Geographic Routing Using Hyperbolic Space
Robert Kleinberg
Computer Science Division
UC Berkeley
Berkeley, CA 94720
Email: rdk@eecs.berkeley.edu
Abstract— We propose a scalable and reliable point-to-
point routing algorithm for ad hoc wireless networks and
sensornets. Our algorithm assigns to each node of the
network a virtual coordinate in the hyperbolic plane, and
performs greedy geographic routing with respect to these
virtual coordinates. Unlike other proposed greedy routing
algorithms based on virtual coordinates, our embedding
guarantees that the greedy algorithm is always successful
in finding a route to the destination, if such a route exists.
We describe a distributed algorithm for computing each
node’s virtual coordinates in the hyperbolic plane, and
for greedily routing packets to a destination point in the
hyperbolic plane. (This destination may be the address of
another node of the network, or it may be an address
associated to a piece of content in a Distributed Hash Table.
In the latter case we prove that the greedy routing strategy
makes a consistent choice of the node responsible for the
address, irrespective of the source address of the request.)
We evaluate the resulting algorithm in terms of both path
stretch and node congestion.
I. INTRODUCTION
This paper proposes a scalable and reliable approach
for point-to-point routing in ad hoc wireless networks
and sensornets. Unlike Internet routing — which is based
on forwarding tables populated by blocks of IP addresses
— sensornet routing must deal with the absence of
a standardized, precomputed address space, while also
obeying the resource limitations of the individual nodes.
This typically excludes implementations which maintain
a large amount of state (e.g. forwarding tables) at each
node. To overcome this limitation, geographic routing
techniques [1], [2] assign coordinates to the nodes of the
network, e.g. based on their locations in the underlying
physical space in which the sensors are deployed. Nodes
then forward messages according to the coordinates of
the destination and the coordinates of their neighbors.
The simplest geographic routing techniques are
greedy, in the sense that nodes always forward messages
to the neighbor which is closest to the destination.
Greedy routing is susceptible to getting stuck in an
infinite loop when there exists a node which is nearer
to the destination than all of its neighbors. The earliest
approaches for overcoming this obstacle [3], [1] were
based on face routing, an idea discussed further in
Section I-A below. Recently, more robust solutions have
been proposed based on the idea of assigning to each
node a virtual coordinate in some metric space X, and
routing messages greedily in this space X [4], [5].
While the efficacy of greedy routing based on virtual
coordinates has been experimentally confirmed [6], [4],
[5], there is a paucity of theorems identifying circum-
stances under which this strategy is guaranteed to work.
In fact, for each of the aforementioned approaches it
is known that greedy routing may fail to find a route
for at least a small percentage of source-destination
pairs. Abstractly, then, we are interested in the following
question: given a graph G and a metric space X, when
can we compute an embedding of G into X such that
greedy routing in X succeeds in defining a path between
every pair of nodes? To make the question more precise,
we may introduce the following definition.
Definition I.1 ([7]). A greedy embedding of an undi-
rected graph G in a metric space (X, d) is a mapping
f : V (G) → X with the following property: for every
pair of distinct vertices s, t ∈ V (G) there exists a vertex
u adjacent to s such that d(f(u), f (t)) < d(f(s), f(t)).
This notion of greedy embedding was defined by
Papadimitriou and Ratajczak [7], who studied the special
case when X is the plane with the Euclidean (i.e. `
2
)
metric. They provided examples of graphs which do not
admit a greedy embedding in the Euclidean plane. In
fact for every finite-dimensional normed vector space V
there exist graphs which have no greedy embedding in
V . (This follows, e.g., from Theorem 1 below.) However,
Papadimitriou and Ratajczak conjectured that every 3-
connected planar graph admits a greedy embedding in
the Euclidean plane. These initial results and conjectures
motivate the following questions.
1) Is there a normed vector space V which admits a
greedy embedding of every graph with n nodes?
If so, how small can we make the dimension of
V ?
2) Is there a finite-dimensional Riemannian manifold
M which admits a greedy embedding of every
finite graph? If so, how small can we make the
dimension of M?
It is trivial to settle Question 1 affirmatively with
dim(V ) = n; here we improve this bound to dim(V ) =
O(log n) and prove that this is the best possible depen-
dence on n.
Theorem 1. If V is a d-dimensional normed vector
space which admits a greedy embedding of every graph
with n nodes, then d = Ω(log n). If d = dlog
3/2
(n)e
then the space `
d
∞
admits a greedy embedding of every
graph with n nodes.
