Fastest-Path Computation

Transcription

1 Fastest-Path Computation DONGHUI ZHANG Coege of Computer & Information Science Northeastern University Synonyms fastest route; driving direction Definition In the United states, ony 9.% of the househods do not have cars. Driving is part of peope s daiy ife. GIS systems ike MapQuest and MapPoint are heaviy reied on to provide driving directions. However, surprisingy enough, existing systems either ignore the driving speed on road networks, or assume the speed remains constant on the road segments. In both cases the users preferred eaving time does not affect the query resut. For instance, MapQuest does not ask the users to input the day and time of driving. However, during rush hours, inbound highways to big cities have much ower speed than usua. So a fastest path computed during non-rush hours, which may consists of some inbound highway segments, may not remain the fastest path during rush hours. Consider a road network modeed as a graph, where each node is a road intersection and each edge is a road segment. Let each edge store a speed pattern, e.g. a piecewise-constant function. For instance, in a working day, during rush hour (say from 7am to 9am) the speed is 0. mies per minute (mpm), and at other times of the day the speed is 1mpm. The Time Interva A Fastest Paths (afp) Query is defined as foows. Given a source node s, an end node e, and a eaving time interva at s, the afp query asks to enumerate a fastest paths, each corresponding to a disjoint sub-interva of eaving time. The union of a sub-intervas shoud cover the entire query time interva. An afp query exampe is: I may eave for work any time between 7am and 9am; pease suggest a fastest paths, e.g. take route A if the eaving time is between 7 and 7:4, and take route B otherwise. It is aso interesting to sove the afp probem with an arriva time interva. For instance: I need to drive from my home to New York Internationa Airport. Pease suggest a fastest paths if the anticipated arriva time is between pm and pm. There are two characteristics that distinguish the afp probem from other shortest/fastest path probems. The query is associated with a eaving/arriva time INTERVAL, not a time instant. In fact if the eaving time were fixed as a time instant, many existing agorithms coud be appied, such as the Dijkstra s shortest path computation and the A* agorithm. Time is continuous. If time were distinct, e.g. one can ony eave precisey at the top of the hours, one coud run existing time-instant agorithms mutipe times. Historica Background Most existing work on path computation has been focused on the shortest-path probem. Severa extensions of the Dijkstra agorithm have been proposed, mainy focusing on the maintenance of the priority queue. The A* agorithm [8] finds a path from a given start node to a given end node by empoying a heuristic estimate. Each node is ranked by an estimate of the best route that goes through that node. A* visits the nodes in order of this heuristic estimate. A survey on shortest-path computation appeared in [11]. Performance anaysis and experimenta resuts regarding the secondary-memory adaptation of shortest path agorithms can be found in [4, 1]. The work in [] contributes on finding the shortest path that satisfies some spatia constraints. A graph index that can be used to prune the search space was proposed in [1]. One promising idea to dea with arge-scae networks is to partition a network into fragments. The boundary nodes, which are nodes having direct inks to other fragments, construct the nodes of a high-eve, smaer graph. This idea of hierarchica path-finding has been expored in the context of computer networks [] and in the context of transportation systems []. In [1], the materiaization trade-off in hierarchica shortest path agorithms is examined. In terms of fastest-path computations, there exists work assuming the discrete mode [1, 9], the fow speed mode [14], or theoretica modes beyond road network [10]. The discrete mode [1, 9] assumes discrete time. The fow speed mode work [14] addresses the fastest path query for a given eaving time instant, not a time interva. The theoretica mode work [10] suggests operations on trave functions that are necessary without investigating how this operations can be supported. To compute fastest paths on a road network with a eaving/arriva time interva and with continuous time, one can utiize a nove extension of the A* agorithm. More detais are given beow. Scientific Fundamentas A simpe extension to the A* agorithm can NOT be used to sove the afp query. Let n 0 be the node to be expanded next and et n 0 have three neighbor nodes, n 1, n and n. A* picks the neighbor node n i (i [1..]) to continue expanding if the trave time from s to n i pus the estimated trave time from n i to e is the smaest. The probem is that since the eaving time is not a fixed time instant, depending on the eaving time instant different neighbors shoud be picked.

