Contemporary Engneerng Scences, Vol. 8, 2015, no. 17, 773-788 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ces.2015.56193 Task Schedulng for Drected Cyclc Graph Usng Matchng Technque W.N.M. Arffn Insttute of Engneerng Mathematcs Unverst Malaysa Perls Kampus Pauh Putra 02600 Arau, Perls, Malaysa S. Salleh Unverst Teknolog Malaysa Center for Industral and Appled Mathematcs, Skuda, Johor, Malaysa Copyrght 2015 W.N.M. Arffn and S. Salleh. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract The schedulng and mappng of task graph to processors s consdered to be the most crucal NP-complete n parallel and dstrbuted computng systems. In ths paper, the theoretcal graph applcaton usng matchng s presented to assgn a number of tasks onto two processors. Ths paper addresses a drected-weghted cyclc graph. The effort s to reduce the graph onto drected acyclc graph. A co-comparablty graph s presented n order to assgn the task onto two processors. Combnng several nnovatve technques lead to an effcent graph-mappng concept, called DCGSmplfy. Our smulaton model found that the proposed technques and algorthms are easy to be mplemented. Keywords: Graph theory, Matchng, Mappng 1 Introducton The problem of schedulng a task graph of parallel program onto a parallel and dstrbuted computng system s a well-defned NP-complete problem that has receved a large amount of attenton. The effcent schedulng of tasks s paramount to maxmzng the benefts of executng an applcaton n a dstrbuted
774 W.N.M. Arffn and S. Salleh and parallel computng. The drected acyclc graph (DAG) structure frequently occurs n many regular and rregular applcatons such as LU decomposton, Gaussan elmnaton, Laplace transforms, fast Fourer transform and nstructon level parallelsm [1]. The produced schedule s judged based on the performance crteron we are tryng to optmze. There are two characterstcs used n order to evaluate a schedulng system, whch are performance and effcency. Many studes have been done on complextes, classfcatons and technques requred for solvng tasks assgnment and schedulng problems. An optmal task assgnment guarantees mnmum turnaround tme for a gven archtecture. Several approaches of optmal task assgnment have been proposed, rangng from graph parttonng based tools to heurstc graph matchng. In an attempt to solve the problem n the general case, a number of heurstcs have been ntroduced. The heurstcs do not guarantee an optmal soluton to the problem, but they try to fnd near-optmal solutons most of the tme [2]. The prevous researches only work on drected acyclc graph. In ths paper, we attempt to solve for drected cyclc graph (DCG). Snce the problem nvolves DCG, therefore a new algorthm to reduce the DCG nto the form of a DAG should be proposed. It s smpler when no cycle exsts n a graph. A DCGSmplfy algorthm s presented to reduce the graph from DCG to DAG. Then, the graph s mapped to the two processors after applyng matchng technque. A smulaton of dfferent number of nodes s mplemented to verfy our algorthm. Ths paper s an extenson of prevous work [3] whch presents a new approach of solvng schedulng task based on matchng technque. The rest of the paper s organzed as follows: n Secton 2 we descrbe some related works. Secton 3 addresses some defntons of classcal graph theory. Then, matchng and two-processor assgnment technque s dscussed n secton 4. The mathematcal modelng based on DCG s ntroduced n secton 5. The results of our work are shown n secton 6. Fnally, we draw some conclusons and dscusson n secton 7. The man contrbuton of ths paper s the novelty reduced technque of drected cyclc graph onto acyclc graph through multcolumn nodes arrangement. 2 Related Work Research on task allocaton and schedulng problems began n the 1960 s, and has become a popular research topc n the past few decades. Mona M. Arafa and Fatma A. Omara [4] proposed two hybrd genetc algorthms, named Crtcal Path Genetc Algorthm (CPGA) and Task Duplcaton Genetc Algorthm (TDGA). The problem nvolved mappng a DAG for a collecton of computatonal tasks and ther data precedence onto a parallel processng system. Both of the algorthms were compared to other exstence algorthms. The expermental results showed that CPGA always outperforms the Mult Crtcal Path algorthm (MCP) and TGDA algorthm outperforms the Duplcaton Schedulng Heurstc algorthm (DSH) n most cases.
