ONLINE ROUTING IN TRIANGULATIONS
∗
PROSENJIT BOSE
†
AND PAT MORIN
†
Abstract. We consider online routing algorithms for routing between the vertices of embedded
planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms,
one that works for all Delaunay triangulations and the other that works for all regular triangula-
tions, (2) a randomized memoryless algorithm that works for all triangulations, (3) an O(1) memory
algorithm that works for all convex subdivisions, (4) an O(1) memory algorithm that approximates
the shortest path in Delaunay triangulations, and (5) theoretical and experimental results on the
competitiveness of these algorithms.
Key words. Routing, Online algorithms, Delaunay triangulations, shortest path, spanning path
1. Introduction. Path finding, or routing, is central to a number of fields in-
cluding geographic information systems, urban planning, robotics, and communica-
tion networks. In many cases, knowledge about the environment in which routing
takes place is not available beforehand, and the vehicle/robot/packet must learn this
information through exploration. Algorithms for routing in these types of environ-
ments are referred to as online [3] routing algorithms.
In this paper we consider online routing in the following abstract setting: The
environment is a planar straight line graph [17], T , with n vertices, whose edges
are weighted by the Euclidean distance between their endpoints, the source v
src
and
destination v
dst
are vertices of T , and a packet can only move on edges of T. Initially,
a packet only knows v
src
, v
dst
, and N (v
src
), where N (v) denotes the set of vertices
adjacent to v.
We classify online routing algorithms based on their use of memory and/or ran-
domization. Define v
cur
as the vertex at which the packet is currently stored. A
routing algorithm is called memoryless if the next step taken by a packet depends
only on v
cur
, v
dst
, and N (v
cur
). An algorithm is randomized if the next step taken by
a packet is chosen randomly from N (v
cur
). A randomized algorithm is memoryless
if the distribution used to choose from N (v
cur
) is a function only of v
cur
, v
dst
, and
N (v
cur
).
The justification for studying the memory requirements of routing algorithms
comes from communication networks, in which memory used by an algorithm results
in header information that travels with a packet. Since this information is only used
for routing purposes and is of no use to the sender or receiver, it effectively results
produces a decrease in communication bandwidth.
For an algorithm A we say that a graph defeats A if there is a source/destination
pair such that a packet never reaches the destination when beginning at the source.
If A finds a path P from v
src
to v
dst
we call P the A path from v
src
to v
dst
. Here
we use the term path in an intuitive sense rather than a strict graph theoretic sense,
since P may visit the same vertex more than once.
In this paper we also consider, as a special case, a class of “well-behaved” trian-
gulations. The Voronoi diagram [16] of S is a partitioning of space into cells such
that all points within a Voronoi cell are closer to the same element p ∈ S than any
other point in S. The Delaunay triangulation is the straight-line face dual of the
∗
This research was supported by the Natural Sciences and Engineering Research Council of
Canada.
†
School of Computer Science, Carleton University, 1125 Colonel By Dr., Ottawa, Canada,
K1S 5B6 ({jit,morin}@scs.carleton.ca)
1
2 P. BOSE AND P. MORIN
Fig. 2.1. Any deterministic memoryless algorithm is defeated by at least one of these graphs.
Voronoi diagram, i.e., two points in S have an edge between them in the Delaunay
triangulation if their Voronoi regions have an edge in common.
In this paper we consider several different routing algorithms and compare their
performance empirically. In particular, we describe
1. a memoryless algorithm that is not defeated by any Delaunay triangulation,
2. a memoryless algorithm that is not defeated by any regular triangulation,
3. a memoryless randomized algorithm that uses 1 random bit per step and is
not defeated by any triangulation,
4. an algorithm that uses O(1) memory that is not defeated by any convex
subdivision,
5. an algorithm for Delaunay triangulations that uses O(1) memory in which a
packet never travels more than a constant times the Euclidean distance between v
src
and v
dst
, and
6. a theoretical and empirical study of the quality (length) of the paths found
by these algorithms.
The first four routing algorithms are described in Section 2. Section 3 presents
theoretical and empirical results on the length of the paths found by these algorithms
and describes our algorithm for Delaunay triangulations. A discussion of related work
is provided in Section 4. Finally, Section 5 summarizes and describes directions for
future research.
