Discrete Dynamic Shortest Path Problems in Transportation Applications

Transcription

1 17 Paper No TRANSPORTATION RESEARCH RECORD 1645 Dscrete Dynamc Shortest Path Problems n Transportaton Applcatons Complexty and Algorthms wth Optmal Run Tme ISMAIL CHABINI A soluton s provded for what appears to be a 3-year-old problem dealng wth the dscovery of the most effcent algorthms possble to compute all-to-one shortest paths n dscrete dynamc networks. Ths problem les at the heart of effcent soluton approaches to dynamc network models that arse n dynamc transportaton systems, such as ntellgent transportaton systems (ITS) applcatons. The all-to-one dynamc shortest paths problem and the one-to-all fastest paths problems are studed. Early results are revsted and new propertes are establshed. The complexty of these problems s establshed, and soluton algorthms optmal for run tme are developed. A new and smple soluton algorthm s proposed for all-to-one, all departure tme ntervals, shortest paths problems. It s proved, theoretcally, that the new soluton algorthm has an optmal run tme complexty that equals the complexty of the problem. Computer mplementatons and expermental evaluatons of varous soluton algorthms support the theoretcal fndngs and demonstrate the effcency of the proposed soluton algorthm. The fndngs should be of maor beneft to research and development actvtes n the feld of dynamc management, n partcular real-tme management, and to control of large-scale ITSs. Department of Cvl and Envronmental Engneerng, Massachusetts Insttute of Technology, Room 1-138, 77 Massachusetts Avenue, Cambrdge, MA The shortest paths problem n networks has been the subect of extensve research for many years (1). The analyss of transportaton networks s one of the many applcaton areas n whch the computaton of shortest paths s one of the most fundamental problems. The maorty of publshed research on shortest paths algorthms, however, dealt wth statc networks that have fxed topology and fxed lnk costs. Interest n the concept of dynamc management of transportaton systems has been ncreasng. New advances have brought renewed nterest n the study of shortest paths problems wth a new twst: Lnk costs generally depend on the entry tme of a lnk. Ths results n a new famly of shortest paths problems known as dynamc, or tme-dependent, shortest paths problems. Chabn (2) dstngushes varous types of dynamc shortest path problems dependng on the followng: (a) fastest versus mnmum cost (or shortest) path problems; (b) dscrete versus contnuous representaton of tme; (c) frst-n-frst-out (FIFO) networks versus non- FIFO networks, n whch one can depart later at the begnnng of one or more arcs and arrve earler at ther end; (d) watng s allowed versus watng s not allowed at nodes; (e) types of shortest path questons asked: one-to-all for a gven departure tme or all departure tmes, and all-to-one for all departure tmes; and ( f ) nteger versus real valued lnk travel tmes and lnk travel costs. In fastest path problems, the cost of a lnk s the travel tme of that lnk. In mnmum cost paths problems, lnk costs can be of general form. Although the fastest paths problem s a partcular case of the mnmum cost paths problem, the dstncton between the two s partcularly mportant for the desgn of effcent soluton algorthms. Tme-dependent margnal travel tmes encountered n system optmum dynamc traffc assgnment models are an example of general form costs. Dependng on how tme s treated, dynamc shortest paths problems can be subdvded nto two types: dscrete and contnuous. In dscrete dynamc networks, tme s modeled as a set of ntegers. Orda and Rom (3) establsh varous propertes for contnuous dynamc networks where tme s treated as real numbers. The results for the oneto-all fastest paths problem studed here are vald for both dscrete and contnuous representatons of tme. It s mportant to note that a dscrete dynamc network can be vewed alternatvely as a statc network obtaned by usng a tmespace expanson representaton. The sze of the equvalent representaton depends on the watng polcy at nodes. As may be expected, t s not the best approach to explctly use the tme-space expanson representaton to compute dynamc shortest paths. The tme-space expanson network has, however, some partcular propertes that can be exploted n the desgn of effcent shortest paths algorthms. The challenge s to dscover partcular propertes of these networks and approprately explot them n desgnng better algorthms. Compared wth that for statc shortest paths problems, the lterature on dynamc versons s very lmted. The man known result s the condton under whch statc shortest path algorthms can be used, at no extra cost, to solve the one-to-all fastest paths problem n dynamc networks: Arc travel tmes must satsfy the FIFO condton. Hdden behnd ths result are varous lmtatons: 1. Ths result apples to the fastest paths problem only and not to the mnmum cost paths problem. 