CDS in Plane Geometric Networks: Short, Small, And Sparse
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1 CDS in Plane Geometric Networks: Short, Small, And Sparse Peng-Jun Wan Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 1 / 38
2 Outline Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 2 / 38
3 Three Parameters of A Virtual Backbone (CDS) G = (V, E ): a (connected) UDG s 2 V : a xed node R: radius of G w.r.t. s. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 3 / 38
4 Three Parameters of A Virtual Backbone (CDS) G = (V, E ): a (connected) UDG s 2 V : a xed node R: radius of G w.r.t. s. U: a CDS of G containing s Size: juj Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 3 / 38
5 Three Parameters of A Virtual Backbone (CDS) G = (V, E ): a (connected) UDG s 2 V : a xed node R: radius of G w.r.t. s. U: a CDS of G containing s Size: juj Radius: Rad (G [U], s) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 3 / 38
6 Three Parameters of A Virtual Backbone (CDS) G = (V, E ): a (connected) UDG s 2 V : a xed node R: radius of G w.r.t. s. U: a CDS of G containing s Size: juj Radius: Rad (G [U], s) Sparsity: (G [U]) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 3 / 38
7 Virtual Backbone (CDS): Small, Short, Sparse Our objective is to construct a CDS U of G with s 2 U such that 1 U is small: juj = Θ (poly (R)) 2 U is short: Rad (G [U], s) = Θ (R) 3 U is sparse: (G [U]) is small, preferably bounded by a constant. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 4 / 38
8 2-Phased Algorithm Phase 1: First- t selection of an MIS I in the BFS ordering w.r.t. s Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 5 / 38
9 2-Phased Algorithm Phase 1: First- t selection of an MIS I in the BFS ordering w.r.t. s Phase 2: Augment I with a set C of connectors to form a CDS Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 5 / 38
10 Roadmap Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 6 / 38
11 Sparsity of Dominators Each node is adjacent to at most 5 dominators. v 11 v 10 v9 v 12 v 3 v 2 v 8 v 13 v 0 v 1 v 7 v 4 v 14 v 5 v 6 v 18 v 15 v 16 v 17 Figure: The layout of 18 dominators in the annulus of radii 1 + 2ε and (1 + 2ε) p 2 + p 3 centered at a node v 0. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 7 / 38
12 Sparsity of Dominators Each node is adjacent to at most 5 dominators. The annulus of radii one and two centered at each node contains at most 18 dominators (Bateman and Erdös) v 11 v 10 v9 v 12 v 3 v 2 v 8 v 13 v 0 v 1 v 7 v 4 v 14 v 5 v 6 v 18 v 15 v 16 v 17 Figure: The layout of 18 dominators in the annulus of radii 1 + 2ε and (1 + 2ε) p 2 + p 3 centered at a node v 0. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 7 / 38
13 Groemer Inequality Theorem Suppose that S is a compact convex set and U is a set of points with mutual distances at least one. Then ju \ Sj area (S) p 3/2 + peri (S) 2 + 1, where area (S) and peri (S) are the area and perimeter of S respectively. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 8 / 38
14 Packing in Disks and Half-Disks Corollary Suppose that S (respectively, S 0 ) is a disk (respectively, half-disk) of radius r, and U is a set of points with mutual distances at least one. Then ju \ Sj p 2π r 2 + πr + 1, 3 U \ S 0 π p 3 r 2 + π r + 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 9 / 38
15 An Upper Bound on The Number of Dominators Since all dominators lie in the disk of radius R centered at s, we have 2π ji j p3 R 2 + πr + 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 10 / 38
16 Roadmap Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 11 / 38
