Trip Planning Queries
F. Li, D. Cheng, M.
Hadjieleftheriou, G.
Kollios, S.-H. Teng
Boston University
Motivation
MapQuest, Google Maps, etc. have
become essential web services.
 Albeit, they provide simple driving
directions given a start and an end point.
 The same is true for vehicle navigation
systems, GPS devices, etc…
 It is time to support more advanced
services!

A Novel Idea

Trip Planning Queries (TPQ):
 Given
a starting location, a destination and
arbitrary points of interest try to find the best
possible trip.

Example:
 Minimize
the total traveling time from Boston
to Providence, while visiting a post office, a
hardware store and a gas station.
Visual Example
Home
Work
Gas station


We can minimize the total distance, time, etc.
We can have different categories of points of
interest (gas stations, hotels, etc.).
Formally
Solve TPQ on metric graphs (e.g.,
transportation networks).
 Given a metric graph G(V, E), a set R of
categories, a starting vertex S and an
ending vertex D, find a vertex traversal T
(or trip) from S to D that visits at least one
vertex from each category in R and has
the minimum possible cost.
 Define the cost of a trip C(T) appropriately.

Observations

TPQ is harder than Traveling Salesman
Problem (TSP):
 Given
any TSP instance assume that every
vertex belongs to its own unique category.

To answer TPQ in practice we need to
develop approximate solutions
The Nearest Neighbor Algorithm
B2
A2
S
D
C2
B3
B1
A1
C1
a 2m+1 - 1 approximation where m is the
total number of categories.
 Yields
The Minimum Distance Algorithm
B2
A2
S
D
C2
B3
B1
A1
 Yields
C1
an m-approximation where m is the
total number of categories.
The Minimum Distance Algorithm
 The
Minimum Distance Algorithm restricts the
search space/region as an ellipse.
p3
p1
S
candidate p
p2
D
search region SR
Other Algorithms
Previous algorithms give approximations
in terms of m.
 Approximations in terms of the maximum
category cardinality r:

 Use

Integer Linear Programming.
Approximations in terms of both m and r:
 Use
the generalized minimum spanning tree.
The Algorithms in Practice

Can we use the previous algorithms in
practice for spatial databases?
 Applications
in Euclidean spaces using R-
trees.
 Applications in Road networks using
specialized indices
NN Algorithm on R-trees
Starting from S locate the nearest
neighbor of S that belongs to any category
in R.
 Iteratively continue until all categories
have been covered.
 Connect the last vertex found with D.
 Use any NN algorithm on R-trees

MD Algorithm on R-trees
We need to locate one point per category
such that L=|Sp| + |pD| is minimized.
 We implement a NN search that tries to
minimize L instead of MinDist when sorting
the R-tree MBRs:

B
M
D
p
A
S
MD Algorithm on Road Network
As before, locate the m points that
minimize the total network distance from S
to D.
 We implement a specialized algorithm for
finding such points on a road network:

 Expand
all paths from S to D.
 Separate into two categories: Paths that have located
a point of interest p and ones that have not.
 The first compete to find a shortest path to p. The
latter compete to find a shortest path from p to D.
 Return the path that minimizes the sum.
An Example in Road Network

We represent the point on a road network by its
distance to the node with smaller index.
n4
n4
p2(3.2)
4.0
p2(3.2)
5.0
D(3.0)
4.0
4.0
4.0
n1 p1(1.0) S(2.0)n2
p3(0.8)
6.0
n3
S->n2->p3->n2->D, 6.6
n1
5.0
p1(1.0) S(2.0)n2
p4(3.2)
D(3.0)
p3(0.8)
6.0
n3
S->n2-> p4 ->D, 5.4
Experimental Evaluation
We used synthetic datasets on real road
networks (Oldenburg) and real datasets
from the state of California.
 We varied the total number of categories
m, the density per category r, and the
network sizes.
 We compare the NN and MD algorithms
on road networks using R-trees.

Datasets
Average Trip Length
Query Cost
Conclusion
Introduced a novel query for spatial
databases.
 Designed four approximation algorithms
with various approximation guarantees.
 Implemented the algorithms in practice
using R-trees for Euclidean spaces and
road networks.
 Contacted a comprehensive experimental
evaluation.

Thank you!