2. Ths result s lmted to forward-search labelng algorthms only. Transportaton applcatons need the computaton of fastest paths from all nodes to a set of destnatons. Hence, statc methods typcally would frst compute fastest paths from all nodes to all nodes and then extract needed solutons. Because the number of destnatons n a transportaton network usually s a small fracton of total number of nodes, a statc approach may not lead to the best soluton algorthm possble. 3. Transportaton applcatons do not necessarly satsfy the FIFO condton. 4. A statc algorthm computes shortest paths for one departure tme nterval only, but shortest paths for all possble departure tme ntervals are generally sought. Traffc management centers n ntellgent transportaton systems (ITS) must operate n real tme. The tme taken to collect data, process them, and broadcast the resultng nformaton may const-

2 Chabn Paper No tute a possble nformaton bottleneck. To avod ths bottleneck, the ITS models and algorthms must run much faster than real tme. Furthermore, transportaton applcatons usually nvolve very-large-sze networks. Typcal real-lfe networks nvolve thousands of lnks and nodes and hundreds of tme ntervals representng the tme dmenson. For nstance, a network model of the cty of Boston contans 25, lnks and 7, nodes. The tme dmenson depends on the length of analyss perod and the dscretzaton nterval. If 15-sec tme dscretzaton s adopted, 48 tme ntervals are requred to model a 2-hour mornng peak perod. Gven these key ssues n ITS and the mportance of shortest paths n ITS models and algorthms, one then must fnd the most effcent methods to solve shortest path problems n dynamc networks. Although other researchers have made valuable contrbutons n addressng ths problem, a queston remaned unanswered for the last 3 decades: What s the exact complexty of these problems and could one dscover algorthms that have the best possble run tme? The man obectve of ths study was to answer ths queston. Three types of shortest path problems n dscrete dynamc networks are addressed n ths study: the one-to-all fastest paths problem departng orgn node at a gven tme nterval, the all-to-one fastest paths problem for all departure tme ntervals, and the all-to-one mnmum cost paths problem for all departure tme ntervals. The latter two problems are of partcular nterest n the context of ITS applcatons. Note that most of the publshed papers on these topcs focus manly on problems of the frst type. The obectve n restudyng the one-to-all verson of the problem essentally was to revst and extend early results. These are establshed by usng alternatve arguments whch are shorter, smpler, and more nsghtful. For nstance, new results on the propertes of the problem are establshed. These are exploted n desgnng new and effcent soluton algorthms. A new all-to-one dscrete dynamc shortest paths algorthm that s demonstrated to be the most effcent soluton algorthm possble s proposed. Numercal tests on large networks support the analytcal fndngs. It s the frst tme that a soluton algorthm wth an optmal computaton tme complexty s proposed for what appears to be a 3-year-old problem frst addressed by Cooke and Halsey (4). DEFINITIONS AND NOTATION Let G (N, A, D, C) be a drected network where N {1,..., n} s the set of nodes and A {(,) NxN} s the set of arcs (or lnks). The number of lnks (arcs) s denoted by m. We denote by D {d (t) (,) A} the set of tme-dependent lnk travel tmes and by C {c (t) (,) A} the set of tme-dependent lnk travel costs. Functons d (t) have nteger-valued doman and postve nteger-valued range. A functon d (t) s then a dscrete and tme-dependent functon that, t s assumed, takes a statc value after a fnte number of tme ntervals M. S {,..., M 1} s the set of departure tme ntervals for whch lnk travel tmes are tme-dependent. Functons c (t) have realvalued range and nteger-valued doman. c (t) s statc when the departure tme s greater than or equal to M 1. We assume that there s no negatve cycle after departure tme nterval M 1. B() denotes