17 Basic Set of Connectors 8u 2 I n fsg, p (u) the least-id neighbor of u in the layer above u. s Layer 0 Layer 1 Layer 2 Layer R Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 12 / 38
18 Basic Set of Connectors 8u 2 I n fsg, p (u) the least-id neighbor of u in the layer above u. output C = fp (u) : u 2 I n fsgg. s Layer 0 Layer 1 Layer 2 Layer R Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 12 / 38
19 Size And Radius jc j ji n fsgj = ji j 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 13 / 38
20 Size And Radius Hence, ji [ C j 2 ji j jc j ji n fsgj = ji j π p 3 R 2 + 2πR + 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 13 / 38
21 Size And Radius Hence, In addition, ji [ C j 2 ji j jc j ji n fsgj = ji j π p 3 R 2 + 2πR + 1. Rad (G [I [ C ], s) 2 (R 1). Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 13 / 38
22 Sparsity Each dominatee is adj. to at most 4 dominators in the layer below itself. v 11 v 10 v 9 v 12 w w 10 v 2 w 9 v 11 3 w 12 w 2 w 3 w 8 v 1 v v 13 w 0 w 1 13 w 7 w v 4 4 v 8 v 7 w 14 w 5 w 6 v 6 w 18 v 14 w 15 v5 w 16 w 17 v18 v 15 v 17 v 16 Figure: The node v 0 is s. It is adjacent to 16 connectors. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 14 / 38
23 Sparsity Each dominatee is adj. to at most 4 dominators in the layer below itself. s is adj. to at most 18 connectors v 11 v 10 v 9 v 12 w w 10 v 2 w 9 v 11 3 w 12 w 2 w 3 w 8 v 1 v v 13 w 0 w 1 13 w 7 w v 4 4 v 8 v 7 w 14 w 5 w 6 v 6 w 18 v 14 w 15 v5 w 16 w 17 v18 v 15 v 17 v 16 Figure: The node v 0 is s. It is adjacent to 16 connectors. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 14 / 38
24 Sparsity Each dominatee is adj. to at most 4 dominators in the layer below itself. s is adj. to at most 18 connectors Each other dominator is adj. to at most 17 connectors in the same or the next layer. v 11 v 10 v 9 v 12 w w 10 v 2 w 9 v 11 3 w 12 w 2 w 3 w 8 v 1 v v 13 w 0 w 1 13 w 7 w v 4 4 v 8 v 7 w 14 w 5 w 6 v 6 w 18 v 14 w 15 v5 w 16 w 17 v18 v 15 v 17 v 16 Figure: The node v 0 is s. It is adjacent to 16 connectors. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 14 / 38
25 Roadmap Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 15 / 38
26 Minimal Cover y 1 y 2 y 3 y 4 y 5 y 2 y 3 y 5 x 1 x 2 x 3 x 4 x 5 x 6 x 7 (a) x 1 x 2 x 3 x 4 x 5 x 6 x 7 (b) Figure: (a) X = fx i : 1 i 7g is covered by Y = fy i : 1 i 5g. (b) fy 2, y 3, y 4 g is a minimal cover of X. The black nodes are private neighbors. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 16 / 38
27 The First Improvement on Connectors for each 0 i R, I i {dominators of depth i in G}; C ; for each 1 i R 1, C i a minimal cover of I i+1 in fp (u) : u 2 I i+1 g; C C [ C i ; output C. s Layer 0 Layer 1 Layer 2 Layer R Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 17 / 38
28 Size, Radius And Sparsity jc j ji j 1, Rad (G [I [ C ], s) 2 (R 1). Lemma For each 2 i < R 1, each dominator in I i is adjacent to at most 12 connectors in C i and at most 11 connectors in C i+1. In addition, jc 0 j 12. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 18 / 38
29 An Equilateral Triangle Property x z y u π/6 q p v Figure: The two circles have unit radius, and 1 kuvk 2. Then, both 4pvx and 4qvy are equilateral. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 19 / 38
30 A Geometric Lemma on Angle Separation Lemma Consider three nodes u, v and w satisfying that 1 < kuwk kuvk 2 and kvwk > 1. If dvuw 2 arcsin , then B (u) \ B (v) B (w). y q z u θ π/6 1 x w p 1 v Figure: If θ 2 arcsin 1 4, then kuyk kuvk, and hence w 2 ux 4upq. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 20 / 38
