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# Greedy embedding problem

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### Greedy embedding problem

1. 1. 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
2. 2. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandSensor Networks and Ad hoc networks Subhas K. Ghosh Greedy Embedding
3. 3. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGeometric 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. Subhas K. Ghosh Greedy Embedding
4. 4. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGeometric routingGreedy 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 Subhas K. Ghosh Greedy Embedding
5. 5. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGreedy 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 Subhas K. Ghosh Greedy Embedding
6. 6. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber band 6Greedy routing : Example s 1 2 8 3 9 4 5 1017 t 18 19 11 12 Subhas K. Ghosh Greedy Embedding
7. 7. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGreedy 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! Subhas K. Ghosh Greedy Embedding
8. 8. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber band 15 6Greedy routing : Example... s 1 2 8 3 9 4 16 520 10 17 18 19 t 13 11 12 Subhas K. Ghosh Greedy Embedding
9. 9. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGreedy 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]. Subhas K. Ghosh Greedy Embedding
10. 10. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGeometric 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 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. Subhas K. Ghosh Greedy Embedding
11. 11. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandVirtual CoordinatesIn [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 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. Subhas K. Ghosh Greedy Embedding
12. 12. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandGraph EmbeddingAn 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 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. Subhas K. Ghosh Greedy Embedding
13. 13. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandPlanar EmbeddingAn 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 planar when no two line-segments on the embedded graph intersect at any point other than their end points. Subhas K. Ghosh Greedy Embedding
14. 14. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandRequired 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 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. Subhas K. Ghosh Greedy Embedding
15. 15. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandTutte Embedding: Rubber band representationLet 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 adrawing 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 . Subhas K. Ghosh Greedy Embedding
16. 16. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandTutte Embedding: Rubber band representationTo 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∈EWe want to ﬁnd the representation with minimum energy, subject to theboundary conditions: minimize E(x) s.t. x(u) = x0 (u), ∀u ∈ S. Subhas K. Ghosh Greedy Embedding
17. 17. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandTutte Embedding: Rubber band representation 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. Subhas K. Ghosh Greedy Embedding
18. 18. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandTutte EmbeddingEvery 3-connected planar graph has a convex embedding in the the Euclideanplane (using Tutte’s rubber band algorithm [Tut60]).TheoremLet 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 such that all interior vertices are in equilibrium. All internal faces of G are then realized as non-overlapping convex polygons. Subhas K. Ghosh Greedy Embedding
19. 19. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandTutte Embedding: example Subhas K. Ghosh Greedy Embedding
20. 20. Introduction Conjecture Contribution Bound Characterization Geometric routing Greedy routing Greedy routing fails! Rubber bandVirtual coordinate: Foundation?Despite the solid grounding of the ideas in geometric graph theory, notheoretical results and guarantees were known for such schemes. Subhas K. Ghosh Greedy Embedding
21. 21. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsGreedy EmbeddingLet d (pu , pv ) denote the Euclidean distance between two points pu and pv .DeﬁnitionGreedy 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)). Subhas K. Ghosh Greedy Embedding
22. 22. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsGreedy 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. Subhas K. Ghosh Greedy Embedding
23. 23. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsGreedy 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]). Subhas K. Ghosh Greedy Embedding
24. 24. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsBoth hypothesis are necessaryPropositionKk,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 Subhas K. Ghosh Greedy Embedding
25. 25. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsBoth hypothesis are necessaryThese 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. Subhas K. Ghosh Greedy Embedding
26. 26. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsImplications 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. Subhas K. Ghosh Greedy Embedding
27. 27. Introduction Conjecture Contribution Bound Characterization Greedy Embedding Conjecture Conjecture - Details Known resultsKnown resultsRecently, 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. Subhas K. Ghosh Greedy Embedding