Task schedulng for drected cyclc graph 775 The ncreasng popularty of graph data n varous domans has led to a renewed nterest n developng effcent graph matchng technques, especally for processng large graphs. Lnhong Zhu et al. [5] studed the problem of approxmate graph matchng n a large attrbuted graph. They proposed a novel structure-aware and attrbute-aware ndex to process approxmate graph matchng n a large attrbuted graph. They use the ndex to fnd a set of best matchng paths. From the best matchng paths, they compute the best matchng answer graph usng a greedy algorthm. Ullman [6] ntroduced a basc algorthm for subgraph somorphsm,.e. exact matchng. Tsa and Fu [7] studed error-correctng somorphsms of attrbuted graphs for mage analyss. They extended ther pattern deformaton model so that numercal attrbutes and probablty dstrbuton can be ntroduced nto prmtves and relatons n non-herarchcal relatonal graphs. Cordella et al. [8] proposed a new algorthm whch can handle subgraph somorphsm test effcently n large graphs. Tong et al. [9,10] proposed an algorthm to fnd best effort matchng n a large graph based on random walk. Another type of work that smlar to Lnhong Zhu et al. whch use ndex to fnd matches n a large data graph effcently, Tan and Patel [11] utlze degree nformaton and neghborhood connectvty to flter unmatched node pars. Zhang et al. [12] proposed another ndexng approach based on graph dstance. Ther methods are not effcent enough to handle graphs wth up to mllons or bllons of nodes and edges. In the preprocessng stage of [12], gven a data graph, they generate a set of ntersectng subgraphs may ncrease sgnfcantly when the sze of data graph grows. Tan et al. [11] used a maxmum weghted bpartte graph matchng algorthm durng the stage of matchng mportant nodes, whch can be costly f the bpartte graph s large. Gutman and Wagner [13] defned the matchng energy of a graph and gave some propertes and asymptotc results of the matchng energy. Shul and Wegen [14] characterzed the connected graph G wth connectvty k has the maxmum matchng energy by ntroducng n-matchng n-partte graph. Mohan and Gupta [15] establshed a methodology for heurstc graph matchng, but restrcted to a small number of test cases. Motvated by the work done by researches [5, 11, 14, 15], a matchng technque to solve task schedulng onto two processor s presented. Compared to the prevous works, ths paper focused on drected cyclc task graph (DCG). 3 Defntons It s mportant to know few types of graph. A drected graph G s an (ordered) par ( V, Esuch ) that V s a fnte set of nodes and E s a set of ordered pars of nodes. That s, f e E, then e ( X, X ) example of the drected graph. and, j X X V j. Fg. 1 shows an
776 W.N.M. Arffn and S. Salleh Fg. 1 A drected graph wth V 1, 2 and (1, 2) E. If G s a drected graph, then a drected cyclc graph (DCG) s a cycle n G conssts of a path from a node to tself. However, a drected acyclc graph (DAG) s a graph whch does not contan any drected cycle. If G ( V, A), s a DAG contanng n nodes, then a topologcal (or ancestral) orderng ( X1,..., Xn ) of the nodes n V s any orderng such that, f ( X, X ) A, then j. Both DCG and DAG are shown n Fg. 2. j (a) (b) Fg. 2 In ths graph (a), 1,2,3,4,1 s a drected cyclc and n (b) s a drected acyclc graph.