the set of nodes havng an outgong arc to node and A() denotes the set of nodes havng an ngong arc from node. G (N, A, D, C ) s called a dscrete dynamc network. Arc travel tmes may possess some propertes useful n studyng and developng algorthms for dynamc networks. A well-celebrated property s the FIFO condton (5), whch may be defned n varous mathematcal forms. For nstance, the FIFO condton s vald f and only f the followng system of nequaltes holds: (,, t), t + d ( t) ( t + 1) + d ( t + 1) When the FIFO condton holds, we say that the dynamc network s FIFO. The FIFO condton also s known as the nonovertakng condton n traffc theory. The above equvalent mathematcal condton smply says that lnk ext tme functons are nondecreasng. Intutvely, ths holds f no overtakng takes place. When the FIFO condton s not satsfed, sometmes t may be preferable to wat a certan amount of tme at the start node of a lnk before embarkng on that lnk. Such watng may or may not be allowable dependng on the applcaton at hand. Therefore, two polces of watng at nodes are consdered: watng s allowed at nodes and watng s not allowed at nodes. The next secton demonstrates that n studyng fastest paths problems, the former watng polcy s a partcular case of the latter one. Shortest paths problems also depend on the type of questons they answer. Two questons are of concern here: What are the shortest paths from one orgn to all destnatons departng at nstant? and What are the shortest paths from all nodes to one destnaton node for all departure tmes? The second queston s perhaps the most relevant one n the context of dynamc management of transportaton systems. Most publshed work has dealt wth the frst queston, wth the excepton of the work by Cooke and Halsey (4) and Zlaskopoulos and Mahmassan (6). Although algorthms that answer the frst queston can be used to answer the second queston as well, such approaches would not be effcent for transportaton networks because only a reduced subset of nodes are actually destnaton nodes. For nstance, a network model for a cty lke Boston contans about 7, nodes, only 1 percent of whch are destnaton nodes. FORMULATIONS AND ALGORITHMS FOR ONE-TO-ALL FASTEST PATHS PROBLEM Watng at Nodes Is Not Allowed Ths s the most studed varaton of the problem (4,5,7). The most celebrated result for ths varaton of the problem s the followng: When the FIFO condton s vald, Dkstra s (8) algorthm can be generalzed to solve the tme-dependent fastest paths problem wth the same tme complexty as the statc shortest paths problem. Dreyfus (7) was the frst to menton ths generalzaton. Later, Kaufman and Smth (5) formally proved that ths generalzaton s vald only f the FIFO condton s satsfed. Ahn and Shn (9) provded an even earler proof. A proof that s more nsghtful and smpler follows. The proof s based on a dfferent formulaton of the problem. Let f denote the mnmum travel tme from orgn node o to node, leavng the orgn node at tme nterval. The key dea s to consder, when wrtng optmalty condtons for node, only those paths that vst prevous node at a tme greater than or equal to f. Mnmum travel tmes are then defned by the followng functonal form: f mn B( ) mn t f t + d( t) Proposton 1: If the FIFO condton s satsfed, the above formulaton of the fastest paths problem s equvalent to the followng equatons:

3 172 Paper No TRANSPORTATION RESEARCH RECORD 1645 f Proof: The equvalence holds because mn t f (t + d (t)) f + d ( f ) f the FIFO condton holds. The formulaton shown n Proposton 1 provdes a strong bass for the development of effcent algorthms. For nstance, t shows that some statc shortest paths algorthms can be extended, at no extra tme, to solve the fastest paths problem f the FIFO condton holds. Ths result s summarzed n the followng proposton: Proposton 2: If the FIFO condton s satsfed, the soluton of a fastest paths problem n such dynamc networks s equvalent to an assocated statc shortest paths problem. The generalzaton of Dkstra s algorthm frst proposed, as a heurstc, by Dreyfus (7), and later proved by Kaufman and Smth (5) and by Ahn and Shn (9), can be vewed as a partcular case of Proposton 2. Note that these generalzatons are subect to restrctons: Only a statc forward labelng process would be permtted, and t s supposed that runnng tme would depend on the number of nodes and lnks only (statc shortest path algorthms may depend on lnk travel costs as well). These restrctons were never made explct n the lterature. Proposton 3: If the FIFO