31 A Geometric Lemma on Angle Separation kuyk kuvk, duvy duyv = duxy, dxvy θ. kvzk = kuvk sin θ 2 sin θ = 4 sin θ 2 cos θ 2 cos θ 2 ) dxvy = 2 arccos kvzk 2 arccos cos θ = θ. 2 y q z u θ π/6 1 x w p 1 v Figure: If θ 2 arcsin 1 4, then kuyk kuvk. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 21 / 38
32 Proof of Sparsity Each dominator u 2 I i is adj. to at most 12 connectors in C i : v 2 w 2 v j' w j' w j'' u w1 v 1 v j'' w k w k 1 v k v k 1 Figure: w 1, w 2,, w k are the connectors in C i adjacent to u. Each v j is a private dominator neighbor of w j in I i +1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 22 / 38
33 Proof of Sparsity Each dominator u 2 I i is adj. to at most 12 connectors in C i : v 2 w 2 v j' w j' w j'' u w1 v 1 v j'' w k w k 1 v k v k 1 Figure: w 1, w 2,, w k are the connectors in C i adjacent to u. Each v j is a private dominator neighbor of w j in I i +1. If k 13, then there exist two dominators v j 0 and v j 00 s.t. \v j 0uv j 00 2π Assume by symmetry that v j 00 is closer to u then v j 0. Then, w j 0 2 B (u) \ B (v j 0) B (v j 00). 13. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 22 / 38
34 Proof of Sparsity Each dominator u 2 I i is adj. to at most 11 connectors in C i+1 : v 0 w 0 Layer i 1 u Layer i w 1 w w k 1 w k Layer i+1 v 1 v 2... v k 1 v k Layer i+2 Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 23 / 38
35 An Example v 10 v 9 v 11 v 12 w w 10 v 2 11 w 9 v 3 w 12 w 2 w 3 w 8 v 1 v v 13 w 0 w 1 13 w 7 w v 4 4 v 8 v 7 w 14 w 5 w 6 v 6 w 18 v 14 w 15 v 5 w 16 w 17 v18 v 15 v 17 v 16 Figure: The node v 0 is s. It is adjacent to 12 connectors. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 24 / 38
36 Roadmap Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 25 / 38
37 An Auxiliary Graph on Dominators G 0 : the graph on dominators in which there is edge between two dominators i they have a common neighbor in G. I 0 s Layer 0 I 1 Layer 1 I 2 Layer 2 I R' Layer R' Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 26 / 38
38 An Auxiliary Graph on Dominators G 0 : the graph on dominators in which there is edge between two dominators i they have a common neighbor in G. R 0 : radius of H w.r.t. s. Then, R 0 R 1. I 0 s Layer 0 I 1 Layer 1 I 2 Layer 2 I R' Layer R' Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 26 / 38
39 An Auxiliary Graph on Dominators G 0 : the graph on dominators in which there is edge between two dominators i they have a common neighbor in G. R 0 : radius of H w.r.t. s. Then, R 0 R 1. I i for 0 i R 0 : the set of dominators of depth i in G 0. I 0 s Layer 0 I 1 Layer 1 I 2 Layer 2 I R' Layer R' Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 26 / 38
40 The Second Improvement on Connectors C ; for each 1 i R 0 1, P i the set of nodes adj. to I i and I i+1, C i a minimal cover in P i of I i+1 ; C C [ C i output C. s I 0 C 0 I 1 C 1 I 2 C R' 1 I R' Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 27 / 38
41 Size, Radius And Sparsity jc j ji j 1, Rad (G [I [ C ], s) = 2R 0 2 (R 1). Lemma jc 0 j 12, and for each 1 i < R 0 at most 11 connectors in C i. 1 each dominator in I i is adjacent to Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 28 / 38
42 Roadmap Three Parameters of CDS Dominators Basic Set of Connectors The First Improvement The Second Improvement The Third Improvement Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 29 / 38
43 Ancestors T : a BFS tree of G rooted at s s 3 p (v) 2 p (v) 1 p (v) v Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 30 / 38
