Fack, Veerle

Vanhove, Stéphanie

English

Ghent University Academic Bibliography

Fast Generation of Many Shortest Path AlternativesStéphanie Vanhove1, Veerle Fack21,2Department of Applied Mathematics and Computer ScienceGhent UniversityKrijgslaan 281 – S9, 9000 Ghent, BelgiumEmail: 1Stephanie.Vanhove@UGent.be, 2Veerle.Fack@UGent.be1. IntroductionRoute  planning  applications  rely  heavily  on  shortest  path  algorithms.  However,  in some cases a ranking of several possible “short” paths is desired, rather than a single shortest path. For example, when trying to optimize multiple parameters, say beautiful scenery as well as travel time, a possible solution is to calculate a ranking of the fastest routes  and  then  pick  the  most  scenic  one  by  hand.  Also  techniques  for  finding dissimilar paths (Akgün and Erkut 2000, Marti et al. 2009) often generate a large set of candidate paths, from which a suitable collection is selected. Figure 1 shows different feasible routes from Ghent to Antwerp, where it is up to the driver to choose one. Figure 1. Different possible routes from Ghent to Antwerp.(Image source: Google Maps)Determining a ranking of shortest paths is called the k shortest paths problem, where k is the number of paths to be calculated. The road network is modeled as a graph, with road segments represented by arcs and intersections by nodes. Arcs have non-negative weight, representing distance or travel time. In this paper we focus on calculating the k simple (i.e. loopless) shortest paths, since loops are usually not useful for path finding in road networks.  Yen's algorithm (Yen 1971) is the basis for many of the currently known algorithms (e.g. Herschberger et al. 2007, Gotthilf and Lewenstein 2009). Path  finding  algorithms  are  often  used  in  interactive  applications,  which  makes them time-critical: a query time of e.g. 12 milliseconds versus 14 seconds can make a big difference. This fact motivates looking for a faster method which does not aim at finding the exact k shortest paths, but still misses only a few of them. Recently some theoretical approximation algorithms were developed (Roddity 2007, Bernstein 2010). In this paper we present a new heuristic that works very well in practice, i.e. which finds a majority of the paths, with only a slight increase in path length, and which is much faster than the exact algorithm.2. Deviation Path AlgorithmsOur  heuristic  is  based  on  Yen's  algorithm,  which  in  its  turn  is  an  example  of  a deviation path algorithm. These are based on the idea that the  i-th shortest path will always deviate at some node from a path previously found. Considering a path P of n nodes, there are only n-1 possible deviation positions: a deviating path D either differs from P immediately from the first node, or it coincides with P up to the 2nd, 3rd, … or n-1th node and then deviates from there on. Figure 2 sketches all possible situations for a path  P  with 5 nodes (note that  D can deviate from a certain node to join  P again later). Figure 2. All possible deviations (dashed lines) from a 5-node path (solid lines).Deviation  path  algorithms  maintain  a  priority  queue  Q of  candidate  paths.  In  every iteration the next shortest path is fetched from Q and added to the results. This path is then used as a basis for calculating new candidate paths which are added to  Q. This process is repeated until k shortest paths are found.Different deviation path algorithms differ in their strategy for calculating new paths. Yen's algorithm calculates new paths from start node s to target node t based on a path P as follows: 1:  For every arc u­v on P: 2:      Remove u­v and all nodes preceding u in P from graph.3:      P'1   subpath from s to u in P.←4:      P'2   shortest path from u to t in modified graph.←5:      P'   append P'← 2 to P'16:      Add P' to Q.7:      Restore graph. The shortest path in line 4 can be calculated using any shortest path algorithm, such as the well-known algorithm of Dijkstra. Since this line is executed many times, this can be very time-consuming. The heuristic we describe in the next section aims at speeding up the algorithm by avoiding these shortest path calculations.3. Heuristic for Calculating DeviationsAt the start  of  the algorithm,  we construct  a  backward shortest  path tree  T,  which stores a shortest path from every node in the graph to the target node t, thus allowing fast retrieval of these paths.Instead of actually computing the shortest path from  u to  t (line 4) we obtain an approximation for this shortest path by looking at the possible deviations obtained by concatenating every outgoing arc  u-x (except  u-v) with the shortest path from x to  t fetched from T. However, since such a path may contain nodes and arcs which were in the meantime removed from the graph, it is necessary to check if the entire path still exists in the graph. Only if such is the case the path is appended to P'1 and added to Q.Figure 3 illustrates this idea.  On the current path  P from  s to  t  (solid lines),  red crosses indicate forbidden nodes and arcs. To find a detour from u to t, we consider the other outgoing arcs from u (dashed lines) and look up the paths from these neighbours to t (dotted lines) in the shortest path tree T. Figure 3. How the heuristic works. ComplexityThe algorithm of Yen has a time complexity of O(k n (m + n log n)), with n the number of nodes and m the number of arcs. However, road networks are sparse so m = ɵ(n), resulting  in  a  time  complexity  of  O(k  n²  log  n).  