Greedy embedding problem

Introduction Conjecture Contribution Bound Characterization

Geometric Routing: Theoretical Foundations, an
Important Conjecture and Some Progress

Subhas K. Ghosh

March 2, 2010

Subhas K. Ghosh Greedy Embedding

Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber band

Sensor Networks and Ad hoc networks



Geometric routing

Ad hoc networks and Distributed Wireless Sensor Networks: no
universally known system of addresses.
Resource limitations: prohibitive to store and maintain large forwarding
tables.
Geometric routing: use geographic coordinates of the nodes as
addresses.
Simplest of routing strategy: recursively select a vertex that is closer to
destination than current vertex.
Point-to-point routing service: enable data centric storage, a Distributed
Hash Table (DHT) is implemented in the sensor network, and data is
stored at the nodes of the network using the DHT.



Geometric routing

Greedy routing
set i := 0, v0 := s
while vi = t do
du := minu∈N (vi ) {d (u, t) : d (u, t) < d (vi , t)}
if ∃u then
set i := i + 1, vi := u
else
fail!
end if
end while



Greedy routing : Example
7 7

15 6 15 6

s
1 1

2 2
8 3 8 3
9 9

4 4

16 16

14 5 14 5

20 20
10 10
17 17
t
18 18

19 19

13 11 13 11

12 12

(a) Graph (b) Forwarding Path

Figure: Greedy Routing On A Graph


6

Greedy routing : Example
s
1

2
8 3
9

4

5

10
17
t
18

19

11

12



Greedy routing : Example...
7 7

15 6 15 6

s
1 1

2 2
8 3 8 3
9 9

4 4

16 16

14 5 14 5

20 20
10 10
17 17

18 18

19 19

t
13 11 13 11

12 12

(a) Graph (b) No Forwarding Path

Figure: No Greedy Routing!


15 6

Greedy routing : Example...
s
1

2
8 3
9

4

16

5

20
10
17

18

19

t
13 11

12



Greedy routing sometimes fails!

Greedy routing sometimes fails to deliver a packet because of the
phenomenon of “voids” (nodes with no neighbor closer to the destination).
This is essentially reaching a local minimum with respect to the distance
functional deﬁned by s, t -pair.
Alternative: face routing [KK00].



Geometric routing is complicated

It is unlikely that future ad hoc networks can rely on the availability of
precise geographic coordinates (GPS is costly and does not work in all
situation).
Importantly, the precise coordinates may be disadvantageous as they do
not account for obstructions or other topological properties of the
network.
How to retain the beneﬁts of geographic routing in the absence of location
information?

Rao, Papadimitriou, Shenker and Stoica (in [RPSS03]) deﬁned a scalable
coordinate-based routing algorithm that does not rely on location information,
and thus can be used in a wide variety of ad hoc and sensor-nets
environments.



Virtual Coordinates

In [RPSS03] the nodes ﬁrst decide on ﬁctitious virtual coordinates in R2 , and
then apply greedy routing based on those. The coordinates are found by a
distributed version of the rubber band algorithm originally due to Tutte [Tut60].

Note that embedding in higher dimension requires storing more
information per node.
On the basis of extensive experimentation in [RPSS03] authors showed
that this approach makes greedy routing much more reliable (works
97.5% cases).
Kleinberg proved 100% success rate by assigning virtual coordinates in
the hyperbolic plane rather than the Euclidean plane.



Graph Embedding

An embedding of an undirected graph G in a metric space (X, d) is a mapping
e : V (G) → X.

1 In this work we will be concerned with a special case when X is the plane
R2 with the Euclidean (i.e. 2 ) metric.
2 The function e then maps each edge of the graph G to the line-segments
joining the images of its end points.



Planar Embedding

An undirected graph G is a planar graph if it can be drawn on a plane so that
no edges intersect.

1 Since we are concerned with R2 we can say that embedding e is planar
when no two line-segments on the embedded graph intersect at any point
other than their end points.



Required Defn.

Let G = (V, E) be a ﬁnite undirected graph with vertex set V (G) and edge set
E(G).
1 A connected acyclic subgraph T of G is a tree. If V (T ) = V (G), then T is
a spanning tree.
2 For x, y ∈ V (G), xy-paths P and Q in G are internally (vertex) disjoint or
openly disjoint if V (P ) ∩ V (Q) = {x, y}.
3 Let p(x, y) denote the maximum number of pair-wise internally disjoint
paths between x, y ∈ V (G).
4 A nontrivial graph G is k-connected if p(u, v) ≥ k for any two distinct
vertices u, v ∈ V (G).
5 The connectivity κ(G) of G is the maximum value of k for which G is
k-connected.



