Greedy embedding problem

Introduction Conjecture Contribution Bound Characterization

Geometric Routing: Theoretical Foundations, anImportant 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: nouniversally known system of addresses.Resource limitations: prohibitive to store and maintain large forwardingtables.Geometric routing: use geographic coordinates of the nodes asaddresses.Simplest of routing strategy: recursively select a vertex that is closer todestination than current vertex.Point-to-point routing service: enable data centric storage, a DistributedHash Table (DHT) is implemented in the sensor network, and data isstored at the nodes of the network using the DHT.



Geometric routing

Greedy routingset i := 0, v0 := swhile vi 6= t do

du := minu∈N (vi) {d (u, t) : d (u, t) < d (vi, t)}if ∃u then

set i := i + 1, vi := uelse

fail!end if

end while



Greedy routing : Example

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Figure: Greedy Routing On A Graph



Greedy routing : Example1

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Figure: Greedy Routing On A Graph



Greedy routing : Example...

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Figure: No Greedy Routing!



Greedy routing : Example...1

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(b) No Forwarding Path

Figure: No Greedy Routing!



Greedy routing sometimes fails!

Greedy routing sometimes fails to deliver a packet because of thephenomenon of “voids” (nodes with no neighbor closer to the destination).This is essentially reaching a local minimum with respect to the distancefunctional 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 ofprecise geographic coordinates (GPS is costly and does not work in allsituation).Importantly, the precise coordinates may be disadvantageous as they donot account for obstructions or other topological properties of thenetwork.

How to retain the benefits of geographic routing in the absence of locationinformation?

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



Virtual Coordinates

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

Note that embedding in higher dimension requires storing moreinformation per node.On the basis of extensive experimentation in [RPSS03] authors showedthat this approach makes greedy routing much more reliable (works97.5% cases).Kleinberg proved 100% success rate by assigning virtual coordinates inthe hyperbolic plane rather than the Euclidean plane.



Graph Embedding

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

1 In this work we will be concerned with a special case when X is the planeR2 with the Euclidean (i.e. `2) metric.

2 The function e then maps each edge of the graph G to the line-segmentsjoining 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 thatno edges intersect.

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



Required Defn.

Let G = (V,E) be a finite undirected graph with vertex set V (G) and edge setE(G).

1 A connected acyclic subgraph T of G is a tree. If V (T ) = V (G), then T isa spanning tree.

2 For x, y ∈ V (G), xy-paths P and Q in G are internally (vertex) disjoint oropenly disjoint if V (P ) ∩ V (Q) = {x, y}.

3 Let p(x, y) denote the maximum number of pair-wise internally disjointpaths between x, y ∈ V (G).

4 A nontrivial graph G is k-connected if p(u, v) ≥ k for any two distinctvertices u, v ∈ V (G).

5 The connectivity κ(G) of G is the maximum value of k for which G isk-connected.



Tutte Embedding: Rubber band representation

Let G = (V,E) be a connected graph and ∅ 6= S ⊆ V . Fix a map x0 : S → R2.We extend this to a map x : V → R2 (a geometric representation of G, or adrawing on paper) as follows:

1 Replace the edges by ideal rubber bands (satisfying Hooke’s Law). Thinkof the nodes in S as nailed to their given position (node u ∈ S tox0(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

E(x) =∑

uv∈E

|x(u)− x(v)|2.

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

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




1 If S 6= ∅, then the function E(x) is strictly convex, asE(x) =

∑uv∈E |x(u)− x(v)|2 =

∑uv∈E

∑2k=1 (xk(u)− xk(v))2.

2 If any of the x(u) tends to infinity, then E(x) tends to infinity ⇒ therepresentation with minimum energy is uniquely determined.

3 If u ∈ V \ S, then at the minimum point the partial derivative of E(x) withrespect 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 itsneighbors.



Tutte Embedding

Every 3-connected planar graph has a convex embedding in the the Euclideanplane (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) ofa convex (n− k)-gon. Let w : E → R+ be an assignment of positive weightsto the internal edges. Then:

There are unique positions p1, . . . , pk ∈ R2 for the interior vertices suchthat all interior vertices are in equilibrium.All internal faces of G are then realized as non-overlapping convexpolygons.



Tutte Embedding: example



Virtual coordinate: Foundation?

Despite the solid grounding of the ideas in geometric graph theory, notheoretical 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 followingproperty: for every pair of non-adjacent vertices s, t ∈ V (G) there exists avertex 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 Ratajczakin [PR05]. They have presented graphs which does not admit a greedyembedding in the Euclidean plane, and conjectured following:

Conjecture

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



Greedy Embedding : Conjecture (Strong)

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

Conjecture

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

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



Both hypothesis are necessary

Proposition

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

v1

v2

v3v4

v5

v6

r

≤ π3

Figure: K1,6 has no greedy embedding



Both hypothesis are necessary

These counterexamples imply that the hypotheses of the conjecture arenecessary, in that there exist counterexamples that are planar but not3-connected (K2,11), or 3-connected but not planar (K3,16); also, they showthat 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 graphswhere greedy routing is guaranteed.Since adding edges only improves the embeddability of a graph, theconjecture extends immediately to any graph with a 3-connected planarsubgraph. 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 circuitgraphs – which is a generalization of 3-connected planar graphs. Similarresult was independently discovered by Angelini, Frati and Grilli [AFG08].

