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Taxonomy of Unit Disk Graphs: Analyzing Power Stretch Factors in Proximity Graphs

This lecture explores the taxonomy of unit disk graphs (UDG) and their applications in network optimization. It discusses minimum energy requirements for unicasting between nodes, the definition and properties of proximity graphs like the Gabriel graph, Relative Neighborhood Graph, and Yao graph, and their respective power stretch factors. Theorems regarding edge relationships in these graphs highlight their performance in communication networks. Understanding these concepts is crucial for optimizing wireless networks and ensuring efficient data transmission.

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Taxonomy of Unit Disk Graphs: Analyzing Power Stretch Factors in Proximity Graphs

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  1. Network Optimization Lectures 15,16 - 1

  2. Taxonomy of TC

  3. 1 1 1 1 Unit Disk Graph • Consisted of a set V of n nodes • Distributed in a two-dimensional plane • Equipped an omnidirectional antenna with the same Pmax • Unit disk graph, UDG(V) V

  4. 0.8 0.8 0.4 0.4 0.5 0.5 UDG( ) G’( ) 0.6 Minimum energy for unicasting a to b: 0.8 b to c: 1.0a to c: 0.5 b to d: 0.4a to d: 1.1 c to d: 0.6 Minimum energy for unicastinga to b: 0.8 b to c: 1.3a to c: 0.5 b to d: 0.4a to d: 1.2 c to d: 1.7 0.8 / 0.8 = 1 1.3 / 1.0 = 1.30.5 / 0.5 = 1 0.4 / 0.4 = 11.2 / 1.1 = 1.09 1.7 / 0.6 = 2.83 G’(V)(UDG(V)) = max {1, 1, 1.09, 1.3, 1, 2.83} = 2.83 Power Stretch Factor (1/2) b b a a V V d d c c

  5. Power Stretch Factor (2/2) • Definition: • PG(u, v): least energy path of two nodes u and v in a graph G • Given a set V of n nodes, the power stretch factor of a subgraph S(V) with respect to UDG(V): Upper bound of size n

  6. Proximity Graphs • Proximity graph G(V) of UDG(V) • Sparser, i.e. G(V)  UDG(V) • Can be constructed locally, e.g. 1-hop locations • Well-known proximity graphs • Gabriel graph, GG(V) • Relative neighborhood graph, RNG(V) • Yao graph, YG(V)

  7. Relative Neighborhood Graph • Definition: Given a set V of nodes, RNG(V) consists of all edges uv such that ||uv||  1 and there is no wV such that ||uw|| < ||uv||, and ||wv|| < ||uv|| w w u v u v uv  RNG(V) uv  RNG(V)

  8. Power Stretch Factor – RNG(1/2) • Theorem: RNG(n) = n – 1 • It was proved that EMST(V)  RNG(V) • Any path between u and v in EMST(V) • contain at most (n - 1) edges • each edge has length at most ||uv|| • PRNG(u,v)  PEMST(u,v)  (n - 1)||u,v|| • RNG(n)  n – 1 3 3 2 2 3 3 3 2 3 2 5 5 u v u v

  9. Power Stretch Factor – RNG(2/2) • Theorem: RNG(n) = n – 1 (cont.) •  = /3 +  •  = /3 – 2 • As  0, length of each edge  ||v1v2|| • As  0, RNG(v1,v2)/ UDG(v1,v2)  (n – 1) • RNG(n) > n – 1 –  Asymptotic analysis n is even n is odd

  10. Gabriel Graph • Definition: Given a set V of nodes, GG(V) consists of all edges uv such that ||uv||  1 and the open disk using uv as diameter does not contain any wV. w w u v u v uv  GG(V) uv  GG(V)

  11. Power Stretch Factor – GG • Theorem: GG(n) = 1 ( = 2, c = 0) i.e.GG(V)  EG2,0(V) v u r

  12. Yao Graph • Definition: Given a set V of nodes, and an integer parameter k 6, • At each u, any k equal-separated rays originated at u defined k cones. • In each cone, choose the closest nodevto u with distance at most one, if there is any, and add a directed link uv. • Ties are broken arbitrarily • YGk(V) is the undirected graph by ignoring the direction of each link. v v v u u u w w w