Tutte Embedding: Rubber band representation

Let G = (V, E) be a connected graph and ∅ = S ⊆ V . Fix a map x0 : S → R2 .
We extend this to a map x : V → R2 (a geometric representation of G, or a
drawing on paper) as follows:
1 Replace the edges by ideal rubber bands (satisfying Hooke’s Law). Think
of the nodes in S as nailed to their given position (node u ∈ S to
x0 (u) ∈ R2 ), but let the other nodes settle in equilibrium.
2 We’ll see that this equilibrium position is uniquely determined.
3 We call it the rubber band representation of G in R2 extending x0 .




To be precise, let x(u) ∈ R2 be the position of node u ∈ V . By definition,
x(u) = x0 (u), ∀u ∈ S. The energy of this representation can be defined as
2
E(x) = |x(u) − x(v)| .
uv∈E

We want to find the representation with minimum energy, subject to the
boundary conditions:

minimize E(x) s.t. x(u) = x0 (u), ∀u ∈ S.




1 If S = ∅, then the function E(x) is strictly convex, as
2 2
E(x) = uv∈E |x(u) − x(v)| = uv∈E k=1 (xk (u) − xk (v))2 .
2 If any of the x(u) tends to inﬁnity, then E(x) tends to inﬁnity ⇒ the
representation with minimum energy is uniquely determined.
3 If u ∈ V S, then at the minimum point the partial derivative of E(x) with
respect to any coordinate of x must be 0 ⇒ v∈neigh(u) (x(u) − x(v)) = 0
4 This equation means that every free node is in the center of gravity of its
neighbors.



Tutte Embedding

Every 3-connected planar graph has a convex embedding in the the Euclidean
plane (using Tutte’s rubber band algorithm [Tut60]).

Theorem
Let G = ({1, ·, n} , E) be a 3-connected, planar graph that has a face
(k + 1, . . . , n) for some k < n. Let pk+1 , . . . , pn be the vertices (in this order) of
a convex (n − k)-gon. Let w : E → R+ be an assignment of positive weights
to the internal edges. Then:
There are unique positions p1 , . . . , pk ∈ R2 for the interior vertices such
that all interior vertices are in equilibrium.
All internal faces of G are then realized as non-overlapping convex
polygons.



Tutte Embedding: example



Virtual coordinate: Foundation?

Despite the solid grounding of the ideas in geometric graph theory, no
theoretical results and guarantees were known for such schemes.


Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known results

Greedy Embedding

Let d (pu , pv ) denote the Euclidean distance between two points pu and pv .

Deﬁnition
Greedy embedding ([PR05]): A greedy embedding x of a graph G = (V, E)
into a metric space (X, d) is a function x : V (G) → X with the following
property: for every pair of non-adjacent vertices s, t ∈ V (G) there exists a
vertex u ∈ V (G) adjacent to s such that d (x (u) , x (t)) < d (x (s) , x (t)).



Greedy Embedding : Conjecture(Weak)

This notion of greedy embedding was deﬁned by Papadimitriou and Ratajczak
in [PR05]. They have presented graphs which does not admit a greedy
embedding in the Euclidean plane, and conjectured following:

Conjecture
(Weak). Every 3-connected planar graph has a greedy embedding in the
Euclidean plane.



Greedy Embedding : Conjecture (Strong)

A convex embedding of a planar graph is a planar embedding with a property
that all faces, including the external faces are convex. Additionally,
Papadimitriou and Ratajczak stated the following stronger form of the
conjecture:

Conjecture
(Strong). Every 3-connected planar graph has a greedy convex embedding in
the Euclidean plane.

Note that every 3-connected planar graph has a convex embedding in the the
Euclidean plane (using Tutte’s rubber band algorithm [Tut60]).



Both hypothesis are necessary

Proposition
Kk,5k+1 has no greedy embedding for k > 0.

v6

v1

v5

r
v2
π
≤ 3

v3
v4

Figure: K1,6 has no greedy embedding



Both hypothesis are necessary

These counterexamples imply that the hypotheses of the conjecture are
necessary, in that there exist counterexamples that are planar but not
3-connected (K2,11 ), or 3-connected but not planar (K3,16 ); also, they show
that high connectivity alone does not guarantee a greedy embedding.



Implications of the conjecture being true

There exists a way to assign virtual coordinates to a large class of graphs
where greedy routing is guaranteed.
Since adding edges only improves the embeddability of a graph, the
conjecture extends immediately to any graph with a 3-connected planar
subgraph. Hence to a even larger class.