condton s satsfed, any forward label settng algorthm based on functonal equatons of Proposton 1 solves the dynamc fastest paths problem and the statc shortest paths problem n the same tme complexty. When the FIFO condton s not satsfed, no algorthm that does not explctly use the tme-space expanson representaton has yet been publshed. On the bass of ths formulaton, such type of soluton algorthms were desgned. Watng at Nodes Is Allowed The case of when watng s allowed at all nodes was analyzed (results can be generalzed f watng s allowed at a subset of nodes only). The man result of ths was that ths varant of the problem s a specal case of the no-watng-s-allowed polcy varant of the problem studed above. Denote by w (t) the maxmum watng tme allowed at node and at departure tme nterval t. Frst note that when watng s allowed at node, the mnmum travel tme possble on arc (,) s gven by the functon D (t) mn w (t) + t s t(s t + d (s)). It s easy to prove that ths functon verfes the FIFO condton f unlmted watng s allowed. Because watng s allowed at nodes, mnmum travel tmes are now gven by the followng functonal form: f mn B( ) f + d f ; mn B( ) mn t f t + D( t) Proposton 4: The watng-s-allowed varant s a partcular case of the watng-s-not-allowed varant studed above. Moreover, f unlmted watng s allowed at all nodes, results of propostons 1, 2, and 3 hold wthout the FIFO requrement on lnk travel tmes. Proof: The above functonal form shows that when watng s allowed at all nodes, ths varant of the fastest paths problem s equvalent to a fastest paths problem wthout watng at nodes and wth lnk delay functons D (t) [nstead of d (t)]. Because D (t) verfes the FIFO condton f unlmted watng s allowed, results of propostons 1, 2, and 3 hold. The fastest paths problem wth no watng s allowed at nodes and FIFO condton not satsfed s the most dffcult varant of the fastest paths problem. ALL-TO-ONE FOR ALL DEPARTURE TIMES FASTEST PATHS PROBLEM As shown above, the all-to-one for all startng tmes fastest paths problem s the most relevant varant of fastest paths problems n the context of dynamc management of transportaton systems. A backward star formulaton of ths problem, when watng s not allowed, s presented here, and a property that s then exploted n developng a new soluton algorthm s exploted. Formulaton Denote by π (t) the fastest travel tme to destnaton q departng node at tme t. The mnmum travel tmes are then defned by the followng functonal form: mn A() d t π t d t q π ( t) + + ; ; q Ths s a well-known optmalty condton that was used by Cooke and Halsey (4) and Zlaskopoulos and Mahmassan (6) to desgn soluton algorthms that have respectvely O(n 3 M 2 ) and O(nmM 2 ) as worst-case runnng tme complextes. An optmal algorthm for ths problem s presented below, based on the followng proposton: Proposton 5: Labels π (t) can be set n a decreasng order of departure tme ntervals. Proof: Because all arc travel tmes are postve ntegers, labels correspondng to tme step t never update labels correspondng to tme steps greater than t. The above result mplctly reflects the acyclc property, along the tme dmenson, of the tme-space expanson of a dscrete dynamc network. Decreasng Order of Tme Algorthm For departure tme ntervals greater than or equal to M 1, the computaton of fastest paths s equvalent to a statc shortest paths problem. A decreasng order of tme (DOT) algorthm s proposed here that s based on the result of Proposton 5. The man loop of the algorthm s carred out n decreasng order of tme. The algorthm assumes that a statc shortest paths procedure, wth an optmal runnng tme SSP, s gven: Step (Intalzaton): π ( t), ( q), π ( t), ( t < M 1) Step 2 (Man Loop): π( M 1) Statc Shortest Paths( d( M 1), q) where π ( t) π ( M 1), ( t M 1, ) q For t M 2down to do: For (, ) A do: π ( t) mn π ( t), d ( t) + π t + d ( t) ( )