Our  heuristic  reduces  this  time complexity to O(k n²). However, since this upper bound is hardly ever reached, the speed-up is much better in practice, as the results will show. Details are omitted here for space reasons.4. Results and discussionWe compared  results  of  the  heuristic  with exact  results  for  several  European road networks. In this paper we present only results for the Navteq Belgian road network, but similar results for other road networks were obtained. 4.1 Quality of the paths foundSince our heuristic approach does not guarantee an exact set  of  k shortest  paths, a comparison of the paths found is necessary. The results can be seen in Figure 4. The value e(k) represents the ranking of the path in an exact set of k shortest paths. E.g. if e(k) = 103 for a given query with k = 100, then the 100th path found by the heuristic is actually the 103rd shortest path, meaning that it “missed” three paths. The value  p(k) shows the percentual weight increase of the path. E.g. if p(k) = 0.68% for k = 100, then the 100th shortest  path found by the heuristic is 0.68% longer than the “real” 100th shortest path.The heuristic performs surprisingly well in some cases, missing very few or even no paths at all. In other cases, more paths are missed, but even then the values for  p(k) remain very small, typically less than 1%. Especially for routing applications, this is very acceptable since a travel time increase of less than 1% can almost be neglected.4.2 Time performanceOf course a heuristic approach is only beneficial if it is significantly faster than the exact algorithm. Time measurements were performed for both the algorithm of Yen and the  heuristic  on an  Intel  dual  core 2.13 GHz machine  with 2 Gigabyte  RAM running  Linux.  The  algorithms  were  implemented,  compiled  and executed  in  Java version 1.6.0_16. The speed-up of the heuristic compared to the exact algorithm of Yen can also be seen in Figure 4.The speed-up is clearly significant, even though it is very variable. For most queries the heuristic is more than 50 times faster than the exact algorithm, often even more than 100 or even 1,000 times faster. This shows that the heuristic is fast enough to be very useful in practice.Figure 4. Results for the Navteq Belgium road network (564,477 vertices and 1,300,765 arcs). For k=100, 1,000 and 10,000 the exact ranking e(k) of the kth path found by the heuristic, the percentual weight increase p(k) and the speedup are shown, each time for 100 random queries. Dots marked in red indicate a lower bound instead of an exact value because of memory limitations.4.3 ConclusionThe above experiments clearly  show that significant  speed-ups can be achieved by compromising only slightly on path quality. The new heuristic thus offers possibilities to serve as a basis for other algorithms and heuristics which make use of a large set of alternative shortest paths.024681012Speedup (k=1,000)NAVTEQ BELGIUMqueryspeedup141642561,0244,0960.00%0.05%0.10%0.15%0.20%0.25%Weight increase (k=100)NAVTEQ BELGIUMqueryweight increase0.00%0.20%0.40%0.60%0.80%Weight increase (k=1,000)NAVTEQ BELGIUMqueryweight increase0.00%0.50%1.00%1.50%2.00%Weight increase (k=10,000)NAVTEQ BELGIUMqueryweight increase9.9710.9711.97Exact ranking (k=1,000)NAVTEQ BELGIUMqueryexact ranking4,0002,0001,0006.647.648.649.6410.64Exact ranking (k=100)NAVTEQ BELGIUMqueryexact ranking1002004008001,60013.2914.2915.2916.29Exact ranking (k=10,000)NAVTEQ BELGIUMqueryexact ranking10,00020,00040,00080,0000246810Speedup (k=100)NAVTEQ BELGIUMqueryspeedup1416642561,024036912Speedup (k=10,000)NAVTEQ BELGIUMqueryspeedup141642561,0244,09616,384AcknowledgementStéphanie  Vanhove is  supported by a  research grant  of  the Research Foundation - Flanders.References Akgün  V  and  Erkut  E,  2000,  On  finding  dissimilar  paths.  European  Journal  of  Operational Research, 121(2):232–246.Bernstein A, 2010, A nearly optimal algorithm for approximating replacement paths and k shortest simple paths in general graphs.  Proceedings of the 21st Annual  ACM-SIAM Symposium on Discrete Algorithms, 742–755.Gotthilf Z and Lewenstein M, 2009, Improved algorithms for the k shortest paths and the replacement paths problems. Information Processing Letters, 109:352–355.Hershberger J, Maxel M, and Suri S, 2007, Finding the k shortest simple paths: a new algorithm and its implementation. ACM Transactions on Algorithms, 3(4):45.Marti  R,  Velarde  JLG  and  Duarte  A,  2009,  Heuristics  for  the  biobjective  path dissimilarity problem. Computers and Operations Research, 36:2905–2912.Roddity L, 2007, On the k-simple shortest paths problem in weighted directed graphs. Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, 920–928.Yen  JY,  1971,  Finding  the  k shortest  loopless  paths  in  a  network.  Management  Science, 17(11):712–716.

Fast generation of many shortest path alternatives

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Fast generation of many shortest path alternatives

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