12,329 research outputs found
Lower Bounds and Approximation Algorithms for Search Space Sizes in Contraction Hierarchies
Contraction hierarchies (CH) is a prominent preprocessing-based technique that accelerates the computation of shortest paths in road networks by reducing the search space size of a bidirectional Dijkstra run. To explain the practical success of CH, several theoretical upper bounds for the maximum search space size were derived in previous work. For example, it was shown that in minor-closed graph families search space sizes in ?(?n) can be achieved (with n denoting the number of nodes in the graph), and search space sizes in ?(h log D) in graphs of highway dimension h and diameter D. In this paper, we primarily focus on lower bounds. We prove that the average search space size in a so called weak CH is in ?(b_?) for ? ? 2/3 where b_? is the size of a smallest ?-balanced node separator. This discovery allows us to describe the first approximation algorithm for the average search space size. Our new lower bound also shows that the ?(?n) bound for minor-closed graph families is tight. Furthermore, we deeper investigate the relationship of CH and the highway dimension and skeleton dimension of the graph, and prove new lower bound and incomparability results. Finally, we discuss how lower bounds for strong CH can be obtained from solving a HittingSet problem defined on a set of carefully chosen subgraphs of the input network
Transit Node Routing Reconsidered
Transit Node Routing (TNR) is a fast and exact distance oracle for road
networks. We show several new results for TNR. First, we give a surprisingly
simple implementation fully based on Contraction Hierarchies that speeds up
preprocessing by an order of magnitude approaching the time for just finding a
CH (which alone has two orders of magnitude larger query time). We also develop
a very effective purely graph theoretical locality filter without any
compromise in query times. Finally, we show that a specialization to the online
many-to-one (or one-to-many) shortest path further speeds up query time by an
order of magnitude. This variant even has better query time than the fastest
known previous methods which need much more space.Comment: 19 pages, submitted to SEA'201
Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms
In the last decade, there has been a substantial amount of research in
finding routing algorithms designed specifically to run on real-world graphs.
In 2010, Abraham et al. showed upper bounds on the query time in terms of a
graph's highway dimension and diameter for the current fastest routing
algorithms, including contraction hierarchies, transit node routing, and hub
labeling. In this paper, we show corresponding lower bounds for the same three
algorithms. We also show how to improve a result by Milosavljevic which lower
bounds the number of shortcuts added in the preprocessing stage for contraction
hierarchies. We relax the assumption of an optimal contraction order (which is
NP-hard to compute), allowing the result to be applicable to real-world
instances. Finally, we give a proof that optimal preprocessing for hub labeling
is NP-hard. Hardness of optimal preprocessing is known for most routing
algorithms, and was suspected to be true for hub labeling
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
PReaCH: A Fast Lightweight Reachability Index using Pruning and Contraction Hierarchies
We develop the data structure PReaCH (for Pruned Reachability Contraction
Hierarchies) which supports reachability queries in a directed graph, i.e., it
supports queries that ask whether two nodes in the graph are connected by a
directed path. PReaCH adapts the contraction hierarchy speedup techniques for
shortest path queries to the reachability setting. The resulting approach is
surprisingly simple and guarantees linear space and near linear preprocessing
time. Orthogonally to that, we improve existing pruning techniques for the
search by gathering more information from a single DFS-traversal of the graph.
PReaCH-indices significantly outperform previous data structures with
comparable preprocessing cost. Methods with faster queries need significantly
more preprocessing time in particular for the most difficult instances
Dynamic Time-Dependent Route Planning in Road Networks with User Preferences
There has been tremendous progress in algorithmic methods for computing
driving directions on road networks. Most of that work focuses on
time-independent route planning, where it is assumed that the cost on each arc
is constant per query. In practice, the current traffic situation significantly
influences the travel time on large parts of the road network, and it changes
over the day. One can distinguish between traffic congestion that can be
predicted using historical traffic data, and congestion due to unpredictable
events, e.g., accidents. In this work, we study the \emph{dynamic and
time-dependent} route planning problem, which takes both prediction (based on
historical data) and live traffic into account. To this end, we propose a
practical algorithm that, while robust to user preferences, is able to
integrate global changes of the time-dependent metric~(e.g., due to traffic
updates or user restrictions) faster than previous approaches, while allowing
subsequent queries that enable interactive applications
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