Theorem([LM08]) Any 3-connected graph G without having a K3,3 minor admits agreedy 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 necessarilyproduce a convex greedy embedding, and in fact the embedding may noteven be a planar one. In this work we consider the convex greedy embeddingconjecture. Other questions that we can ask:

1 What is the least dimension of a normed vector space V where everygraph 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 spanningtree (wt(T )) and weight of the minimum weight spanning tree (wt(MST))satisfies wt(T )/wt(MST) ≤ (|V | − 1)1−δ, 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 edgeweights may produce greedy embedding of G.



Weak greedy embedding

In order to obtain this result we consider a weaker notion of greedyembedding. Weak greedy embedding allows path finding algorithm toproceed 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 planarembedding of G with the following property: for every pair of non-adjacentvertices s, t ∈ V (G) there exists a vertex u ∈ V (G) adjacent to s such thatd (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 = ∅ thenreturn failure.

else∀v ∈ B: WEAK− GREEDY (v, t, β).

end ifend 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 recursiveprocedure returns success.H(s, βs) ⊆ G induced by all vertices and edges of βs–weak greedyst–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 uniqueβs–weak greedy st–paths for all possible terminal vertex t ∈ V − {s} froms.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 any3-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 itslength 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 anyspanning tree T of G in its embedding wt(T ), anda 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 anda lower bound on βmax. Let dmax(G) = maxu,v∈V d(u, v) be the diameterof 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 · d2max(G).




Definition (Increasing and decreasing sequence)

For a βs–weak greedy st–path Pst = {s = u0, u1, . . . , uk = t}, an orderedvertex sequence {ui0 , . . . , uir

} of Pst is an increasing sequence of length r ifd(ui0 , t) ≤ . . . ≤ d(uir

, t) holds. Similarly, an ordered sequence of vertices{ui0 , . . . , uir

} of Pst is a decreasing sequence of length r ifd(ui0 , t) ≥ . . . ≥ d(uir

, t) holds. Usually, we will refer any maximal (by propertyof monotonically non-decreasing or non-increasing) sequence of vertices asincreasing 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 thatreaches 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 dindicates 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

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

β + 1β − 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

d(1− 1γ

) ≤ wt(dec(k, d, γ)) ≤ dk(1 +1γ

)

Lemma

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

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

βk − 1β − 1

)




Finally we derive upper and lower bounds on the the weight of T (s, βs) as:

dmin(G) (βmax − 1) (|V | − 1) ≤ wt(Ts) ≤ 2dmax(G)

(β|V |−1max − 1βmax − 1

)

Using wt(MST) ≤ wt(Ts) ≤ wt(T ), we have:

dmax(G) ≤ wt(MST) ≤ wt(Ts) ≤ 2dmax(G)

(β|V |−1max − 1βmax − 1

)

Or, (β|V |−1max − 1βmax − 1

)≥ 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:

βmax ≤√

2 · dmax(G)dmin(G)

+ 1 ≤√

2 · d(G) + 1 ≤ 2√

2 · d(G)




It follows

Theorem

Let G = (V,E) be any three connected planar graph. Then G has a β-weakgreedy 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

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F G

H

(a)

A

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F G

H

(b)

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

β ∈ [1, 2√

2 · d(G)]

In Tutte embedding of a 3–connected planar graph G with arbitrary weights onthe edges, bound on β depends entirely on the choice of the edge weights inthe 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) ifembedding x : V → R2 of G is such that the maximum weight spanning tree(T ) and minimum weight spanning tree (MST) satisfies,wt(T )/wt(MST) ≤ (|V | − 1)1−δ, for some 0 < δ ≤ 1, then embedding x is aconvex 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)5 · d2

max(G)≤ wt(T )

wt(MST)

And hence, βmax ≤(

5 · dmax(G) · d(G)2 · (|V | − 1)

)·(

wt(T )wt(MST)

)+ 1

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 Gis a convex greedy embedding then in embedding x the maximum weightspanning tree (T ) and minimum weight spanning tree (MST) satisfies:wt(T )/wt(MST) ≤ (|V | − 1)1−δ, for some 0 < δ ≤ 1.



Proof outline

FactA graph is 3-connected and planar if and only if each edge is in exactly twonon-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. letwt(MST) = 1. Since T is a spanning tree it has (|V | − 1) edges, andhence 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 Gis not an end vertex of P . A sequence S = (G0, {xiPiyi : i = 1, . . . , k}) is anear-decomposition of G if:

1 G0 is a subdivision of K4,2 xiPiyi 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 acommon 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 benon-separating cycles of G such that C1 ∩ C2 = uev. Then there exists anear-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 faceF1 and another vertex v′ from face F2. First consider K4, which has fourfaces, 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 MSTthen other edges are of length 0, as wt(e) ≥ 1 and wt(MST) = 1. If e is notselected in MST - then it can be easily seen that either wt(MST) > 1, or thedrawing is not planar - a contradiction.

u

v

v′

u′

K4 Planar convex embedding of K4

u

v

u′

v′

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



Concluding remarks

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

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

We believe that answer to this question will help in resolving convex greedyembedding 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.


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