2. Four Simple Algorithms. In this section we describe four online routing
algorithms and prove theorems about which types of graphs never defeat them. We
begin with the simplest (memoryless) algorithms and proceed to the more complex
algorithms.
However, before beginning we should note that deterministic memoryless algo-
rithms have some inherent limitations. This can be seen by considering the 8 graphs
in Fig. 2.1 with the center vertex as the destination. Since the corresponding cor-
ner vertices of all graphs look the same, a deterministic memoryless algorithm must
make the same decision at the corner vertices of each graph. But for any possible
combination of decisions, at least one of the graphs will defeat the algorithm.
Thus, there is no hope of finding a deterministic memoryless algorithm that works
for all embedded planar graphs (or even all 2-connected planar graphs).
2.1. Greedy Routing. The greedy routing (GR) algorithm always moves the
packet to the neighbor gdy(v
cur
) of v
cur
that minimizes dist(gdy(v
cur
), v
dst
), where
dist(p, q) denotes the Euclidean distance between p and q. In the case of ties, one of
the vertices is chosen arbitrarily. The greedy routing algorithm can be defeated by a
triangulation T in two ways: (1) the packet can get trapped moving back and forth
ONLINE ROUTING IN TRIANGULATIONS 3
v
dst
v
src
v
dst
v
src
(a) (b)
Fig. 2.2. Triangulations that defeat the greedy routing algorithm.
v
v
dst
h
u
u
h
v
Fig. 2.3. The proof of Theorem 2.1.
on an edge of the triangulation (Fig. 2.2.a), or (2) the packet can get trapped on a
cycle of three or more vertices (Fig. 2.2.b). However, as the following theorem shows,
neither of these situations can occur if T is a Delaunay triangulation.
Theorem 2.1. There is no point set whose Delaunay triangulation defeats the
greedy routing algorithm.
Proof. We proceed by showing that every vertex v of T has a neighbor that is
strictly closer to v
dst
than v is. Thus, at each routing step, the packet gets closer to
v
dst
and therefore, after at most n steps, reaches v
dst
. Refer to Fig. 2.3.
Consider the Voronoi diagram [16] VD(T ) of the vertices of T and let e be the
first edge of VD(T ) intersected by the directed line segment (v, v
dst
). Note that e is on
the boundary of two Voronoi cells, one for v and one for some other vertex u, and the
supporting line of e partitions the plane into two open half planes h
v
= {p : dist(p, v) <
dist(p, u)} and h
u
= {p : dist(p, u) < dist(p, v)}. Since the Voronoi diagram is the
straight line face dual of the Delaunay triangulation, the edge (u, v) ∈ T . Also, by
the choice of e, v
dst
∈ h
u
, i.e., dist(u, v
dst
) < dist(v, v
dst
).
2.2. Compass Routing. The compass routing (CR) algorithm always moves
the packet to the vertex cmp(v
cur
) that minimizes the angle
6
v
dst
, v
cur
, cmp(v
cur
)
over all vertices adjacent to v
cur
. Here the angle is taken to be the smaller of the two
angles as measured in the clockwise and counterclockwise directions. In the case of
ties, one of the (at most 2) vertices is chosen using some arbitrary deterministic rule.
One might initially believe (as we did) that compass routing can always be used
to find a path between any two vertices in a triangulation. However, the triangulation
in Fig.
2.4 defeats compass routing. When starting from one of the vertices on the
outer face of T , and routing to v
dst
, the compass routing algorithm gets trapped on
4 P. BOSE AND P. MORIN
v
dst
v
src
Fig. 2.4. A triangulation that defeats the compass routing algorithm.
R1
R
2
R
3
u
v
v
dst
Fig. 2.5. The proof of Lemma 2.2.
the cycle shown in bold. The following lemma shows that any triangulation that
defeats compass routing causes the packet to get trapped in a cycle.
Lemma 2.2. Let T be a triangulation that defeats compass routing, and let v
dst
be a vertex such that compass routing fails to route a packet to v
dst
when given some
other vertex as the source. Then there exists a cycle C = v
0
, . . . , v
k−1
(k ≥ 3) in T
such that cmp(v
i
) = v
i+1
for all 0 ≤ i < k.
1
Proof. Since T defeats compass routing, and the compass routing algorithm makes
the same decision each time it visits a vertex, either there is an edge (u, v) such that
cmp(u) = v and cmp(v) = u, or there is the situation described in the lemma. We
prove that there can be no such edge (u, v). Suppose such an edge (u, v) does exist.
