The VLDB Journal (2017) 26:373–397
Finding lowest-cost paths in settings with safe and preferred zones
· Jianzhong Qi
· Christian S. Jensen
· Rui Zhang
· Yuan Li
Received: 6 April 2016 / Revised: 11 January 2017 / Accepted: 16 January 2017 / Published online: 30 January 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We deﬁne and study Euclidean and spatial net-
work variants of a new path ﬁnding problem: given a set of
safe or preferred zones with zero or low cost, ﬁnd paths that
minimize the cost of travel from an origin to a destination.
In this problem, the entire space is passable, with preference
given to safe or preferred zones. Existing algorithms for prob-
lems that involve unsafe regions to be avoided strictly are not
effective for this new problem. To solve the Euclidean vari-
ant, we devise a transformation of the continuous data space
with safe zones into a discrete graph upon which shortest
path algorithms apply. A naive transformation yields a large
graph that is expensive to search. In contrast, our transforma-
tion exploits properties of hyperbolas in Euclidean space to
safely eliminate graph edges, thus improving performance
without affecting correctness. To solve the spatial network
variant, we propose a different graph-to-graph transforma-
tion that identiﬁes critical points that serve the same purpose
as do the hyperbolas, thus also avoiding the extraneous edges.
Christian S. Jensen
University of Melbourne, Melbourne, Australia
Aalborg University, Aalborg, Denmark
Latrobe University, Melbourne, Australia
Having solved the problem for safe zones with zero costs,
we extend the transformations to the weighted version of the
problem, where travel in preferred zones has nonzero costs.
Experiments on both real and synthetic data show that our
approaches outperform baseline approaches by more than an
order of magnitude in graph construction time, storage space,
and query response time.
Keywords Path ﬁnding · Safest path · Safe zones ·
Preferred zones · Hyperbola
1.1 Motivation and contribution overview
Shortest path computation has been studied extensively.
However, in some scenarios, the shortest path may not be
the desired one. In hazardous environments, it can be life
critical to minimize the distance traveled in unsafe regions.
For example, a person who drives a long distance through the
desert may try to travel via villages (“safe zones”) because a
breakdown in an unpopulated region can be life threatening.
The traditional shortest path from an origin to a destination
is likely to differ substantially from a “shortest” path that
is based on a preference to travel as little as possible out-
side populated regions. In a more familiar scenario, a tourist
who plans to walk to a given destination may prefer a path
that visits interesting streets and blocks, e.g., with interesting
houses, galleries, or other sights, as much as possible. Here,
traveling in interesting regions (“safe zones”) is merely pre-
ferred, and the tourist is unlikely to choose the “safest path”
if it comes at the cost of a very long walk.
In the ﬁrst of the above two scenarios, we may assign zero
cost to travel in safe zones, while in the second, we may