2 One possibe soution is to expand a such neighbors simutaneousy. However, expanding a picked neighbors may resut in an exponentia number of paths being expanded regardess of the size of the answer set. Instead, [7] proposed a new agorithm caed IntA- FastestPaths. The main idea of the agorithm is summarized beow: 1. Maintain a priority queue of expanded paths, each of which starts with s. For each path s n i, maintain T (,s n i ) + T est (n i e) as a piecewiseinear function of eaving time interva I. Here, T (,s n i ) is the trave time from s to n i, measured as a function of eaving time. T est (n i e) is a ower bound estimation function of the trave time from n i to the end node e. A straightforward ower-bound estimator is d euc (n i,e)/v max, which is the Eucidean distance between n i and e, divided by the max speed in the network. A better estimator caed the boundary node estimator which is described ater.. Simiar to the A* Agorithm, in each iteration pick a path from the priority queue to expand. Pick the path, whose maintained function s minimum vaue during I is the minimum among a paths.. Maintain a specia trave-time function caed the ower border function. It is the ower border of trave time functions for a identified paths (i.e. paths aready picked from the priority queue) that end to e. In other words, for any time instant I, the ower border function has a vaue equa to the minimum vaue of a trave time functions of identified paths from s to e. This function consists of mutipe trave time functions, each corresponding to some path from s to e and some subinterva of I during which this path is the fastest. 4. Stop either when there is no more path eft in the priority queue, or if the path picked to be expand next has a minimum vaue no ess than the maximum vaue of the ower border function. Report the ower border function as the answer to the afp query. Beow is a running exampe. The exampe invoves a simpe road network given in Figure 1. The goa is to find the fastest path from s to e at some time during I =[:0-7:0]. Initiay, the priority queue contains ony one entry, which corresponds to the unexpanded node s. It has two neighbors e and n. It can be derived that, T ( [:0-7:0), s e) = min and T ( [:0-7:0),s n) =, [:0-:4) (7:00 ) +, [:4-7:00), [7:00-7:0] s e 1 n s e: [-8):1/ s n: [-7):1/, [7-8):1 n e: [-7:08):1/, [7:08-8):1/10 Fastest-Path Computation. Figure 1 A simpe road network. Distances are given on the edges. Speed patterns (#mpm) are given at the right of the network. As expressed in step 1 of Agorithm IntAFastestPaths, in the priority queue, the paths are ordered not by T (), but by T () + T est (). The functions of the two paths are compared in Figure. Here, T est (n e) = 1min, since d euc (n,e) = 1 mie and v max = 1mpm. 7 T()+Test() s > e s > n :0 :4 7:00 7:0 Fastest-Path Computation. Figure Comparison of the functions T () + T est () associated with paths s e and s n. According to step, the path s n to be expanded next, since its minimum vaue,, is smaer than the minimum vaue,, of the path s e. In genera, to expand a path s n, first a the required information for n and its adjacent nodes needs to be retrieved, Then, for each neighbor n j of n the foowing steps need to be foowed: Given the trave time function for the path s n and the eaving time interva I from s, determine the time interva during which the trave time function for the road segment n n j is needed. Determine the time instants t 1, t,... I at which the resuting function, i.e. the trave time function for the path s n j, T ( I, s n j ), changes from one inear function to another. For each time interva [t 1,t ),..., determine the corresponding inear function of the resuting function T ( I, s n j ). In this exampe, the time interva for n e is determined to be [:, 7:07] as shown in Figure. At time :0 (start of I), the trave time aong the path s n is minutes. Therefore, the start of the eaving time interva for n e, i.e. the start of arriva time interva to n, is :0+min = :. Simiary, the end of the eaving time interva is 7:0+min = 7:07. During the time interva [:-7:07], the trave time