  13. Power Stretch Factor – YGk (1/4) • Theorem: For any integer k 6, • Proof. • Let  = 1/(1-(2sin/k)) • Construct a path u~v in YGk(V) • Prove by induction that P(u~v) < ||uv|| on the number of its edges • Initial: if uv  YGk(V), p(u~v) = uv • Assumption: the claim is true for any path with l edges. • Induction: prove it is also true for any path wit l + 1 edges

  14. Power Stretch Factor – YGk (2/4) • Proof. (cont.) • If uv  YGk(V), p(u~v) = uv • Otherwise, exist a node w in the same cone of v, which is a neighbor of u in YGk(V) such that p(u~v) = p(uw)p(w~v)  uwv is not acute  uwv is acute

  15. The claim Power Stretch Factor – YGk (3/4) • Proof. (Case 1:  uwv is not acute) • ||uw||2 + ||wv||2  ||uv||2 • k  6    /3 • ||uw|| < ||uv||  ||uw||/||uv||  1 • ||wv|| < ||uv||  ||wv||/||uv||  1 • So, ||uw|| + ||wv||  ||uv||, for any   2 • P(u~v) = ||uw|| + P(w~v) = ||uw|| + ||wv||  ||uv||

  16. Power Stretch Factor – YGk (4/4) • Proof. (Case 2:  uwv is acute) • We know ||uw||  ||uv|| • Max length of vw is achieved when ||uw|| = ||uv|| • wuv<  = 2/k • ||wv||  2sin(/2)||uv|| = 2sin(/k)||uv|| • P(u~v)  ||uw|| + ||wv|| • P(u~v)  ||uw|| +  (2sin(/k)||uv||) • P(u~v)  ||uv||

  17. Comparison (1/3)  Node degree of RNG is bound by 6 if no two neighbors having the same distance to a node u v Unbound case of RNG

  18. Comparison (2/3) • YGk has bounded out degree, but some nodes may have a very large in-degree. v w v w u u

  19. Comparison (3/3) • Property (relationships) • EMST(V)  RNG(V) • RNG(V)  GG(V) • RNG(V)  YGk(V), for any k  6 • Property (sparseness) • |RNG(V)|  3n – 10 • |GG(V)|  3n – 8 • |YGk(V)|  nk • Average node degrees are all bounded by a constant

  20. Some Variations • GYGk(V): First Yao then Gabriel graph • YGGk(V): First Gabriel then Yao graph • Second phase can further improve the sparseness • Power stretch factors remains the same of YGk(V) • Out in-degrees are still unbounded

  21. Yao and Sink (YG*k) • In each cone of a node u, recursively construct a spanning towards node u

  22. Yao and Sink (YG*k) w x z v u

  23. Yao and Sink (YG*k) w x z v u

  24. Yao and Sink (YG*k) • Node z w x z v u

  25. Yao and Sink (YG*k) w x z v u

  26. Yao and Sink (YG*k) • Replace the directed star consisting of all links towards a node u by a directed tree T(u) as the sink

  27. Yao and Sink (YG*k) YGk YG*k

  28. Ordered Yao Graphs General idea: apply YGk on GG according to some ordering on nodes

  29. OrderYaoGG • Consist of three phases • Construct GG • Compute local ordering of nodes in GG • Construct YGk on GG according to the local ordering

  30. OrderYaoGG • Phase 1: each node self-constructs its neighbors in GG n messages

  31. OrderYaoGG • Phase 2: Each node compute its local ordering in GG Initially, the order is set 0, i.e. unordered.

  32. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  33. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  34. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  35. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  36. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  37. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  38. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  39. OrderYaoGG • Phase 2: Each node compute its local ordering in GG

  40. OrderYaoGG • Phase 3: Construct YGk on GG according to the local ordering

  41. OrderYaoGG • Phase 3: Construct YGk on GG according to the local ordering

  42. OrderYaoGG • Phase 3: Construct YGk on GG according to the local ordering

  43. OrderYaoGG • Phase 3: Construct YGk on GG according to the local ordering

  44. OrderYaoGG

  45. SYaoGG

  46. Comparison of Various Yao Graphs

  47. Yao and Sink (YG*k)

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