Task schedulng for drected cyclc graph 777 The co-comparablty graph, G ( V, A ) s an undrected graph, that can be defned as V V and A {(, j) } f there s no path from to j or from j to n G. In the mathematcal dscplne of graph theory, a matchng or ndependent edge set n a graph s a set of edges wthout common vertces. It may also be an entre graph consstng of edges wthout common vertces. Let a task graph G be a DCG composed of N nodes n 1, n 2, n 3,..., n N. Each node s termed a task of the graph whch n turn s a set of nstructons that must be executed. A node has one or more nputs. A node wth no parent s called source node whle a node wth no chld s called destnaton node. The graph also has E drected edges representng the communcaton between the nodes. The weght on an edge s called the communcaton cost of the edge and denoted by cn, n ). 4 Matchng and two-processors assgnment ( j Matchng has several applcatons n the real world. One of good examples s the personnel assgnment applcaton. The problem can be modeled usng matchng f a bpartte graph G s constructed wth bpartton ( XY, ), as shown n Fg. 3 where X x, x,..., x and Y y, y,..., y 1 2 n 1 2 n. A node x s connected to y j f and only f the canddate x s qualfed for job y j. The problem now becomes one of determnng whether G havng a perfect matchng. Fg. 3 Personnel assgnment problem usng matchng. The matchng technque presents an nterestng relatonshp between the two processors schedulng problem and the sze of maxmum matchng n the complement of the task graph. Ths relatonshp was establshed by Fuj et al. [16] could be expanded to nclude cases where communcaton s consdered. Matchng can be descrbed as gven an undrected graph G ( V, A), where V s a set of nodes and A s a set of edges. A subset T of A s called a matchng n G f ts elements are edges and no two are adjacent n G. The two ends of an edge n T are sad to be matched under T. A matchng M s a maxmum matchng f G has no matchng T wth T T.
778 W.N.M. Arffn and S. Salleh The problem that we want to nvestgate n ths paper s to schedule a task graph onto two processors P 1 and P 2. We present a task graph, a drected cyclc graph ntally. The graph s then reduced to a drected acyclc graph, G ( V, A) where V s a set of nodes and A s a set of arcs (edges). We can say that v Ss a maxmal task n S f there does not exst task v Ssuch that v precedes u n G. In order to obtan the relatonshp between matchng and schedulng n ths study, we propose a co-comparablty graph, G ( V, A ). Ths paper s an extenson of our prevous work n [3] whch consder a problem of fndng an optmal schedule for G onto two processors wth communcaton cost, whch s to fnd the maxmum number of dsjont task pars, can be reduced to the maxmum matchng problem n G. A co-comparablty graph s presented n our work n order to show the matchng technque before mappng the nodes onto two processors. In ths paper, we verfy our proposed algorthm through the smulaton model. 5 The model for task schedulng problem In ths paper, a task graph of weghted-drected cyclc graph s presented. The man goal of our study s to obtan a mnmum total completon tme of a task schedulng. In order to acheve the goal, a good matchng algorthm and mappng strategy of task that wll be assgned to processor should be mplemented. Our approach s shown n Fg. 4. Fg. 4 The flowchart of DCGSmplfy technque.