The proof of Theorem 1 follows easily from standard
metric embedding techniques [8]. However, in light of
the first half of the theorem it is perhaps surprising that
Question 2 also has an affirmative answer: there exists
a finite-dimensional manifold which admits a greedy
embedding of every finite graph.
Theorem 2. Every connected finite graph has a greedy
embedding in the hyperbolic plane.
As will become apparent from the proof of Theorem 2,
our embedding technique is well suited to distributed
implementations, since the task of computing virtual
coordinates requires only the following sub-tasks:
1) Computing a spanning tree of the network.
2) Determining this tree’s maximum degree, d.
3) Naming each node of the network with a node
of the infinite d-regular tree, by identifying the
spanning tree with a subtree of the infinite d-
regular tree,
One of the chief reasons for requiring a point-to-point
routing service in a sensor network is to enable data-
centric storage [9], in which a Distributed Hash Table
(DHT) is implemented in the sensor network, and data
is stored at the nodes of the network using the DHT.
Our embedding and routing technique is well-suited
to data-centric storage, using a generalization of the
Content Addressable Network (CAN) paradigm [10].
More precisely, the put and get operations of the DHT
can be implemented by hashing the key to a point p
in the hyperbolic plane, greedily routing to a node x
whose virual coordinate is nearer to p than any of its
neighbors, and either storing or retrieving the value from
this node. In order for this to constitute a reliable DHT
implementation, one must ensure that the choice of the
node x does not depend on the source of the put or
get operation. We prove this in Theorem 3.
A. Related Work
1) Geometric routing in sensor networks: The earliest
papers on geometric routing in sensor networks [3], [1]
assumed that the nodes of the network know their own
positions in the underlying physical space which the
network inhabits, and they proposed routing strategies
based on greedy routing with respect to these coor-
dinates. As we have already observed, greedy routing
does not guarantee point-to-point connectivity in the
network because messages can become trapped in local
minima of the distance-to-destination function. (Such
local minima are sometimes referred to as “lakes” or
“voids”, since they tend to arise when there is a large
empty space in the region occupied by the nodes.)
To deal with this inadequacy of greedy routing, the
algorithms in [3], [1] and many subsequent papers use
face routing to escape from such local minima by routing
around the perimeter of a face in a planar subgraph of
the network until greedy routing can be resumed. This
is guaranteed to succeed in defining a route between
every pair of nodes, provided that nodes know their
own locations and that there is a distributed algorithm
to compute a connected planar subgraph of the net-
work. In [1] this “planarization” step is accomplished
by assuming that the underlying network is a unit disk
graph and using the relative neighborhood graph or the
Gabriel graph. An alternative planarization of unit disk
graphs with better stretch properties is introduced in [11].
For more sophisticated cost measures, [2] presents a
combination of greedy routing and face routing which
is asymptotically optimal for bounded-degree unit disk
graphs. Surprisingly, the first distributed planarization
algorithm for general graphs was discovered only very
recently [12].
All of the foregoing work assumed that nodes were
aware of their physical locations, an assumption which
is often violated in practice for various reasons. (For
instance, the most widely used node, the Berkeley
mote [13], has no GPS antenna.) In addition, implemen-
tations of face routing are either based on non-rigorous
heuristics or on complicated planarization procedures,
in many cases tailored to restrictive assumptions about
the nature of the underlying graph (e.g. that it is a
unit-disk graph). To overcome these shortcomings, two
recent papers [4], [5] proposing routing algorithms which
assign virtual coordinates in a metric space X and
forward messages using greedy routing in X. In [5]
the metric space is the Euclidean plane, and virtual
coordinates are assigned using a distributed version of
Tutte’s “rubber band” algorithm [14] for finding convex
embeddings of graphs. In [4] the graph is embedded in
2
R
d
for some value of d much smaller than the network
size, by identifying d beacon nodes and representing
each node by the vector of distances to those beacons.
The distance function on R
d
used in [4] is actually
not a metric; it is a modification of the `
1
norm, fine-
tuned for the objective of sensornet routing. Both [4]
and [5] provide substantial experimental support for the
efficacy of their proposed embedding techniques — both
algorithms are successful in finding a route from the
source to the destination more than 95% of the time
— but neither of them has a provable guarantee. Our
Theorem 2 complements the results of [5] by proving
that one may achieve 100% success rate with greedy
routing by assigning virtual coordinates in the hyperbolic
plane rather than the Euclidean plane. Our Theorem 1
complements the results of [4] by proving that one may
also achieve 100% success rate by embedding in R
d
where d = O(log n) and using the `
∞
norm rather
than the `
1
norm, though our d-dimensional embedding
(unlike theirs) is not beacon-based and fails to meet the
simplicity requirement which is a cornerstone of work
in [4].