4 Chabn Paper No Proposton 6: Algorthm DOT solves for the all-to-one fastest paths problem, wth a seral computaton tme n θ(ssp + nm + mm ). Proof: The optmalty of the algorthm follows from Proposton 5. The order analyss of runnng tme follows n a straghtforward manner by countng the number of operatons appearng n the statements of algorthm DOT. Proposton 7: The complexty of the all-to-one fastest paths problem, for all startng tmes, s n Ω(SSP + nm + mm ). Hence, algorthm DOT has an optmal runnng tme complexty (no algorthm wth a better runnng tme complexty can be found). Proof: The problem has the above complexty because every soluton algorthm has to access all arc data (mm ), ntalze network node labels because fastest paths for all departure tmes are sought (nm ), and compute an all-to-one fastest paths tree for departure tme ntervals beyond departure tme M 1 (SSP). Usng results of Proposton 6, algorthm DOT s then optmal. ALL-TO-ONE FOR ALL DEPARTURE TIMES MINIMUM COST PATHS PROBLEM In ths secton, the results establshed n prevous sectons are expanded to the mnmum cost paths problem. Formulaton Let C (t) denote the mnmum travel cost from node, departng at tme t, to destnaton q. Mnmum travel costs are then defned by the followng functonal form: mn A() c t C t d t q C ( t) + + ; ; q These equatons extend the optmalty condtons descrbed n prevous sectons for the fastest paths problem. Here they are used to extend results establshed for the fastest paths problem. In partcular, the exact complexty of the mnmum cost paths problem s determned and an extenson of algorthm DOT that runs n an optmal tme s shown. Proposton 8: Labels C (t) can be set n a decreasng order of departure tme ntervals. Proof: Because all arc travel tmes are postve ntegers, labels correspondng to tme steps t do not depend on labels correspondng to tme ntervals smaller than t. Ths result mplctly reflects the acyclc property, along the tme dmenson, of the tme-space expanson of a dscrete dynamc network. Note that lnk travel costs can take any real number value, ncludng negatve numbers. Extenson of Algorthm DOT To Compute Mnmum Cost Paths Note that for departure tme ntervals greater than or equal to M 1, lnk travel tmes and costs become statc. Labels C (M 1) are then the mnmum travel costs for all departures takng place at a tme greater or equal to M 1. Moreover, solvng for these labels s equvalent to solvng a statc shortest paths problem usng c (M 1) as lnk dstances. Statc shortest paths algorthms to be used to solve for C (M 1) depend on the assumptons on c (M 1). An assumpton requred by all statc shortest paths algorthms s that there s no negatve cycle n the network; otherwse, one can crculate an nfnte number of tmes leadng to an nfnte decrease n label values. Label settng algorthms requre that c (M 1) are nonnegatve. Label correctng algorthms work even for nonnegatve lnk dstances. The algorthm proposed below assumes the avalablty of an approprate statc shortest paths procedure. If many alternatves are avalable, one should use a procedure wth the fastest run tme f the obectve s to obtan algorthms wth optmal run tme. An extenson of algorthm DOT s proposed for mnmum cost paths and s based on the result of Proposton 8. The man loop of the algorthm s carred out n decreasng order of tme. The algorthm assumes that a vald statc shortest paths procedure, wth an optmal runnng tme SSP, s avalable. Ths s not a restrcton because very effcent statc shortest path solvers are wdely avalable and are based on a vast body of research results publshed durng the last 4 decades. The algorthm DOT for mnmum cost paths s as follows: Step (Intalzaton): C ( t), ( q), C ( t), ( t < M 1) C( M 1) Statc Shortest Paths ( c( M 1), q) where C ( t) C ( M 1), ( t M 1, ) Step 2 (Man Loop): For t M 2down to do: For (, ) A do: ( ) C ( t) mn C ( t), c ( t) + C t + d ( t) q Proposton 9 and Proposton 1 respectvely generalze Proposton 6 and Proposton 7. The proofs are not provded because these are straghtforward generalzatons of respectve proofs gven n Secton 4. Proposton 9: Algorthm DOT solves for the all-to-one mnmum cost paths problem, wth a seral computaton tme n θ(ssp + nm + mm). Proposton 1: The complexty of the all-to-one shortest paths problem, for all startng tmes, s n Ω(SSP + nm + mm). Hence, the above extenson of algorthm DOT has an optmal runnng tme complexty (no algorthm wth a better runnng tme complexty can be found). COMPUTER IMPLEMENTATIONS AND EXPERIMENTAL EVALUATION The four mplemented soluton algorthms for the all-to-one fastest paths problem are algorthm DOT and three dynamc adaptatons of label-correctng algorthms usng three types of data structures for node canddates lst: the Deque data structure (1) as descrbed n Zlaskopoulos and Mahmassan (6), the two-queue data structure (11), and a three-queue data structure that extends the noton of the two-queue data structure (12). All algorthms were coded n C++. Algorthm DOT has very attractve propertes. Frst, ts computer codng s a very easy task. Second, algorthm DOT does not need complcated data structures; a basc two-dmensonal array suffces.