Known results

Recently, greedy embedding conjecture has been proved in [LM08]. In [LM08]
authors construct a greedy embedding into the Euclidean plane for all circuit
graphs – which is a generalization of 3-connected planar graphs. Similar
result was independently discovered by Angelini, Frati and Grilli [AFG08].

Theorem
([LM08]) Any 3-connected graph G without having a K3,3 minor admits a
greedy embedding into the Euclidean plane.


Introduction Conjecture Contribution Bound Characterization Open Problem Weak greedy

Open Problem

The greedy embedding algorithm presented in [LM08, AFG08] not necessarily
produce a convex greedy embedding, and in fact the embedding may not
even be a planar one. In this work we consider the convex greedy embedding
conjecture. Other questions that we can ask:
1 What is the least dimension of a normed vector space V where every
graph with n nodes has a greedy embedding?
2 Why Tutte embedding improves delivery success?



Contribution

Given a 3-connected planar graph G = (V, E),
1 An embedding x : V → R2 of G is a planar convex greedy embedding if
and only if, in the embedding x, weight of the maximum weight spanning
tree (wt(T )) and weight of the minimum weight spanning tree (wt(MST))
1−δ
satisﬁes wt(T )/wt(MST) ≤ (|V | − 1) , for some 0 < δ ≤ 1.
√
2 G has a β-weak greedy convex embedding in R2 with β ∈ [1, 2 2 · d(G)].
Also, this bound is achieved by Tutte embedding.
3 We show a possibility that Tutte embedding with correct choice of edge
weights may produce greedy embedding of G.



Weak greedy embedding

In order to obtain this result we consider a weaker notion of greedy
embedding. Weak greedy embedding allows path ﬁnding algorithm to
proceed as long as local optima is bounded by a factor. Formally,

Deﬁnition (Weak greedy embedding)
Let β ≥ 1. A β–weak greedy embedding x of a graph G = (V, E) is a planar
embedding of G with the following property: for every pair of non-adjacent
vertices s, t ∈ V (G) there exists a vertex u ∈ V (G) adjacent to s such that
d (x (u) , x (t)) < β · d (x (s) , x (t)).



Weak greedy routing

Algorithm WEAK − GREEDY (s, t, β)
if s = t then
return success.
else
∆
B = {v : (s, v) ∈ E and d(v, t) < β · d(s, t)}.
if B = ∅ then
return failure.
else
∀v ∈ B: WEAK − GREEDY (v, t, β).
end if
end if


Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a Theorem

Outline

For vertex s deﬁne βs : min∀t∈V −{s} at least one branch of this recursive
procedure returns success.
H(s, βs ) ⊆ G induced by all vertices and edges of βs –weak greedy
st–paths for all possible terminal vertex t ∈ V − {s}.
Let T (s, βs ) be any spanning tree of H(s, βs ). T (s, βs ) has unique
βs –weak greedy st–paths for all possible terminal vertex t ∈ V − {s} from
s.
We will call Ts = T (s, βs ) optimal weak greedy tree w.r.t vertex s.
Deﬁne βmax = maxs∈V {βs }.
In following our objective will be to obtain a bound on βmax for any
3-connected planar graph G under embedding x.
To obtain this bound we will use the properties of weak greedy trees.



Bounding the weight of trees: Outline

In the planar convex embedding of G, let weight of an edge e = uv be its
length i.e. wt(e) = d(u, v).
Deﬁne wt(T (s, βs )) = e∈E(T (s,βs )) wt(e).
We obtain a lower and upper bound on the weight of T (s, βs ).
On the other hand we also obtain a upper bound on the weight of any
spanning tree T of G in its embedding wt(T ), and
a lower bound on the weight of any minimum spanning tree MST of G,
wt(MST).



Bounding the weight of trees: Outline...

Surely wt(MST) ≤ wt(Ts ) ≤ wt(T ), and from this we derive an upper and
a lower bound on βmax . Let dmax (G) = maxu,v∈V d(u, v) be the diameter
of G, and let minimum edge length in embedding of G be dmin (G).
√
We derive that, wt(T ) ≤ 2 · (|V | − 1) · dmax (G).
We also show that, dmax (G) ≤ wt(MST) ≤ 2.5 · d2 (G).
max




Deﬁnition (Increasing and decreasing sequence)

For a βs –weak greedy st–path Pst = {s = u0 , u1 , . . . , uk = t}, an ordered
vertex sequence {ui0 , . . . , uir } of Pst is an increasing sequence of length r if
d(ui0 , t) ≤ . . . ≤ d(uir , t) holds. Similarly, an ordered sequence of vertices
{ui0 , . . . , uir } of Pst is a decreasing sequence of length r if
d(ui0 , t) ≥ . . . ≥ d(uir , t) holds. Usually, we will refer any maximal (by property
of monotonically non-decreasing or non-increasing) sequence of vertices as
increasing or decreasing sequence.

