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Exploring Yao Graphs: Connectivity, Spanners, and Dominance in Radio Networks

This document examines properties of Yao graphs, particularly focusing on the Symmetric Yao Graph's connectivity for over six sectors and its implications in graph theory. The exercises explore the relationship between spanners, weak spanners, and power spanners, outlining proofs through induction and counterexamples. The construction of the HL-Graph demonstrates node dominance and layer priorities. Through detailed diagrams and examples, the paper illustrates the behavior of nodes in different layers and the edges established within those layers, aiming for a better understanding of network topology in radio communication.

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Exploring Yao Graphs: Connectivity, Spanners, and Dominance in Radio Networks

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  1. Algorithms for Radio NetworksExercise 6 Stefan Rührup sr@upb.de

  2. The Yao-Family Yao-Graph nearest neighborin each sector Spanner ⊇ SparsY Sparsified Yao-Graph use only the shortestingoing edges weak- & power-Spanner, constant in-degree ⊇SymmY Symmetric Yao-Graph only symmetric edges not a spanner, nor weak spanner, nor power-spanner

  3. Exercise 12 • Prove that the Symmetric Yao Graph is connected for k>6 sectors! SymmY is not a Spanner: vm um only symmetric edges v2 u2 v1  u1

  4. u v Exercise 12 • Prove that the Symmetric Yao Graph is connected for k>6 sectors! Proof: Induction over the distance of vertices • Edge connecting the closest pair is part of SymmY • Consider two nodes u,v: • Either u and v are connected orthere is a path from u to w and a path from w to v Case 1: Case 2:  w  V: |u-w| < |u-v| k>6  |v-w| < |u-v| w u v

  5. Exercise 13 • Which implications are true (for appropriate constants c,c’,c’’ and d  2)? • Spanner  weak c’-Spanner • weak c’-Spanner  c-Spanner • weak c’-Spanner  (c’’,d)-Power Spanner • (c’’,d)-Power Spanner  weak c’-Spanner • c-Spanner  (c’’,d)-Power Spanner • (c’’,d)-Power Spanner  c-Spanner If an implication is not true, then give a counterexample!

  6. v3 v2 v4 1/2 1/3 1 v1 1 vn Exercise 13 Weak Spanner, but not a Spanner: Spanner X Koch Curve Weak Spanner X Power Spanner, but not a Weak Spanner: Power Spanner Circular node chain with |vi - vi+1| = 1/i

  7. Construction of the HL Graph • Every node on layer i is dominated by some node in layer i+1 • No nodes may dominate each other (the priority decides) • Edges are inserted in the publication radius of each node L1 publication radius L1 domination radius L1 node L0 node L1 edge

  8. Radii and Edges of the HL Graph • Li-1 publication radius ≥ Li domination radius:  ≥  > 1 • Layer-i edges are established in between r1 layer-1 domination radius L1 node L0/L1 edge r0 layer-0 domination radius L0 node L0 edge L1 edge L1 node · r0 L1 node

  9. Exercise 14 • Draw the HL Graph of v1,...,v6 for ==2! v1 v2 v6 1+ 2+2 v5 v3 v4

  10. Exercise 14 layer-0-edges of v2 v1 v6 v2 L0-domination v5 v3 v4 L0-publication/L1-domination

  11. Exercise 14 No other node with higher rank within the domination radius: 6 v1 v6 v2 3 4 L0-domination v5 v3 2 1 v4 5 L0-publication/L1-domination

  12. Exercise 14 node v1 has highest priority and becomes layer-1 node 6 v1 L0-domination v6 v2 3 4 L0-publication/L1-domination v5 v3 2 1 v4 5

  13. Exercise 14 V(L0) = {v1,...,v6} V(L1) = {v1,v4} V(L2) = {v1} 6 v1 layer-0 edges layer-1 edges v6 v2 3 4 v5 v3 2 1 v4 5

  14. Exercise 14 The same node set with other priorities: v5 is dominated by v3, so it will remain on layer 0 and v6 can become layer-1 node. 1 v1 V(L0) = {v1,...,v6} V(L1) = {v3,v6} V(L2) = {v3} v6 v2 4 3 layer-0 edges layer-1 edges v5 v3 5 6 v4 2

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