On the etwor Parttonng of Large Urban Transportaton etwors Hamdeh Etemadna and Khaled Abdelghany Abstract Ths paper ams at developng a traffc networ parttonng mechansm for dstrbuted traffc management applcatons. The mechansm can be used to partton a typcal urban transportaton networ such that: a) the nter-flow among the resultng subnetwors s mnmzed; b) the subnetwors are balanced n terms of ther szes/flow actvtes; and c) each subnetwor s connected. Two heurstcs are presented. The frst heurstc adopts a recursve teratve procedure to determne the networ s sparsest cuts that mantan the balance and connectvty requrements. The second heurstc adopts a greedy recursve networ coarsenng technque to determne the most flowndependent subnetwors. Parttonng result of these two heurstcs s evaluated usng hypothetcal networs wth dfferent confguratons. Keywords Dstrbuted Traffc Management, Parttonng, etwor Coarsenng, Sparsest Cut. etwor I. ITRODUCTIO RAFFIC management n large congested urban areas s Tconsderably challengng. It requres the development of traffc management capabltes that effectvely allevate recurrent and non-recurrent congeston. The evoluton of Advanced Traffc Management Systems (ATMS) over the last two decades has brought consderable attenton to the development of real-tme traffc management systems for congested urban networs. The goal s to provde roadway networ managers wth the capablty to develop effcent real-tme traffc management strateges to allevate recurrent and non-recurrent congeston stuatons. Consderable research effort has been devoted over the last two decades to study the problem of real-tme traffc networ management. A holstc approach to develop realtme management capabltes would follow a centralzed archtecture through whch the entre traffc networ s managed usng one central controller. However, the large networ szes mae t algorthmcally dffcult to develop a global optmal traffc management scheme, especally f ths scheme needs to be generated n real-tme []. In contrast to the centralzed archtecture, an alternatve approach s to adopt decentralzed traffc management archtecture [-6]. The qualty of a traffc management Hamdeh. Etemadna s wth the Penn State Unversty,Unversty Par, PA, 68 USA (correspondng author; phone: 84-86-996; e- mal:hze@psu.edu ). Khaled. Abdelghany, s a Assocate Professor at Southern Methodst Unversty, Dalals TX 76 USA. He s the char of the Cvl and Envronmental Engneerng Department (e-mal: haled@psu.edu). scheme usng a dstrbuted archtecture manly depends on the boundares of the local subnetwors and ther nterfaces wth the man traffc movements n the networ. Most of the exstng research assumng that the boundares of the local subnetwors are predefned. In ths paper, we nvestgate the problem of networ parttonng to defne the optmal (near-optmal) boundares for subnetwors n dstrbuted traffc management applcatons. The paper s organzed as follows. A lterature revew of the networ partton problem s presented n the next secton. A mathematcal defnton of the networ parttonng problem s then presented n secton III. Secton IV descrbes two heurstcs that are developed to solve ths problem. The results of a set of experments and performance of the heurstcs are also presented n secton V. Fnally, summary and research extensons are dscussed n secton VI. II. LITERATURE REVIEW etwor parttonng problem has been studed over the past few decades n the graph theory lterature [7]. For example, the mnmum multcut problem, the sparsest cut problem, the α- balanced cut problem, and the bsecton cut problem are dfferent versons of the networ parttonng problem. The networ parttonng problem has been proven to be P-hard [8] and [9]. Therefore, an approxmate soluton that s based on solvng an approxmaton of the problem s proposed [8,,, ]. For example, the dual of the lnear approxmaton of the sparsest cut problem s the maxmum concurrent flow (MCF) problem []. Another approach that adopts a multlevel recursve heurstc s proposed by Hauc and Borrello [], Wchlund and Enar [4] and Karyps and Kumar [7]. The heurstc recursvely creates a coarser verson of the networ through combnng ts nodes to reduce the networ sze. A refnng strategy s then used to modfy the boundares obtaned from the coarsenng step []. Addtonal networ parttonng methodologes developed to target the data clusterng problem can be found n Asano et al. [6], Xu et al. [7], Jan and Dubes [8], Gersho and Gray [9], and Inaba et al. []. Based on ths bref survey, we cannot drectly apply the exstng approaches for the problem consdered n ths paper. III. THE MULTI-WAY BALACED ETWORK PARTITIOIG PROBLEM Gven a hghway networ G(, A), s the set of nodes and A s the set of lns. The networ s assumed to be managed by a set of dstrbuted controllers C. Each controller c C manages the traffc n a subnetwor 4
