32 graphs Local search procedure is applied to the best solution by principles better then stopped In this procedure, the pair of the same label vertices in each graph Gi which is randomly selected will be swapped in the its alignment vector to improve the suitability of the weights of the relevant edges If after swapped, scoring function is increasable, the getting solution will replace the best solution and stop the search procedure of iteration to update the pheromone Greedy GAVEO ACO-MGA A permutation of the two node labeled A is illustrated in Fig 5, where alignment vectors are column vectors; the letters are the label of the corresponding components Score -40 -35 -570 -1055 Time 0.6 2.3 17 Score -20 65 45 1132 Time Score 249 123.8 501 696.1 1087.7 1479.7 2484.1 7288.5 Time 33.6 231.5 481.2 1266 Table Comparison of the score and runtime with the data sets including 4, 8, 16 and 32 graphs, and the average number of the vertices of each set is 50 nodes Method/Number of 16 32 graphs -1144 -4704 -31004 -155508 Score Greedy 4.8 11.3 49 210.8 Time -101 -75 -10872 -33698 Score GAVEO 1164 2739.1 6921.3 16340.8 Time Score 684.9 3337.6 1273.1 -18642.9 ACO-MGA 763.4 6523.5 12670.5 28859.8 Time Fig A permutation of the two same label nodes in Local Search procedure Comment The experimental results show that: • In the two cases the graphs have average 20 vertices or 50 vertices, the runtime of Greedy algorithms is very little than the other two algorithms However, the results of this algorithm are very low in comparison to GAVEO and ACO-MGA ACO-MGA algorithm performs as specified in Fig for the case of apply local search procedure 389 TABLE Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 4,8,16 and 32 graphs, with the average number of vertices of each graph is 50 vertices and runtime is 600s Method/Number of 16 32 graphs GAVEO -107 -77 -5282 -96123 Score ACO-MGA Score 672.9 2898.4 744.8 -16945.8 • The ACO-MGA algorithm results better algorithm GAVEO more With the graphs have average 20 vertices, the runtime of ACO-MGA is faster than GAVEO but when the number of vertices in the graphs increases, the runtime of GAVEO is faster in case the number of graph is over However, the experiments in the next section shows in the same running time, the ACO-MGA still give much better score than GAVEO The comparison of score of ACO-MGA algorithm and GAVEO algorithm on data sets consist of 32 graphs with the average number of vertices of each graph is 20 vertices when increasing time from 50s to 200s is as fig.6 B Comparing evolution algorithm and ACO-MGA algorithm in the same runtime Since Greedy algorithm has short runtime, but it has low score, in this article we only conducted experiments to compare the performance of evolutionary algorithms and the ACO-MGA algorithm with the same runtime The experiments performed on the same data set and the same runtime to compare the score of two algorithms First experiment, running on the data sets consist of 8, 16 and 32 graphs, each graph has average of 20 vertices and runtime alternately is 50s, 150s and 200s Experimental results are shown in Table 3, Table and Table The second experiment, run on data sets consist of 4, 8, 16 and 32 graphs, each graph has average of 50 nodes and runtime alternately is 200s, 300s and 600s The results of this experiment are presented in Tables 6, and The better results shown in bold Fig Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 32 graphs, with the average number of vertices of each graph is 20 vertices and runtime is 50,150 and 200s Comment The above results showed that in the same runtime, the new algorithm gives much better results than GAVEO TABLE Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 8, 16 and 32 graphs, with the average number of vertices of each graph is 20 vertices and runtime is 50s Method/Number of 16 32 graphs 57 46 -1327 GAVEO Score ACO-MGA Score 689.1 2004.1 6511.2 V CONCLUSION MGA problem is a new approach to analysis the structure of biological molecules, so far there have been two commonly algorithms solved it Greedy algorithm is a heuristic algorithm, so it is outstanding in runtime but not effective TABLE Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 8, 16 and 32 graphs, with the average number of vertices of each graph is 20 vertices and runtime is 150s Method/Number of 16 32 graphs GAVEO 75 35 953 Score ACO-MGA Score 689.7 2180.9 7166.1 Our new algorithms called ACO-MGA has much better results than GAVEO when run on the same data set and the same runtime When the number of vertices of the graph increases, the duration of local search in ACO-MGA also increases, so the runtime of ACO-MGA is longer than GAVEO in some cases In the future can improve the local search technique to reduce the running time and increase the efficiency of the algorithm TABLE Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 8, 16 and 32 graphs, with the average number of vertices of each graph is 20 vertices and runtime is 200s Method/Number of 16 32 graphs 74 -38 1254 GAVEO Score ACO-MGA Score 689.9 2261.6 10059.6 ACKNOWLEDGEMENT TABLE Comparison of results of ACO-MGA algorithm and GAVEO algorithm with data sets consist of 8, 16 and 32 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graph indexing method, Proc of 23th International Conference on Data Engineering, 966-975 391 ... an alignment In each iteration, each ant will perform iterative process to buil the vectors a = (a1,… ,an) for an alignment A as follows: Ants randomly select a real node on the construction graph. .. random walk procedure satisfy the constraints Ω Then, those solutions are evaluated and used for updating the pheromone trails as reinforcement learning information that helps ant colony constructs... on reinforcement learning information at pheromone trail and For each alignment matrix A of multigraph G, the scoring function s(A) is defined as (1): 387 heuristic information follow random walk