It is straightforward to observe that if an st–path is βs –weak greedy for βs > 1,
then it has a monotonically non-decreasing sequence of vertices. However,
every st–path must have a trailing monotonically decreasing sequence that
reaches t.




We will call an increasing sequence {ui0 , . . . , uir } of Pst a β-increasing
sequence of length r if it is maximal and for j = 1, . . . , r, d(uij , t) ≤ βd(uij−1 , t)
holds (with equality for at least one j). We will denote it as inc(r, d, β), where d
indicates d(ui0 , t) = d.

Lemma
Let inc(k, d, β) = {ui0 , . . . , uik } be a β-increasing sequence of length k from a
βs –weak greedy st–path such that d(ui0 , t) = d. Then

β+1
d(β k − 1) ≤ wt(inc(k, d, β)) ≤ d(β k − 1)
β−1

Where wt(inc(k, d, β)) is the sum of the weight of the edges of inc(k, d, β).




Lemma
Let dec(k, d, γ) = {ui0 , . . . , uik } be a γ-decreasing sequence of length k such
that d(ui0 , t) = d. Then
1 1
d(1 − ) ≤ wt(dec(k, d, γ)) ≤ dk(1 + )
γ γ

Lemma
Let P (k, β) be a k length β–weak greedy st–path such that t is a leaf vertex of
the tree Ts . Then

βk − 1
dmin (G) · k · (β − 1) ≤ wt(P (k, β)) ≤ 2 · dmax (G) ·
β−1




Finally we derive upper and lower bounds on the the weight of T (s, βs ) as:
|V |−1
βmax − 1
dmin (G) (βmax − 1) (|V | − 1) ≤ wt(Ts ) ≤ 2dmax (G)
βmax − 1

Using wt(MST) ≤ wt(Ts ) ≤ wt(T ), we have:
|V |−1
βmax − 1
dmax (G) ≤ wt(MST) ≤ wt(Ts ) ≤ 2dmax (G)
βmax − 1

Or,
|V |−1
βmax − 1 1
≥
βmax − 1 2

And this holds for any βmax > 1 when |V | ≥ 3.




Using wt(MST) ≤ wt(Ts ) ≤ wt(T ), we also have:
√
dmin (G) (βmax − 1) (|V | − 1) ≤ wt(Ts ) ≤ wt(T ) ≤ 2 · (|V | − 1) · dmax (G)

Now using d(G) = dmax (G)/dmin (G) we have:
√ dmax (G) √ √
βmax ≤ 2· + 1 ≤ 2 · d(G) + 1 ≤ 2 2 · d(G)
dmin (G)




It follows
Theorem

Let G = (V, E) be any three connected planar√graph. Then G has a β-weak
greedy convex embedding in R2 with β ∈ [1, 2 2 · d(G)]. Also, this bound is
achieved by Tutte embedding.



However, this is not very good

F G F G

C
B C
B D
A D
A

E H E H

(a) (b)

Figure: Illustration of Tutte embedding of a cube: (a)Equal edge weights, (b)Unequal
edge weights

√
β ∈ [1, 2 2 · d(G)]
In Tutte embedding of a 3–connected planar graph G with arbitrary weights on
the edges, bound on β depends entirely on the choice of the edge weights in
the Tutte embedding.


Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be close

When weight of T and MST are close

Theorem (If weights are close it must be greedy)

For sufﬁciently large |V | for a 3-connected planar graph G = (V, E) if
embedding x : V → R2 of G is such that the maximum weight spanning tree
(T ) and minimum weight spanning tree (MST) satisﬁes,
1−δ
wt(T )/wt(MST) ≤ (|V | − 1) , for some 0 < δ ≤ 1, then embedding x is a
convex greedy embedding of G.



When weight of T and MST are close

Proof: wt(MST) ≤ wt(Ts ) ≤ wt(T ) and wt(MST) > 0,

2 · dmin (G) · (βmax − 1) · (|V | − 1) wt(T )
≤
5 · d2 (G)
max wt(MST)

5 · dmax (G) · d(G) wt(T )
And hence, βmax ≤ · +1
2 · (|V | − 1) wt(MST)

then for sufﬁciently large |V |, βmax → 1 from above (note that βmax > 1).