G c ( c, A c ), where c s the set of nodes and A c s the set of lns n ths subnetwor. Defne f j as the vehcular flow durng the horzon of nterest for ln (, j). We defne the bnary decson varable x c whch s equal to one f ntersecton belongs to subnetwor G c ( c ), and zero otherwse. Also, we defne the bnary decson varable y jc whch s equal to one f ln (, j) belongs to subnetwor G c and zero otherwse. The problem s to determne the optmal parttonng of the networ nto a predefned number of subnetwors such that the subnetwors are flowndependent. In other words, the partton mnmzes nterflow among adjacent subnetwors n order to reduce the need for traffc management schemes that requre ntensve communcaton/coordnaton among adjacent controllers. In addton, the networ parttonng should mantan a balance n terms of the amount of traffc management actvtes requred by each local controller. For nstance, the parttonng should be conducted such that the subnetwors are relatvely close n sze (spatal balance), and/or they are smlar n terms of the amount of served traffc wthn ther boundares (flow balance). The balance crteron would lely enable the decentralzed archtecture to acheve the real-tme computatonal requrement as t dstrbutes the computatonal effort, assocated wth developng the traffc management actvtes, equally among controllers. The mathematcal program gven below s used to formulate ths problem. The objectve functon mnmzes the flow on the cut lns as llustrated n (). The nteger decson varable z j defnes the assgnment of the lns to the dfferent subnetwors. It s equal to one for non-cut lns where both ntersectons and j belong to the same subnetwor ( c and j c ), and s equal to two f a ln s a cut ln ( c and j c where c c ). Constrants n () ensure that each ntersecton belongs to only one subnetwor. Constrants n () and (4) relate the decson varables X, Y, and Z. In (), the constrant ensures the connectvty of each subnetwor. We defne the parameter b mj whch s equal to one f ntersecton m falls n the magnary rectangle where ntersectons and j are two corners along ts dagonal, and m s connected to any of the ntersectons n the subnetwor that contans and j. Constrants n (6) and (7) mantan the spatal and flow balance requrements, respectvely. The parameter δ and γ defne the maxmum acceptable dfference n the szes and served flows of the resultng subnetwors. Constrants n (8) and (9) state the bnary condtons of the decson varables x c and y jc, respectvely. The decson varable z j s {, } nteger as shown n constrans (. Mnmze f j (z j ) Subject to: j j () z j = c y jc (, j) c, j, c C (4) x mc b mj x c + x jc (m,, j) c, j, c C δ x c j j f j γ j () + δ c C (6) j f j. y jc c C j j f j + γ y jc (,) (, j), j, c C (8) x c (,), c C (9) z j {, } (, j), j () IV. SOLUTIO APPROACH In ths secton, we present two heurstcs to effcently solve ths problem. We consder the case n whch the networ s requred to be parttoned nto -spatally balanced subnetwors wth a balance rato α =. Fgure llustrates the man steps of the frst heurstc (sparsest cut heurstc), whch conssts of two man teratve loops. The outer loop recursvely solves the balanced cut problem for dfferent values of the balance rato α, whle the nner loop apples a greedy procedure to obtan the balanced cut for a specfc value of α. The cut resultng from the frst teraton of the outer loop dvdes the networ nto two components wth balance rato, where the smaller component s one of the requred parttons. In the next teraton, the remanng part of the networ s parttoned usng a value of α that s equal to (. e., α = ). The process s recursvely appled untl the remanng part of the networ s dvded nto two components (α =.), mplyng that all parttons are obtaned. The nner loop s based on the wor descrbed n [7], whch mplements an teratve greedy procedure to obtan the balanced cut for a gven α. In each teraton, the all pars unt-demand sparsest cut problem s solved to determne the mnmum densty cut (.e., two components wth mnmum flow nterface) []. Comparng the szes of the resultng components, f the αbalance rato s satsfed, the smaller component s reported as one partton, and the outer loop s agan actvated to partton the remanng part of the networ wth an updated value of α. (7) c x c =, c C () y jc x c + x jc (, j) c, j, c C () 4
