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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

On k-Path Covers and their Applications.

The Vldb Journal, no. 1 (2016)

Cited by: 7|Views231
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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
Tables
  • 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
Download tables as Excel
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.
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