5 174 Paper No TRANSPORTATION RESEARCH RECORD 1645 TABLE 1 Computaton Tmes for 3, Nodes, 9, Lnks, and Dfferent Tme Intervals for All-to-One Fastest Paths Problem TABLE 3 Computaton Tmes for 1, Lnks, 6 Tmes Steps, and Varyng Number of Nodes for All-to- One Fastest Paths Problem Fnally, algorthm DOT has an optmal runnng tme and does not suffer from worst-case behavor that s typcal to other soluton algorthms such as label-correctng algorthms. An extensve evaluaton of the algorthms was carred out. Other known algorthms were mplemented and evaluated. Label-correctng algorthms generally had the closest performances to algorthm DOT. Performance of label-correctng algorthms depends on network topology and the dynamcs of lnk travel tmes. These algorthms present best-case as well as worst-case behavors. Tables 1, 2, and 3 show results for nstances of the problem n whch label-correctng algorthms gave ther best computaton tmes. These nstances correspond to very sparse networks that contan fewer cycles and then are less dffcult to solve by usng label-correctng algorthms. These nstances were generated by usng a dscrete dynamc network generator that was developed to test computer codes on networks havng dfferent topologes, number of cycles, denstes, and lnk travel tmes. Runnng tmes are wall clock tmes obtaned on an SGI Indy workstaton. To gve an ndcaton of how fast algorthm DOT s, Step 2 typcally requres only three tmes of what s requred as computng tme to ntalze node labels to nfnty on a network composed of 3, nodes and 9, arcs. As may be expected from the above theoretcal analyss, algorthm DOT has proved to be better than label-correctng algorthms. Because numercal results presented here correspond to best cases of label-correctng algorthms, algorthm DOT always should perform better than label-correctng methods. TABLE 2 Computaton Tmes for 8 Nodes, 6 Tmes Steps, and Varyng Number of Lnks for All-to-One Fastest Paths Problem Table 1 and 2 show that algorthm run tmes are lnear n the number of nodes and lnks. For algorthm DOT ths s a general behavor as demonstrated theoretcally above. Label-correctng methods may lead to a hgher than lnear runnng tme ncrease as a functon of the number of nodes and lnks. The obectve here was not to analyze label-correctng algorthms but nstead to evaluate the relatve performance of algorthm DOT and label-correctng methods. The results n Table 1 and 2 are consstent wth the obectve to report only on problem nstances n whch label-correctng algorthms gave ther best run tmes. (The obectve was not to compare algorthm DOT to worse-case behavors of label-correctng algorthms). Results of Table 3 show that the run tmes of label-correctng algorthms decrease as a functon of the number of nodes. At frst glance, ths may seem ncorrect. However, because the number of lnks s constant, as the number of nodes ncreases the network tends toward a tree. Ths led to a reducton n the number of node revsts and hence led to a reducton n runnng tme. Algorthm DOT, conversely, has an ncreasng run tme as a functon of the number of nodes, as shown theoretcally. Table 4 compares two versons of algorthm DOT: the fastest paths verson and the mnmum cost paths verson. As may be expected, run tmes are almost smlar. The latter verson takes an extra small fracton of run tme. Ths s attrbutable to the extra tme requred to manage the lnk travel costs array. Algorthm DOT also was expermented wth on a real-lfe network emanatng from the hghway network of the cty of Amsterdam. The obectve was to test the effects on run tmes of the length of dscretzaton ntervals. The travel tmes used were obtaned from the output results of an analytcal dynamc traffc assgnment model developed by Chabn and He (13) and He (14). Table 5 summarzes the results of ths experence. Table 5 shows that run tme of algorthm DOT s proportonal to the number of tme ntervals or, equv- TABLE 4 Computaton Tmes for 1, Lnks, 6 Tmes Steps, and Varyng Number of Nodes of Algorthm DOT for Fastest and Mnmum Cost Path Problems