= α = α tr = = = & S = all pars unt-demand sparsest cut S, S etwor : G(, A). of Parttons : S = S S = α Partton = G, A) = G( S S, A ) ( tr tr Fgure The α-balanced mult-way parttonng The second heurstc adopts a networ coarsenng approach smlar to the one presented n []. However, the new heurstc ams at mantanng the balanced requrements for any number of parttons. It also smultaneously generates all subnetwors rather than solvng a networ bsecton problem at each teraton. The coarsenng heurstc starts by determnng a centrod node for each partton. Dfferent strateges could be used to select the centrod nodes. A search process could be actvated to dentfy nodes wth the hghest flows, and spatally dstrbuted n the networ. Startng from a centrod node and adoptng a greedy heavy ln matchng strategy, the adjacent node wth the maxmum ln flow s combned wth the centrod formng a coarser node. The networ s then updated to provde the next-level coarser networ, as llustrated n Fgure. To mantan the balance among the parttons, ths process s sequentally rotated among all centrods. The heurstcs termnates when every node n the orgnal networ s combned wth a centrod node resultng n the fnal coarsened networ. In the fnal coarsened networ each node represents a partton. Ths coarsened node s refned bac to retreve ts related subnetwor of nodes and lns. de A Centrod Before combnng node A All parttons are generated Greedy Procedure for the α- Balanced Cut + = S S = S S S > α Fgure Example of the heavy ln matchng strategy + = S After combnng node A V. RESULTS AD AALYSIS A set of experments are conducted to compare the performance of the two developed heurstcs. Table llustrates the networ parttonng results usng the two heurstcs. A hypothetcal networ wth radal topology and symmetrc flow patterns are assumed. Two dfferent ln flow patterns are consdered. The frst pattern represents the case n whch the domnant flows are along the radal corrdors n the networ, whle the second pattern consders a case n whch the domnant flows are along the networ s rng corrdors. In both cases, the domnant flow value s assumed to be fve tmes that of the flow value assgned to all other lns. The flow assgned to each ln s ndcated by ts thcness as llustrated n the gven networ setches. Total ntra-flow and nter-flow as percentages of the total ln flows are recorded. The coeffcent of varaton (CoV) wth respect to the number of nodes n the dfferent subnetwors s recorded to measure the spatal balance. Smlarly, the CoV of the subnetwors ln flows s recorded to measure ther flow balance. As shown n the table, the sparsest cut heurstc provdes the optmal parttonng pattern for all cases where the networ s symmetrcally parttoned (e.g., two, four, eght, and sxteen parttons). For the coarsenng heurstc, centrods are selected manually consderng the networ s symmetrc flow pattern. For example, n the radal flow pattern wth two parttons case, the centrods are selected on two opposte radal corrdors. As llustrated, the coarsenng heurstc s able to provde the optmal soluton for some of the tested cases. It obtans the optmal parttons for the crcular flow pattern wth two, fve, ten, and twenty parttons. However, t fals to provde the optmal soluton for some other tested cases. For example, n the case of radal flow pattern wth two parttons, whle both heurstcs acheve perfect balance, the total ntra-flow of the coarsenng heurstc s recorded to be 87.% compared to 97.% that s obtaned by the sparsest cut heurstc. The dfference n the two solutons s explaned by ther underlyng approaches. The sparsest cut heurstc prmarly searches for the mnmum cuts n the networ, whle the coarsenng heurstc searches for components wth runnng tme, the coarsenng heurstc sgnfcantly outperforms the sparsest cut heurstc n all tested cases. For nstance, n the crcular flow case wth parttons, an executon tme of.4 seconds s recorded for the maxmum flows. As shown n the twopartton case, the cut obtaned by the coarsenng heurstc s not the mnmum cut as obtaned by the sparsest cut heurstc. However, the coarsenng heurstc sgnfcantly outperforms the sparsest cut heurstc n all tested cases n runnng tme. For nstance the executon tme n the crcular flow wth parttons recorded to be.4 and.7 seconds for the coarsenng heurstc and sparsest cut heurstc, respectvely. VI. SUMMARY Ths paper presents a methodology for parttonng urban transportaton networs for dstrbuted traffc management applcatons. The methodology parttons the networ such that: a) the nter-flow among the resultng networ components s mnmzed, b) these networ components are balanced n terms of ther szes and/or traffc flows wthn ther boundares, and c) each networ component s connected. Two heurstcs 44