Task schedulng for drected cyclc graph 779 Our developed algorthm s very smple and constructed usng the followng steps: Step1: Identfy the exstence of cycle(s) n the task graph. Step2: All the nodes are arranged n two columns lke wrapped-butterfly topology, nspred by DPllar [17]. Step3: A source and destnaton nodes are chosen from the graph. Step4: Obtan the path from source to destnaton nodes such that they do not share common ntermedate nodes. The produced graph now s a drected acyclc graph. Step5: An edge s created when there s no communcaton between two nodes. We call t as a co-comparablty graph. Step6: The connected nodes are then mapped onto two processors smultaneously, whle the other tasks are freely mapped to any avalable processor. These steps are constructed to solve the problem, named Model 1. The hypothess on DCGSmplfy s effcent to construct and the co-comparablty graph based on the DCGSmplfy s effcent and scalable to dfferent szes of graphs. The next secton wll dscuss the mplementaton of the algorthm and the task graph wth same communcaton cost and computatonal cost. 5.1 Model 1 The proposed algorthm nvolves few stages of soluton. Frstly, we defne the source and destnaton nodes (n our smulaton, we named both nodes as source and target). Next, we want to obtan a DAG by choosng the ntermedate nodes. Then, a co-comparablty graph s obtaned and matchng technque s appled to the task graph. The DCGSmplfy algorthm s outlned n Algorthm 1. Algorthm 1: DAG and apply matchng technque Gven a graph G( V, E ), obtan a DAG, G( V, A ). Construct ts co-comparablty graph, G ( V, A ). In G, obtan a set of edges n maxmum matchng M. Let S 1 V, M 1 M and 1. for 1 to V M f there exsts a maxmal task u n of M,then u S S {} u 1 M 1 M S that s not ncluded n any of the task pars f there exsts a task par ( uv, ) n M such that u and v are maxmal n S, then
780 W.N.M. Arffn and S. Salleh ( uv, ) S S { u, v} 1 M M { u, v} 1 f there are two task pars ( u1, u2) and ( v 1, v 2 ) n S, u 2 and v 2 are ndependent, then ( u, v ) 1 1 S S { u, v } 1 1 1 M M {( u, v )} {( u, v )} 1 2 2 1 1 n M such that u 1 and v 1 are maxmal //Map the task graph onto processors. Each can be mapped accordng to ts ndexed number or can be mapped smultaneously by two processors when s a par of tasks. An arbtrary task graph, wth eght number of nodes s proposed as n Fg. 5. To mprove the flexblty of the propose task graph graph, multcolumn (multlayer) of the graph s constructed by replcatng the sngle graph, tmes. In ths study, two-column of task graph,.e. s presented. The problem s stated as follows: Gven a graph G(8, E, 2), how the tasks can be mapped onto two avalable processors? Fg. 5 Model 1: A DCG wth eght number of nodes. To solve the problem, a matchng technque s presented for the drected multcolumn graph. In our approach, we begn by arrangng the nodes n two-column. The purpose of dong ths s as alternatve way of fndng a drected acyclc graph. The nodes of graph as shown n Fg. 5 s then arranged as n Fg. 6. The communcaton (edges) between the nodes also presented n Fg. 6.
Task schedulng for drected cyclc graph 781 s t Fg. 6 Nodes are arranged n duplcated column/mutcolumn. The next step requres us to fnd the drected acyclc graph. As an example, we want to fnd the shortest path from the source node 1, denote as s to the destnaton node 8, denote as t. So, the path from s to t must be found such that all nodes must be chosen and they must not share common ntermedate nodes. As can be seen from Fg. 6, there are three nodes can be chosen from the source node 1. Let say, node 2 s chosen from node 1. Next, from node 2, node 3, 5 and 7 can be reached. Next, node 3 s chosen. From node 3, only node 4 can be chosen. Next, node 5 s chosen from node 4. After that, node 7 s chosen from node 5. Note that there are two nodes that are not chosed yet. So, agan from the source node, node 6 s chosen. From node 6, ether node 5 or node 8 can be chosen. Notce that, same ntermedate nodes are not allowed. Thus, node 5 s strctly not avalable to be chosed. Next, from node 6, node 8 can be reached, whch s the destnaton node. Now, the drected acyclc graph s obtaned. It s llustrated as n Fg. 7. Fg. 7 The reduced drected cyclc graph to drected acyclc graph.