3 T(, s > n) Retrieve the inear function of T (, s n) at time t. Let it be α + β. :0 :4: 7:00 7:07:07 Fastest-Path Computation. Figure The time interva, [:-7:07], during which the speed on n e is needed. on n e, T ( [:-7:07], n e) is {, if [:-7:0) 10 7 (7:08 ), if [7:0-7:07] There are two cases that trigger the resuting trave time function T (, s n e) to change from one inear function to another. In the first, simpe case the function T (, s n) changes. The time instants at which the resuting function changes are the ones at which T (, s n) changes. In Figure 4, these correspond to time instants :0, :4 and 7:00. In the second, trickier case, the changes of the resuting function are triggered by the changes of T (, n e), e.g. at time 7:0. In this exampe, one can determine that at time 7:0, T (, s n e) changes. The reason is that if one eaves s at 7:0, since the trave time on s n is minutes, one wi arrive at n at 7:0. At that time the trave time function of n e changes. To find the time instant 7:0, compute the intersection of the function T (, s n) with a 1 o ine passing through the point (7:0, 0). The time instant 7:0 is the eaving time corresponding to that intersection point. Trave time 9 T(, s => n > e) T(, n > e) T(, s => n) :0 :4 : 7:00 7:0 7:0 7:07 Retrieve the inear function of T (, n e) at time t = t + (α t + β). Let it be γ + δ. Compute a new inear function (α + β) + (γ ( + α + β) + δ)), which can be re-written as (α γ + α + γ) + (β γ + β + δ). This is the inear function as part of T (, s n e), for the time interva from t to the next identified time instant. For instance, the combined function T ( I, n e), which is shown in Figure 4, is computed as foows. At t =:0, the first inear function is a constant function. So t = t + =:. The second inear function starting with : is another constant function. So the combined function is 9, which is vaid unti the next identified time instant. At t =:4, the first inear function is (7:00 )+. Therefore t =:4+=7:00. The second inear function is. The combined function is (7:00 ) +. At t =7:00, the first function is constant. At t =7:00+=7:0, the second function is. So the combined function is. Finay, at t =7:0, the first function is, and at t =7:0+=7:0, the second function as 10 7(7:08 ). And thus the combined function is +(10 7(7:08 ( + ))) = 1 7 (7:0 ). After the expansion, the priority queue contains two functions, as shown in Figure. Note that in both functions, the ower bound estimation part is 0, since both paths aready end with e. T()+Test() 9 T(, s => n > e) :0 :4 7:00 7:0 7:0 T(, s > e) Fastest-Path Computation. Figure The two functions in the priority queue. Fastest-Path Computation. Figure 4 The time instants at which T (,s n e) changes to another inear function, and the T (, s n e) function. Now that a the four time instants :0, :4, 7:00, and 7:0 have been determined, the 4-piece function T ( I, s n e) can be derived by combining T (, s n) and T (n e). For each, T (, s n e) is equa to T (, s n) pus T (, n e), where is the time at which node n is reached. That is, = + T (, s n). The foowing agorithm shoud be used to expand a path, for every identified time instant t {t 1,t,...} (e.g. :0): The next step of Agorithm IntAFastestPaths is to pick the path s n e, as its minimum vaue (min) is gobay the smaest in the queue. An important question that arises here is when to stop expanding, as expanding a paths to the end node is prohibitivey expensive. The agorithm terminates when the next path has a minimum vaue no ess than the maximum vaue of the maintained ower border function. When there is ony one identified path that ends with e, the ower border function is the function of this path. In Figure, T (, s n e) is the ower border function. As each new path ending with e is identified, its function is combined with the previous ower border

4 T()+Test() 9 T(, s => n > e) :0 :4 7:00 7:0 7:0 :8:0 7:0: T(, s > e) Fastest-Path Computation. Figure The ower border and the resut for Query. function. E.g. in Figure the new ower border function, after the function T (, s e) is removed from the priority queue, is shown as the thick poyine. The agorithm can terminate if the next path to be expanded has a minimum vaue no ess than the maximum vaue of the ower border function (in this case, ).Since the maximum vaue of the ower border keeps decreasing, whie the minimum trave time of paths in the priority queue keeps increasing, the agorithm IntAFastestPaths is expected to terminate very fast. In this exampe, the set of a fastest paths from s to e when [:0-7:0] is: s e, if [:0-:8:0) s n e, if [:8:0-7:0:) s e, if [7:0:-7:0] Key