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# On k-Path Covers and their Applications.

The Vldb Journal, no. 1 (2016)

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Abstract

For a directed graph G with vertex set V we call a subset C ⊆ V a k-(All-)Path Cover if C contains a node from any path consisting of k nodes. This paper considers the problem of constructing small k-Path Covers in the context of road networks with millions of nodes and edges. In many application scenarios the set C and its induced overla...More

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Introduction

- The massive acquisition of geospatial data in the course of collaborative projects like OpenStreetMap (OSM)1 or by companies like Google or TomTom has lead to a dramatic growth of data to be handled in Spatial Network Databases (SNDB).
- SNDBs manage geographic entities located in an underlying road network supporting efficient data retrieval operations, in particular taking into account connectivity properties of the road network.
- Google/Bing/ Yahoo Maps are all incarnations of SNDBs. Let them first look at a few applications for SNDBs which can benefit from small k-Path Covers

Highlights

- The massive acquisition of geospatial data in the course of collaborative projects like OpenStreetMap (OSM)1 or by companies like Google or TomTom has lead to a dramatic growth of data to be handled in Spatial Network Databases (SNDB)
- As k-Path Cover is NP-hard [4] and k-ShortestPath Cover turns out to be NP-hard as well, we investigate approximation algorithms based on low VC-dimension of the underlying set system
- Let us start with the pruning approach for constructing sets C covering all paths consisting of k nodes: In Table 2 we first examine how different node orders affect the quality and the running time of the cover construction
- We introduced the k-All-Path Cover optimization problem with the goal of computing compact yet faithful synopses of the vertex set of road networks
- Even for covering all paths consisting of k nodes, we could construct surprisingly small cover sets in reasonable time for moderate values of k
- While other route planners require a relatively expensive customization phase to adapt to personalized metrics, our approach allows to incorporate them in the overlay graph on the fly

Results

- The main contributions of the paper are the following:

The authors generalize the work of [12] by considering all paths in the network instead of only shortest paths and devise efficient algorithms to compute small k-Path Covers. - As k-Path Cover is NP-hard [4] and k-ShortestPath Cover turns out to be NP-hard as well, the authors investigate approximation algorithms based on low VC-dimension of the underlying set system.
- In Section 4 the authors develop practical algorithms to construct k-All-Path Covers and k-ShortestPath Covers as well as instance-based lower bounds.
- The authors conclude with an experimental section showing the practicability of the developed algorithms and in particular looking at the use case of personalized route planning

Conclusion

- The authors introduced the k-All-Path Cover optimization problem with the goal of computing compact yet faithful synopses of the vertex set of road networks.
- While other route planners require a relatively expensive customization phase to adapt to personalized metrics, the approach allows to incorporate them in the overlay graph on the fly.
- This leads to a speed-up of an order of magnitude compared to Dijkstra’s algorithm.
- This might be a good starting point to achieve query times in the milliseconds range even for personalized route queries

- Table1: Benchmark graphs (M = 106)
- Table2: k-APC: Influence of different node orders on cover size and lower bounds (lb) for k = 16. The columns are from left to right: graph, order, lower bound size, lower bound construction time, cover size, relative cover size, cover construction time
- Table3: k-APC: Approximation ratios (apx) and construction times for comp-inc order and varying values of k. ’perc’ describes the fraction of nodes in V that are contained in the cover C
- Table4: k-SPC: Sampling vs. Pruning: Comparison for USA and GER
- Table5: Ratio of edges on a k-sampled path compared to all edges in the original path. Values are averaged over 1000 random queries
- Table6: Personalized Route Planning/Preprocessing. Construction of k-APC and overlay graph: size of k-APC, number of edges in the overlay graph, time to construct the overlay graph, total time (k-APC and overlay graph construction), avg. and maximum degree in overlay graph
- Table7: Personalized Route Planning Queries on GER: 8 metrics, random source-target pairs, random weights w1, . . . , w8. Averages for 100 random queries: Dijkstra baseline, search for access nodes, search in overlay multigraph, total search time, speed-up vs. Dijkstra
- Table8: Search times when increasing the number of metrics (random weights). BW graph, k = 20. Average of 100 random queries, speed-up compared to plain Dijkstra

Related work

- Tao et al in [12] considered the problem of computing a k-Shortest-Path Cover, that is, a set of nodes C ⊆ V such that C contains at least one node from every shortest path (under some fixed metric) consisting of k nodes. More concretely, they could, for example, construct a set C which was only 15% of the size of V for k = 16 and the US road network. As one application example they showed how to use such a small C to accelerate shortest path queries (for a fixed metric) via the overlay graph induced by C (and in combination with additional speed-up techniques like reach [9]). We want to note, though, that the achievable query times have meanwhile been superseded by current speed-up techniques like Transit Nodes [3], Contraction Hierarchies (CH) [8] or Hub Labels [2] which answer queries faster by several orders of magnitude. A fundamental restriction of all these techniques (including [12]) is that they all rely on fixed edge weights for the preprocessing stage. If edge weights change (e.g. for a different user profile), the preprocessing has to be revised or even redone from scratch again, making all these approaches unsuitable for the use-case where every query comes with a different user profile.

Reference

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- D. Delling, A. V. Goldberg, T. Pajor, and R. F. F. Werneck. Customizable route planning. In P. M. Pardalos and S. Rebennack, editors, Symposium on Experimental Algorithms (SEA), pages 376–387.
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- D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. In Symposium on Computational Geometry (SCG), pages 61–71, New York, NY, USA, 1986. ACM.
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- V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264–280, 1971.

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