782 W.N.M. Arffn and S. Salleh Next, we need to map the tasks onto processor. In order to map the tasks onto two processors, the co-comparablty of the graph should be obtaned. Based on the defnton stated n prevous secton, an edge s created when there s no path or communcaton from u to v. Snce there s no path between nodes 5-6, 5-8 and 7-8, therefore we create a path to connect node 5 and node 6 also node 7 and node 8. Fnally, the co-comparablty graph can be llustrated as follows: Fg. 8 The co-comparablty of the drected acyclc graph. From Fg. 8, we can map the nodes onto processors. Node 5 and node 6 are mapped onto two dfferent processors, and the same to node 7 and node 8. The remanders are freely assgned to avalable processor. 6 Results To evaluate our proposed algorthm, we have mplemented t usng Intel(R) Core(TM) 5 (2.3 GHz) usng Javascrpt programmng language wth D3 lbrary. The algorthm s appled to test for several number of nodes. In ths paper, we show our smulaton model for eght, twelve, sxteen and twenty number of nodes. Our smulaton model s able to solve for a large number of nodes. After we run the smulaton model, we obtaned the results as follows: (a)
Task schedulng for drected cyclc graph 783 (b) (c) (d) Fg. 9 The DCG to DAG colored wth red lne and matchng colored wth blue lne for 8, 12, 16 and 20 nodes n (a), (b), (c) and (d) respectvely. Fg. 9 (a) shows eght nodes wth communcaton cost. For example, we have cn ( 1, n2) 6. Intally we have a drected cyclc graph. Then, our algorthm tres to fnd the drected acyclc graph (red lne). Then, we apply the co-comparablty and
784 W.N.M. Arffn and S. Salleh matchng colored wth blue lne. Based on Fg. 9, we map the nodes onto two processors as follows: (a) (b) (c)
Task schedulng for drected cyclc graph 785 (d) Fg. 10 The schedule length for (a) eght nodes, (b) twelve nodes, (c) sxteen nodes and (d) twenty nodes. From the results obtaned n Fg. 10, we can tabulate the total length (communcaton cost) to complete all the tasks as shown n Table 1. Table 1. Total length of completng all tasks. Number of Nodes (Tasks) Total Length(Unt) 8 40 12 16 20 35 57 72 7 Concluson and Dscusson Based on the work done n ths paper, we can conclude that the technque of matchng for a drected cyclc graph onto two processors s easy to mplement and
786 W.N.M. Arffn and S. Salleh effcent. Ths s the novelty of ths paper. We begn fndng the soluton from the drected cyclc graph. The qualty technques used for fndng the acyclc graph and co-comparablty graph from the drected cyclc graph led to an effcent graph-mappng concept. The technque s vald and applcable for larger number of nodes. Our next paper wll put an attenton on graph parttonng and map t onto n 2processors. Acknowledgements. Ths work was supported by the Mnstry of Educaton, Malaysa for Research Acculturaton Grant Scheme (RAGS) under project code 9018-00036. References [1] S. Baskyar and C. Dcknson, Schedulng Drected A-cyclc Task Graphs on a Bounded Set of Heterogeneous Processors Usng Task Duplcaton, Journal Parallel Dstrb. Comput., 65 (2005), 911-921. http://dx.do.org/10.1016/j.jpdc.2005.01.006 [2] H. El-Rewn and T.G. Lews, Introducton to Parallel Computng, Prentce- Hall, 1994. [3] W. N. M. Arffn and S. Salleh, The Matchng Technque of Drected Cyclc Graph for Task Assgnment Problem, AIP Conference Proceedngs, 1635 (2014), 387. http://dx.do.org/10.1063/1.4903612 [4] Mona M. Arafa and Fatma A. Omara, Genetc Algorthms for Task Schedulng Problem, Journal Parallel Dstrb. Comput., 70 (2010), 13-22. http://dx.do.org/10.1016/j.jpdc.2009.09.009 [5] Lnhong Zhu, Wee Keong Ng, James Cheng, Structure and Attrbute Index for Approxmate Graph Matchng n Large Graphs, Informaton Systems, 36 (2011), 958-972. http://dx.do.org/10.1016/j.s.2011.03.009 [6] J.R. Ullmann, An Algorthm for Subgraph Isomorphsm, The Journal of the ACM, 23 (1976), no. 1, 31-42. http://dx.do.org/10.1145/321921.321925 [7] W. H. Tsa, K. Fu, Error-correctng Isomorphsms of Attrbuted Relatonal Graphs for Pattern Analyss, IEEE Transacton on Systems, Man and Cybernetcs, 9 (1979), no. 12, 757-768. http://dx.do.org/10.1109/tsmc.1979.4310127
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788 W.N.M. Arffn and S. Salleh [17] Y. Lao, J. Yn, D. Yn, L. Goa, D. Pllar: Dual-port Server Interconnecton Network for Large Scale Data Centers, Computer Networks, 56 (2012), 2132-2147. http://dx.do.org/10.1016/j.comnet.2012.02.016 Receved: July 11, 2015; Publshed: September 1, 2015