6 Chabn Paper No alently, s nversely proportonal to the length of a tme nterval for a gven analyss-perod length. CONCLUSIONS TABLE 5 Computaton Tmes for Amsterdam Hghway Network wth 196 Nodes, 38 Lnks, and Analyss Perod of 2.75 Hours Two types of shortest paths problems n dscrete dynamc networks were studed: the one-to-all for one departure tme nterval and the all-to-one for all departure tme ntervals. Early results were revsted. New propertes were establshed and exploted n desgnng soluton algorthms. A new soluton algorthm was proposed for all-to-one all departure tme ntervals, fastest and mnmum cost path problems. It was proved, analytcally, that the proposed algorthm s the most effcent soluton method possble. Extensve expermental evaluaton supports analytcal results. It s expected that these fndngs wll be of maor beneft to research and development actvtes n the area of dynamc management of transportaton systems. On an SGI Indy workstaton, the new algorthm, algorthm DOT, computes shortest paths from all nodes to one destnaton node for all departure tmes wthn.5 sec for dynamc networks composed of 3, nodes, 9, arcs, and 9 tmes ntervals. The number of destnaton nodes for such transportaton networks typcally s n the order of 3. Hence, the determnaton of shortest paths would requre 2.5 mn of computaton tme on an SGI Indy workstaton. To acheve faster run tme, hgh performance mplementatons of algorthm DOT are the subect of ongong research. REFERENCES 1. Deo, N., and C. Y. Pang. Shortest Path Algorthms: Taxonomy and Annotaton. Networks, Vol. 14, 1984, pp Chabn, I. A New Short Paths Algorthm for Dscrete Dynamc Networks. Proc., 8th IFAC Symposum on Transport Systems, 1997, pp Orda, A., and R. Rom. Mnmum Wat Paths n Tme-Dependent Networks. Networks, Vol. 21, 1991, pp Cooke, K. L., and E. Halsey. The Shortest Route Through a Network wth Tme-Dependent Internodal Transt Tmes. Journal of Mathematcal Analyss Applcatons, Vol. 14, 1966, pp Kaufman, D. E., and R. L. Smth. Fastest Paths n Tme-Dependent Networks for Intellgent Vehcle-Hghway Systems Applcaton. IVHS Journal, Vol. 1, 1993, pp Zlaskopoulos, A. K., and H. S. Mahmassan. Tme-Dependent Shortest Path Algorthm for Real-Tme Intellgent Vehcle/Hghway System. In Transportaton Research Record 148, TRB, Natonal Research Councl, Washngton, D.C., 1993, pp Dreyfus, S. E., An Apprasal of Some Shortest-Path Algorthms. Operatons Research, Vol. 17, 1969, pp Dkstra, E. W. A Note on Two Problems n Connecton wth Graphs. Numersche Mathematk, Vol. 1, 1959, pp Ahn, B. H., and J. Y. Shn. Vehcle Routng wth Tme Wndows and Tme-Varyng Congeston. Journal of the Operatonal Research Socety, Vol. 42, 1991, pp Pape, U. Implementaton and Effcency of Moore-Algorthms for the Shortest Route Problem. Mathematcal Programmng, Vol. 7, 1974, pp Gallo, G., and S. Pallotno. Shortest Paths Algorthms. Annals of Operatons Research, Vol. 13, 1988, pp Chabn, I. Fastest Paths n Dynamc Networks. Transportaton Scence (n press). 13. Chabn, I., and Y. He. An Analytcal Approach to Dynamc Traffc Assgnment wth Applcatons to Intellgent Transportaton Systems. Presented at Optmzaton Days, Montreal, May He, Y. A Flow-Based Approach to Dynamc Traffc Assgnment Problem: Formulatons, Algorthms and Computer Implementatons. M.S. thess. Massachusetts Insttute of Technology, Cambrdge, Publcaton of ths paper sponsored by Commttee on Transportaton Network Modelng.