28. 28. Introduction Conjecture Contribution Bound Characterization Open Problem Weak greedyOpen ProblemThe 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 every graph with n nodes has a greedy embedding? 2 Why Tutte embedding improves delivery success? Subhas K. Ghosh Greedy Embedding
29. 29. Introduction Conjecture Contribution Bound Characterization Open Problem Weak greedyContributionGiven 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. Subhas K. Ghosh Greedy Embedding
30. 30. Introduction Conjecture Contribution Bound Characterization Open Problem Weak greedyWeak greedy embeddingIn 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)). Subhas K. Ghosh Greedy Embedding
31. 31. Introduction Conjecture Contribution Bound Characterization Open Problem Weak greedyWeak 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 Subhas K. Ghosh Greedy Embedding
32. 32. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremOutline 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. Subhas K. Ghosh Greedy Embedding
33. 33. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding 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). Subhas K. Ghosh Greedy Embedding
34. 34. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding 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 Subhas K. Ghosh Greedy Embedding
35. 35. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...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. Subhas K. Ghosh Greedy Embedding
36. 36. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...We will call an increasing sequence {ui0 , . . . , uir } of Pst a β-increasingsequence 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.LemmaLet inc(k, d, β) = {ui0 , . . . , uik } be a β-increasing sequence of length k from aβs –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−1Where wt(inc(k, d, β)) is the sum of the weight of the edges of inc(k, d, β). Subhas K. Ghosh Greedy Embedding
37. 37. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...LemmaLet dec(k, d, γ) = {ui0 , . . . , uik } be a γ-decreasing sequence of length k suchthat d(ui0 , t) = d. Then 1 1 d(1 − ) ≤ wt(dec(k, d, γ)) ≤ dk(1 + ) γ γLemmaLet P (k, β) be a k length β–weak greedy st–path such that t is a leaf vertex ofthe tree Ts . Then βk − 1 dmin (G) · k · (β − 1) ≤ wt(P (k, β)) ≤ 2 · dmax (G) · β−1 Subhas K. Ghosh Greedy Embedding
38. 38. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...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 − 1Using wt(MST) ≤ wt(Ts ) ≤ wt(T ), we have: |V |−1 βmax − 1 dmax (G) ≤ wt(MST) ≤ wt(Ts ) ≤ 2dmax (G) βmax − 1Or, |V |−1 βmax − 1 1 ≥ βmax − 1 2And this holds for any βmax > 1 when |V | ≥ 3. Subhas K. Ghosh Greedy Embedding
39. 39. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...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) Subhas K. Ghosh Greedy Embedding
40. 40. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremBounding the weight of trees: Outline...It followsTheoremLet 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 isachieved by Tutte embedding. Subhas K. Ghosh Greedy Embedding
41. 41. Introduction Conjecture Contribution Bound Characterization Trees of all kind ... and a TheoremHowever, 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)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. Subhas K. Ghosh Greedy Embedding
42. 42. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeWhen weight of T and MST are closeTheorem (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, 1−δwt(T )/wt(MST) ≤ (|V | − 1) , for some 0 < δ ≤ 1, then embedding x is aconvex greedy embedding of G. Subhas K. Ghosh Greedy Embedding
43. 43. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeWhen weight of T and MST are closeProof: 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). Subhas K. Ghosh Greedy Embedding
44. 44. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeMore interesting directionTheorem (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: 1−δwt(T )/wt(MST) ≤ (|V | − 1) , for some 0 < δ ≤ 1. Subhas K. Ghosh Greedy Embedding
45. 45. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeProof outlineFactA 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. 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. Subhas K. Ghosh Greedy Embedding
46. 46. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeProof 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 , {xi Pi yi : i = 1, . . . , k}) is anear-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. Subhas K. Ghosh Greedy Embedding
47. 47. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeProof outline...LemmaLet 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 . Subhas K. Ghosh Greedy Embedding
48. 48. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeProof 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 , vmust 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 u u v v v v u K4 Planar convex embedding of K4 Figure: Illustration to the proof of Case - 1 for K4 Subhas K. Ghosh Greedy Embedding
49. 49. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be closeConcluding remarksWith 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. Subhas K. Ghosh Greedy Embedding
50. 50. Introduction Conjecture Contribution Bound Characterization When weights are close When weights must be close Subhas K. Ghosh Greedy Embedding
51. 51. Appendix ReferenceReference: 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. Subhas K. Ghosh Greedy Embedding