2) Greedy embeddings: Greedy embeddings of
graphs were defined by Papadimitriou and Ratajczak
in [7], a paper primarily concerned with the question
of whether every 3-connected planar graph has a greedy
embedding in the Euclidean plane. While this conjecture
remains unsolved, their paper contains many interesting
related theorems and examples. Most relevant to the
present work is their theorem that every 3-connected
planar graph has an embedding in R
3
which satisfies
the definition of a greedy embedding when “distance” is
defined by the function d(~x, ~y) = −~x · ~y. This is not a
greedy embedding because the distance function is not
a metric on R
3
. However, this type of embedding is just
as useful as a greedy embedding from the standpoint of
geometric routing in sensor networks, since it is based
on a distance function which is easily computable from
the virual coordinates of the two points in question. Our
Theorem 2 improves on the three-dimensional embed-
ding from [7] in two respects:
• The embedding can be defined for any connected
graph G, not only for graphs which contain a planar
3-connected subgraph.
• The task of computing the embedding easily lends
itself to distributed computation, since it only re-
quires computing a spanning tree of the graph
and identifying this tree with a subtree of a com-
plete d-regular tree for some d. In contrast, the
three-dimensional embedding from [7] requires first
computing a 3-connected planar subgraph of the
graph, then computing a Koebe-Andre’ev-Steinitz
representation of this subgraph, i.e. identifying the
subgraph with the edge graph of a 3-dimensional
convex polytope whose edges are all tangent to a
unit sphere. It is hard to see how these steps could
be performed in a decentralized manner.
3) Metric embeddings: The study of low-distortion
embeddings of metric spaces has had a profound impact
on algorithmic theory in the last decade. The notion
of greedy embeddings studied here is related to order-
preservation properties of metric embeddings rather than
their distortion properties. While these ordinal embed-
ding questions have received comparably less attention, a
number of prior papers have studied them for their use in
nearest-neighbor search, data visualization, compression,
and clustering [15], [16]. Bilu and Linial [17] studied
monotone maps of metric spaces — i.e. those which
preserve the relative ordering of distances — and proved
that there exist graphs with n vertices having no mono-
tone embedding into a vector space of dimension o(n).
In this paper, we have seen that this negative result does
not prevent us from basing wireless routing algorithms
on low-dimensional ordinal embeddings.
B. Outline
The rest of this paper is organized as follows. Sec-
tion II is concerned with greedy routing in hyperbolic
space, including the proof of Theorem 2. In Section III
we turn our attention to normed vector spaces and
prove Theorem 1. Section IV presents the results of an
experimental evaluation of our algorithm, to determine
its performance in terms of stretch and congestion. We
finish with some open questions.
II. GREEDILY EMBEDDING GRAPHS IN HYPERBOLIC
SPACE
In this section we establish Theorem 2, which asserts
that every connected finite graph has a greedy embedding
in the hyperbolic plane. We start by making two obser-
vations which are trivial consequences of the definition
of greedy embedding.
Observation II.1. If H ⊆ G is a subgraph containing
all vertices of G, then every greedy embedding of H is
also a greedy embedding of G.
Observation II.2. If T is a tree and T
0
⊆ T is a subtree,
then every greedy embedding of T in a metric space X
restricts to a greedy embedding of T
0
.
Proof. Let f : T → X be a greedy embedding. For
any two nodes s, t ∈ T
0
, we know that s and t may be
joined in T by a path s = s
0
, s
1
, . . . , s
k
= t such that
3
the function d(f(s
i
, t)) is monotonically decreasing as
i goes from 0 to k. This path must lie in T
0
, since T
0
is a subtree of T and the endpoints of the path belong
to T
0
. Thus s has a neighbor s
1
in T
0
which satisfies
d(s
1
, t) < d(s, t).
Our strategy for proving Theorem 2 is to exhibit, for
each d ≥ 3, a greedy embedding of the infinite d-regular
tree into the hyperbolic plane. Since every finite tree is a
subtree of the infinite d-regular tree for sufficiently large
d, this implies (using Observation II.2) that every finite
tree has a greedy embedding into the hyperbolic plane.
Now, given an arbitrary connected finite graph G, we can
take a spanning tree T ⊆ G, find a greedy embedding
of T into the hyperbolic plane, and use Observation II.1
to conclude that this is also a greedy embedding of G.