Then there is a triangle (u, v, w) in T such that w is in the same half-plane bounded
by the line through u and v as v
dst
. Referring to Fig. 2.5, the vertex w must be in
the one of the regions 1, 2, or 3. But this is a contradiction, since if w is in region 1,
then cmp(v) = w, if w is in region 2, then cmp(u) = w (and cmp(v) = w), and if w
is in region 3, then cmp(u) = w.
We call such a cycle, C, a trapping cycle in T for v
dst
. Next we characterize
trapping cycles in terms of a visibility property of triangulations. Let t
1
and t
2
be
two triangles in T . Then we say that t
1
obscures t
2
if there exists a ray originating at
v
dst
that strikes t
1
first and then t
2
. Let u and v be any two vertices of T such that
cmp(u) = v. Then define 4uv as the triangle of T that is contained in the closed
half-plane bounded by the line through uv and that contains v
dst
. We obtain the
following useful characterization of trapping cycles.
Lemma 2.3. Let T be a triangulation that defeats compass routing and let C =
v
0
, . . . , v
k−1
be a trapping cycle in T for vertex v
dst
. Then 4v
i
v
i+1
is either identical
to, or obscures 4v
i−1
v
i
, for all 0 ≤ i < k.
Proof. Refer to Fig. 2.6. Assume that 4v
i
v
i+1
and 4v
i−1
v
i
are not identical,
otherwise the lemma is trivially true. Let w be the third vertex of 4v
i
v
i+1
. Then w
cannot lie in the cone defined by v
dst
, v
i
and v
i+1
, otherwise we would have cmp(v
i
) =
1
Here and henceforth, all subscripts are assumed to be taken modk.
ONLINE ROUTING IN TRIANGULATIONS 5
v
i−1
v
i
v
i+1
w
v
dst
Fig. 2.6. The proof of Lemma 2.3.
ccw (v)
cw(v)
v
dst
v
Fig. 2.7. Definition of cw (v) and ccw(v).
w. But then the line segment joining w and v
i+1
obscures v
i
and hence 4v
i
v
i+1
obscures 4v
i−1
v
i
.
A regular triangulation [18] is a triangulation obtained by orthogonal projection
of the faces of the lower hull of a 3-dimensional polytope onto the plane. Note that
the Delaunay triangulation is a special case of a regular triangulation in which the
vertices of the polytope all lie on a paraboloid. Edelsbrunner [8] showed that if T is a
regular triangulation, then T has no set of triangles that obscure each other cyclically
from any viewpoint. This result, combined with Lemma 2.3, yields our main result
on compass routing.
Theorem 2.4. There is no regular triangulation that defeats the compass routing
algorithm.
2.3. Randomized Compass Routing. In this section, we consider a random-
ized routing algorithm that is not defeated by any triangulation. Let cw(v) be the
vertex in N (v) that minimizes the clockwise angle
6
v
dst
, v, cw(v) and let ccw(v) be
the vertex in N (v) that minimizes the counterclockwise angle
6
v
dst
, v, ccw(v) (See
Fig. 2.7). Then the randomized compass routing (RCR) algorithm moves the packet
to one of {cw(v
cur
), ccw(v
cur
)} with equal probability.
Before we can make statements about which triangulations defeat randomized
compass routing, we must define what it means for a triangulation to defeat a ran-
domized algorithm. We say that a triangulation T defeats a (randomized) routing
algorithm if there exists a pair of vertices v
src
and v
dst
of T such that a packet orig-
inating at v
src
with destination v
dst
has probability 0 of reaching v
dst
in any finite
number of steps.
Note that, since randomized compass routing is memoryless, proving that a trian-
gulation T does not defeat randomized compass routing implies that a packet reaches
its destination with probability 1. The following theorem shows the versatility of
randomized compass routing.
Theorem 2.5. There is no triangulation that defeats the randomized compass
routing algorithm.
Proof. Assume, by way of contradiction that a triangulation T exists that defeats
the randomized compass routing algorithm. Then there is a vertex v
dst
of T and a
minimal set S of vertices such that: (1) v
dst
/∈ S, (2) the subgraph H of T induced
by S is connected, and (3) for every v ∈ S, cw(v) ∈ S and ccw(v) ∈ S.