More interesting direction

Theorem (If G is greedy then weights are close)

Given a 3-connected planar graph G = (V, E), if embedding x : V → R2 of G
is a convex greedy embedding then in embedding x the maximum weight
spanning tree (T ) and minimum weight spanning tree (MST) satisﬁes:
1−δ
wt(T )/wt(MST) ≤ (|V | − 1) , for some 0 < δ ≤ 1.



Proof outline

Fact
A graph is 3-connected and planar if and only if each edge is in exactly two
non-separating induced cycles [Kel78]

1 Consider two cases: (Case - 1) e is on two internal faces F and F , and
(Case - 2) e is on the boundary face (inﬁnite face is the other one).
2 Let us also assume that wt(T )/wt(MST) ≥ (|V | − 1). W.l.o.g. let
wt(MST) = 1. Since T is a spanning tree it has (|V | − 1) edges, and
hence has at least one edge e ∈ T of weight wt(e) ≥ 1.



Proof outline...

For a graph G, a thread is a path P of G such that any degree 2 vertex x of G
is not an end vertex of P . A sequence S = (G0 , {xi Pi yi : i = 1, . . . , k}) is an
ear-decomposition of G if:
1 G0 is a subdivision of K4 ,
2 xi Pi yi is a path with end-vertices xi and yi such that Gi = Gi−1 ∪ Pi is a
subgraph of G, and Gi−1 ∩ Pi = {xi , yi }, but xi , yi do not belong to a
common thread of Gi−1 for i = 1, . . . , k, and
3 Gk = G.



Proof outline...

Lemma
Let G be a 3–connected graph, e = uv ∈ E(G). Let C1 and C2 be
non-separating cycles of G such that C1 ∩ C2 = uev. Then there exists an
ear-decomposition of G such that C1 ∪ C2 ⊂ G0 .



Proof outline...

Say, e = uv is on two internal faces F1 and F2 . Consider a vertex u from face
F1 and another vertex v from face F2 . First consider K4 , which has four
faces, and exactly one planar convex embedding. However, vertices u, v, u , v
must be spanned by the MST using exactly 3 edges. If e is chosen in the MST
then other edges are of length 0, as wt(e) ≥ 1 and wt(MST) = 1. If e is not
selected in MST - then it can be easily seen that either wt(MST) > 1, or the
drawing is not planar - a contradiction.
u u u

v
v v v u
K4 Planar convex embedding of K4

Figure: Illustration to the proof of Case - 1 for K4



Concluding remarks

With our result, and the example presented above we can ask following
question:

For every 3–connected planar graph G, is it possible to choose edge weights
in the Tutte embedding such that we obtain a greedy convex embedding?

We believe that answer to this question will help in resolving convex greedy
embedding conjecture of Papadimitriou and Ratajczak.


Appendix Reference

Reference: I
Patrizio Angelini, Fabrizio Frati, and Luca Grilli.
An algorithm to construct greedy drawings of triangulations.
In 16th International Symposium on Graph Drawing (GD ’08), 2008.
To appear.

Alexander Kelmans.
The concept of a vertex in a matroid, the non-separating cycles, and a new criterion for graph planarity.
In Algebraic Methods in Graph Theory, Colloq. Math. Soc. Janos Bolyai, (Szeged, Hungary, 1978)NorthHolland, 1:345–388, 1978.

Brad Karp and H. T. Kung.
GPSR: greedy perimeter stateless routing for wireless networks.
In MobiCom ’00: Proceedings of the 6th annual international conference on Mobile computing and networking, pages 243–254, New York, NY,
USA, 2000. ACM Press.

Tom Leighton and Ankur Moitra.
Some results on greedy embeddings in metric spaces.
In FOCS ’08: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS’08), Washington, DC, USA, 2008.
IEEE Computer Society.

Christos H. Papadimitriou and David Ratajczak.
On a conjecture related to geometric routing.
Theor. Comput. Sci., 344(1):3–14, 2005.

Ananth Rao, Christos Papadimitriou, Scott Shenker, and Ion Stoica.
Geographic routing without location information.
In MobiCom ’03: Proceedings of the 9th annual international conference on Mobile computing and networking, pages 96–108, New York, NY, USA,
2003. ACM Press.

W. T. Tutte.
Convex Representations of Graphs.
Proc. London Math. Soc., s3-10(1):304–320, 1960.


Greedy embedding problem

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