Appications The key appication of fastest-path computation is road navigation systems. Exampe systems are mapquest. com, oca.ive.com, and MapPoint. Such systems can produce better driving directions if integrated with traffic patterns and fastest-path computation techniques. Future Directions To speed up the cacuation, the road network shoud be partitioned. At the index eve, each partition is treated as a singe network node and the detais within each partition are omitted. Pre-computation are performed to cacuate the trave time from each input edge to each output edge. The partitioning can be performed hierarchicay. Another direction is that, to simpify the computed fastest paths, the agorithm shoud be extended to aow the users to provide a maximum number changes in road names. Cross References 1. Dynamic Trave Time Maps. Modeing Road Networks. Query Processing in Road Network Databases 4. Routing Agorithms. Trip Panning Queries in Road Network Databases Finay, the boundary-node estimator, which is a owerbound trave time from n i to e and which is used to improve the efficiency of the agorithm, is described beow. Partition the space into non-overapping ces []. A boundary node [] of a ce is a node directy connected with some other node in a different ce. That is, any path inking a node in a ce C 1 with some node in a different ce C must go through at east two boundary nodes, one in C 1 and one in C. For each pair of ces, (C 1, C ), pre-compute the fastest trave time (function) from each boundary node in C 1 to each boundary node in C. For each node inside a ce, pre-compute the fastest trave time from and to each boundary node. At query time, n i and e are given. Since any path from n i to e must go through some boundary node in the ce of n i and through some boundary node in the ce of e, a ower-bound estimator can be derived as the summation of three parts: (1) the fastest time from n i to its nearest boundary node, () the fastest time from some boundary node in n i s ce to some boundary node in e s ce, and () the fastest time from e s nearest boundary node to e.. Vehice Routing Agorithms Recommended Reading [1] I. Chabini. Discrete Dynamic Shortest Path Probems in Transportation Appications. Transportation Research Record, 14:170 17, [] V.P. Chakka, A. Everspaugh, and J.M. Pate. Indexing Large Trajectory Data Sets With SETI. In Biennia Conf. on Innovative Data Systems Research (CIDR), 00. [] Y. Huang, N. Jing, and E. Rundensteiner. Spatia Joins Using R-trees: Breadth-First Traversa with Goba Optimizations. In VLDB, pages 9 40, [4] B. Jiang. I/O-Efficiency of Shortest Path Agorithms: An Anaysis. In ICDE, pages 1 19, 199. [] N. Jing, Y.-W. Huang, and E. A. Rundensteiner. Hierarchica Encoded Path Views for Path Query Processing: An Optima Mode and Its Performance Evauation. TKDE, 10():409 4, [] F. Kamoun and L. Keinrock. Hierarchica Routing for Large Networks: Performance Evauation and Optimization. Computer Networks, 1:1 174, 1977.

5 [7] E. Kanouas, Y. Du, T. Xia, and D. Zhang. Finding Fastest Paths on A Road Network with Speed Patterns. In ICDE, 00. [8] R.-M. Kung, E. N. Hanson, Y. E. Ioannidis, T. K. Seis, L. D. Shapiro, and M. Stonebraker. Heuristic Search in Data Base Systems. In Expert Database Systems Workshop (EDS), pages 7 48, [9] K. Nachtiga. Time depending shortest-path probems with appications to raiway networks. European Journa of Operationa Research, 8:14 1, 199. [10] A. Orda and R. Rom. Minimum Weight Paths in Time-Dependent Networks. Networks: An Internationa Journa, 1, [11] S. Paottino and M. G. Scuteà. Shortest Path Agorithms in Transportation Modes: Cassica and Innovative Aspects. In P. Marcotte and S. Nguyen, editors, Equiibrium and Advanced Transportation Modeing, pages Kuwer Academic Pubishers, [1] S. Shekhar, A. Fetterer, and B. Goya. Materiaization Trade-Offs in Hierarchica Shortest Path Agorithms. In SSTD, pages , [1] S. Shekhar, A. Kohi, and M. Coye. Path Computation Agorithms for Advanced Traveer Information System (ATIS). In ICDE, pages 1 9, 199. [14] K. Sung, M.G.H. Be, M. Seong, and S. Park. Shortest paths in a network with time-dependent fow speeds. European Journa of Operationa Research, 11(1): 9, 000. [1] J. L. Zhao and A. Zaki. Spatia Data Traversa in Road Map Databases: A Graph Indexing Approach. In Proc. of Int. Conf. on Information and Knowedge Management (CIKM), pages, 1994.