are developed. The frst heurstc adopts a recursve approach n whch the networ s parttoned along ts sparsest cuts tll the balance condton s acheved. The second heurstc adopts a greedy-based networ coarsenng methodology. The results show that both heurstcs provde optmal/near-optmal soluton qualty. Several extensons are consdered for ths research wor. For example, we wll examne the effect of adoptng the presented heurstcs on the performance of the real world transportaton networs. We wll also examne the performance of the traffc management schemes usng a dstrbuted archtecture. Fnally, we wll consder mplementng a heurstc approach that combnes the coarsenng and sparsest cut technques. The trade-off between the level of coarsenng and the qualty of the soluton wll be nvestgated. VII. REFERECES [] Peeta, S. and A. Zlasopoulos. Foundatons of Dynamc Traffc Assgnment: The Past, the Present and the Future, etwors and Spatal Economcs, Vol.,, pp. -6. [] Cuena, J., J. Hernandez, and M. Molna. Knowledge-Based Models for Adaptve Traffc Management Systems, Transportaton Research Part C: Emergng Technologes, Vol.,., 99, pp. - 7. [] Hawas, Y. and H. Mahmassan. A decentralzed scheme for real-tme route gudance n congested vehcular networs, Yoohama, Japan, Proceedngs of the Second World Congress on Intellgent Transport Systems, 99. [4] Pavls, Y. and M. Papageorgou. Smple decentralzed feedbac strateges for route gudance n traffc networs, Transportaton Scence, Vol., 999, pp. 64-78. [] Log, F., and S. G. Rtche. A mult-agent archtecture for cooperatve nter-jursdctonal traffc congeston management, Transportaton Research C, Vol...,, pp. 7-7. [6] Hernandez, J., S. Ossows, and A. Garca-Serrano. Multagent archtectures for ntellgent traffc management systems, Transport Research Part C: Emergng Technologes, Vol.,, pp.47-6. [7] Shmoys, D. B. Cut problems and ther applcaton to dvdeand-conquer. In Dort S. Hochbaum, Approxmaton Algorthms for P-hard Problems, 997, pp. 9. [8] Matula, D. W. and F. Shahroh. Sparsest cuts and bottlenecs n graphs. Journal of Dscrete Appled Mathematcs, Vol. 7, 99, pp. -. [9] Garey, M.R. and D.S. Johnson. Computers and Intractablty: A Gude to the Theory of P-Completeness, W. H. Freeman & Co., ew Yor, Y, 979. [] Lnal,., E. London, and Y. Rabnovch. The geometry f graphs and some of ts algorthmc applcatons. Combnatorca, Vol., 99, pp. -46. [] Garg., V. V. Vazran, and M. Yannaas. Approxmate max-flow mn-(mult)cut theorems and ther applcatons. SIAM J. Comput., : 994, pp. -. [] Rao, S. Fndng near optmal separators n planar graphs. In Proceedngs of the 8 th Annual IEEE Symposum on Foundaton of Computer Scence, 987, pp. -7. [] Hauc, S. and G. Borrello. An evaluaton of bparttonng technque. In: Proc. Chapel Hll Conference on Advanced Research n VLSI, 99. [4] Wchlund, S. and J. A. Enar. On Multlevel Crcut Parttonng. In: Intl. Conference on Computer Aded Desgn, 998. [] Karyps, G. and V. Kumar. Multlevel -way hypergraph parttonng, VLSI Desgn, Vol...,, pp.8-. [6] Asano, T., B. Bhattacharya, M. Kel, and F. Yao. Clusterng algorthms based on mnmum and maxmum spannng trees. In Proceedngs of the 4th Annual Symposum on Computatonal Geometry, 988, pp. 7. [7] Xu, Y., V. Olman, and D. Xu. Mnmum spannng trees for gene expresson data clusterng. Genome Informatcs, Vol.,, pp. 4. [8] Jan A. K. and R. C. Dubes. Algorthms for Clusterng Data. Prentce Hall, Englewood Clffs,.J., 988. [9] Gersho, A. and R.M. Gray. Vector Quantzaton and Sgnal Compresson. Kluwer Academc, Boston,99. [] Inaba, M.,. Katoh, and H. Ima. Applcatons of Weghted Vorono Dagrams and Randomzaton to Varance-Based - clusterng, Proc. th Ann. ACM Symp. Computatonal Geometry, June 994, pp. -9. [] Matula, D. W. Concurrent flow and concurrent connectvty n graphs. In: Alav, Y., et al (Ends), Graph Theory and ts Applcaton to Algorthm and Computer Scence. Wley ew Yor, Y, 98, pp. 4-9. 4
Table etwor Parttonng for a Radal etwor wth Two Dfferent Flow Patterns Radal Flow Pattern Coarsenng Heurstc. of Parttons: de Balance: Runnng Tme (Sec): 87... 9.8.. 4 86..7... 8. 7...47. 8 9. 6 8. Sparsest Cut Heurstc. of Parttons: de Balance: Runnng Tme (Sec): 97.. 7. 9..4.8 4 9.88 86..7.8. 8 9 8. 6 8. Crcular Flow Pattern Coarsenng Heurstc. of Parttons: de Balance: Runnng Tme (Sec): 94.66.4. 89..69... 88..79. 77..6. 7. 9.77.8.9.4 68.7..4 Sparsest Cut Heurstc. of Parttons: de Balance: Runnng Tme (Sec): 94.66.4.6 87.6.4..6 46. 88..79.78 77..6 7.7 67.8.8.4.9.4 68.7..7 46