Thus we have reduced Theorem 2 to the problem of
constucting a greedy embedding of the infinite d-regular
tree into the hyperbolic plane, for each d ≥ 3.
The next subsection will review some basic facts
about the geometry of the hyperbolic plane, which are
necessary for the construction of these greedy embed-
dings. The following subsection will construct a greedy
embedding for the infinite 3-regular tree. Finally, in the
last subsection, we will cover the case d > 3.
A. The hyperbolic plane and its isometries
In this section we recall some facts about the geometry
of the hyperbolic plane H. For an excellent exposition of
these and other properties of H, see the second chapter
of [18].
Two standard models of H are the upper half-plane
model and the Pointcar´e disk model. In the upper half-
plane model H is represented by the set of points
(x, y) ∈ R
2
satisfying y > 0, endowed with the
Riemannian metric
ds
2
=
dx
2
+ dy
2
y
2
.
In the Poincar´e disk model, H is represented by the
set of points satisfying x
2
+ y
2
< 1, endowed with the
Riemannian metric
ds
2
=
4(dx
2
+ dy
2
)
(1 − x
2
− y
2
)
2
.
In both cases, we will refer to points of the hyperbolic
plane using complex coordinates, by treating a point
(x, y) ∈ R
2
as a complex number z = x + yi. The
hyperbolic plane has a boundary circle ∂H “at infinity”
which may be identified with the circle |z| = 1 in
the Poincar´e model; in the upper half plane model the
boundary circle ∂H is identified with the one-point
compactification of the real line. In both cases, the points
of the boundary circle are referred to as “points at
infinity” or “ideal points”.
Every geodesic of the hyperbolic plane is represented
by an arc of a circle which meets ∂H in two ideal points
p, q. (A straight line is considered to be a degenerate
case of a circle. In the upper half plane model, we treat
a vertical line as a circle which intersects ∂H in two
points: the point at infinity and one point of the real
line.) For distinct points p, q, r ∈ ∂H, the hyperbolic
geodesics joining p to q, q to r, and r to p enclose a
region of H bounded by a curvilinear triangle; we call
this region the ideal triangle with vertices p, q, r.
Every orientation-preserving isometry of H is repre-
seted by a M¨obius transformation
z 7→
az + b
cz + d
for some complex coefficients a, b, c, d such that ad 6=
bc. Representing this transformation by a complex matrix
a b
c d
,
it turns out that the composition of two M¨obius transfor-
mations is represented by the product of the correspond-
ing matrices. This defines a mapping from the group
GL
2
(C) of invertible 2-by-2 complex matrices to the
group of M¨obius transformations of C. The kernel of this
mapping consists of all scalar multiples of the identity, so
the group of M¨obius transformations is isomorphic to the
quotient of GL
2
(C) by the subgroup of scalar multiples
of the identity. This quotient is denoted by P GL
2
(C).
A M¨obius transformation is an isometry of the upper
half-plane model of H if and only if it preserves the
upper half-plane as a point set. It is easy to check that
every 2-by-2 real matrix with determinant 1 represents a
M¨obius transformation preserving the upper half-plane,
and in fact the converse is true: every M¨obius trans-
formation preserving the upper half-plane is represented
by an element of GL
2
(R) with determinant 1. Thus
the isometry group of H is identified with the group
P SL
2
(R) of two-by-two real matrices with determinant
1, modulo {±I}. This group acts transitively on the set
of clockwise-ordered triples of points of ∂H.
The mapping z 7→
1+iz
1−iz
defines an isometry from
the upper half-plane model of H to the Poincar´e disk
model of H. Denoting this M¨obius transformation by
M, the inner automorphism i
M
of the M¨obius group
P GL
2
(C) maps P SL
2
(R) to another subgroup which
is the isometry group of the Poincar´e disk. We will
identify isometries of the Poincar´e disk with elements of
P SL
2
(R) using this inner automorphism of P GL
2
(C).
4
The subgroup P SL
2
(Z) ⊂ P SL
2
(R) consists of
all transformations represented by two-by-two integer
matrices. This subgroup is generated by the matrices
a =
0 −1
1 0
; b =
0 1
−1 1
.
These matrices satisfy a
2
= b
3
= −I, so they represent
elements of order 2 and 3, respectively, in P SL
2
(Z).
(Recall that −I is identified with I in P SL
2
(Z).) In
the Poincar´e disk model, these transformations a, b are
represented by the mappings
a : z 7→ −z
and
b : z 7→
(1 + 2i)z + 1
z + (1 − 2i)
.