44 Ancestors T : a BFS tree of G rooted at s p i (v) for v 6= s: the i-th ancestor of v in T. s 3 p (v) 2 p (v) 1 p (v) v Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 30 / 38
45 The Third Improvement on Connectors k: a positive integer parameter i p (u) C ; for each dominator u 6= s, i min j : p j (u) 2 I ; C C [ p j (u) : 1 j min fi 1, kg ; output C. i p (u) k p (u) (a) u u (b) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 31 / 38
46 Size Hence, jc j k (ji j 1). ji [ C j ji j + k (ji j 1) = (k + 1) ji j k 2π (k + 1) p3 R 2 + πr + 1 k 2π = (k + 1) p3 R 2 + πr + 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 32 / 38
47 Radius Rad (G [I [ C ], s) (1 + 1/k) R Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 33 / 38
48 Radius Rad (G [I [ C ], s) (1 + 1/k) R Denote H = G [I [ C ]. It is su cient to show that for each dominator u, dist H (u, s) dist G (u, s). k Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 33 / 38
49 Radius Rad (G [I [ C ], s) (1 + 1/k) R Denote H = G [I [ C ]. It is su cient to show that for each dominator u, dist H (u, s) dist G (u, s). k Induction on dist G (u, s): Trivial if dist G (u, s) = 0. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 33 / 38
50 Radius Rad (G [I [ C ], s) (1 + 1/k) R Denote H = G [I [ C ]. It is su cient to show that for each dominator u, dist H (u, s) dist G (u, s). k Induction on dist G (u, s): Trivial if dist G (u, s) = 0. So, we assume that dist G (u, s) > k and let i = min j : p j (u) 2 I. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 33 / 38
51 Induction: Case 1 i p (u)=v u Case 1: i k + 1. Let v = p i (u). Then, dist H (u, s) dist H (v, s) + i dist G (v, s) + i k < (dist G (v, s) + i) k = dist G (u, s). k Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 34 / 38
52 Induction: Case 2 v k p (u) Case 2: i > k + 1 and p k (u) is adjacent to some dominator v at the same layer as p k+1 (u). dist H (u, s) dist H (v, s) + k dist G (v, s) + k + 1 k < (dist G (v, s) + k + 1) k = dist G (u, s). k u Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 35 / 38
53 Induction: Case 3 v k p (u) u Case 3: i > k + 1 and p k (u) is adj. to some dominator v at the same layer as itself. Then, dist H (u, s) dist H (v, s) + k dist G (v, s) + (k + 1) k = (dist G (v, s) + k) k = dist G (u, s). k Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 36 / 38
54 Sparsity v u (H) 2 p 3πk 2 + 3πk π/ p 3. q(v) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 37 / 38
55 Sparsity v u (H) 2 p 3πk 2 + 3πk π/ p 3. For each v 2 C, q (v) denotes the closest descendant dominator of v; for each v 2 I, q (v) = v. q(v) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 37 / 38
56 Sparsity v u (H) 2 p 3πk 2 + 3πk π/ p 3. For each v 2 C, q (v) denotes the closest descendant dominator of v; for each v 2 I, q (v) = v. Then, dist G (v, q (v)) k. q(v) Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 37 / 38
57 Sparsity v u (H) 2 p 3πk 2 + 3πk π/ p 3. For each v 2 C, q (v) denotes the closest descendant dominator of v; for each v 2 I, q (v) = v. Then, dist G (v, q (v)) k. Consider a node u 2 I [ C. For each v 2 N H (u), q(v) dist G (u, q (v)) dist G (u, v) + dist G (v, q (v)) k + 1. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 37 / 38
58 Sparsity v q(v) u Let S 1 (u) (resp. S 2 (u), S 3 (u)) be the set of dominators which are at most k 1 (resp. k, k + 1) hops away from u. Then, jn H (u)j 3 js 1 (u)j + 2 js 2 (u) n S 1 (u)j + js 3 (u) n S 2 (u)j = js 1 (u)j + js 2 (u)j + js 3 (u)j 2π p3 i 2 + πi + 1 k+1 i=k 1 = 2 p 3πk 2 + 3πk π p 3. Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 38 / 38
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