If ∆ is the ideal triangle with vertices 0, 1, i, then
a exchanges ∆ with its complex conjugate
∆, and b
preserves ∆ while cyclically permuting its vertices. As g
ranges over all elements of P SL
2
(Z), the ideal triangles
g(∆) form a tiling of H. The dual of this tiling is a 3-
regular tree T which may be described as follows. Let
u = (2 −
√
3)i and v = −u. Let e be the pair (u, v).
Then the vertex set of T is {g(u) : g ∈ P SL
2
(Z)}
and the edge set of T is {g(e) : g ∈ P SL
2
(Z)}. The
group P SL
2
(Z) acts both vertex-transitively and edge-
transitively on T . In fact P SL
2
(Z) is isomorphic to the
group of automorphisms of T which preserve the cyclic
ordering of the edges around every vertex. The tiling and
its dual tree T are illustrated in Figure 1, which also
illustrates the equivalent picture in the upper half-plane
model.
B. A greedy embedding of the infinite 3-regular tree
Recall that Figure 1 illustrates an infinite tiling of H
and an infinite 3-regular tree T dual to the tiling, both
of which are invariant under the action of the discrete
isometry group P SL
2
(Z). We claim that this embedding
of the tree T is a greedy embedding. To verify this, it suf-
fices to show that for every path s = s
0
, s
1
, . . . , s
k
= t
in T , the inequality d(s
0
, t) > d(s
1
, t) is satisfied. Recall
that the action of P SL
2
(Z) on T is edge-transitive, so
we may assume without loss of generality that s
0
= u
and s
1
= v. If we delete edge (u, v) from T , this
separates T into two subtrees consisting of the nodes
corresponding to points in the upper half-plane and those
in the lower half-plane. Since v lies on the path from u
to t, we may conclude that t is in the lower half-plane.
To prove our assertion that d(u, t) > d(v, t), we will
prove the following more general fact: for any z in the
lower half of the unit disk, d(u, z) > d(v, z). To prove
it, let L be a geodesic segment from u to z and let y be
the intersection of L with the real line. We have
d(u, z) = d(u, y) + d(y, z).
The geodesic segment from v to z lies entirely in the
lower half-plane, hence it does not contain y; conse-
quently
d(v, z) < d(v, y) + d(y, z).
But d(v, y) = d(u, y) since complex conjugation is an
isometry of H. Hence d(v, z) < d(u, y) + d(y, z) =
d(u, z), as claimed.
C. A greedy embedding of the infinite d-regular tree
To construct a greedy embedding of the infinite d-
regular tree for any d > 3, we use the same ideas but
the symmetry group Γ which plays the role of P SL
2
(Z)
is defined implicitly by the geometry rather than by
specifying its elements explicitly in terms of their matrix
coefficients.
Let P be the interior of the ideal polygon whose
vertices are the complex d-th roots of unity. Note that
P is preserved by the mapping ρ(z) = e
2πi/d
z, which
generates a cyclic subgroup of order d in the hyperbolic
isometry group P SL
2
(R). Since P SL
2
(R) acts transi-
tively on clockwise-ordered triples of ideal points, we
may take any side of the ideal polygon ∂P and find a
hyperbolic isometry σ which maps its endpoints to 1
and −1 while mapping the midpoint of the arc joining
these endpoints to −i. Now consider the two hyperbolic
isometries
a : z 7→ −z
b : z 7→ σ(ρ(σ
−1
(z))).
Isometry a rotates the disk by a half-turn interchanging
P with its complex conjugate P . Isometry b preserves
the set P while cyclically permuting its corners. These
two transformations generate a discrete subgroup Γ ⊂
P SL
2
(R) such that the sets {g(P ) : g ∈ Γ} constitute
a tiling of the hyperbolic plane by ideal d-gons. The dual
of this tiling is a complete d-regular tree T whose nodes
are all the points {g(σ(0)) : g ∈ Γ}. The complex
numbers u = σ(0), v = −u are pure imaginary, and
every edge of T is the image of the edge (u, v) under
a suitable transformation g ∈ Γ. See Figure 2 for an
illustration of the d = 4 case.
The proof that T is greedily embedded in the hyper-
bolic plane works as before: the action of Γ on T is
edge-transitive, so it suffices to consider a path in T
whose first hop is (u, v) and whose endpoint t lies in
the lower half-plane. The proof that d(u, z) > d(v, z)
for every z in the lower half-plane is exactly the same.
5
