Lokale Netzstrukturen Exercise 5. Juli 19, 2017

Transcription

1 Lokale Netzstrukturen Exercise 5 Juli 19, 2017

2 Ex 1 a) Definition The undirected degree 8 Yao graph over a node set V R 2, denoted YK 8 (V ), is defined as follows. For any node v V partition the plane into 8 equal cones centered at v and s.t. the boundaries of two cones are parallel to the x-axis (see Fig. 1). For each such cone which contains at least one other node x v, add the undirected edge vu to YK 8 (V ), where u is the nearest neighbor of v in that cone w.r.t. Euclidean distance (ties can be broken arbitrarily). Figure: Illustration of YK 8

3 Ex 1 a) (cont d) Ex Prove that EMST(V ) YK 8 (V ). Preliminary observation: If q and q are two nodes in C l (p), then qq < max { pq, pq }. Consider pqq : Since qpq = π/4 < π/3, its opposite side qq cannot be longest side of the triangle.

4 Ex 1 a) (cont d) Ex Prove that EMST(V ) YK 8 (V ). Proof: Let EMST be the set of edges forming EMST (V ) and YK the set of edges in YK 8 (V ). Show: Any edge vw EMST not in YK can be replaced by an edge in YK s.t. MST property is maintained This proves the claim since all edges from EMST can be replaced by edges from YK Let vw EMST \ YK and assume w C l (v).

5 Ex 1 a) (cont d) Proof (cont d): N l (v) There is a nearest neighbor u N l (v), s.t. uv YK. It hols that u w and vu vw, because vw EMST \ YK. Delete vw from EMST and obtain two subtrees one containing v and the other containing w Note that u and w must be in the same component. If not, then uw would be a shorter connecting edge for the two subtrees than vw (by preliminary observation), contradicting the fact that EMST is a MST. Hence, u and w are in same subtree and adding edge vu to EMST \ {vw} results in a spanning tree with total weight no greater than that of EMST.

6 Ex 1 b) Definition (Minimal Dominating Set) A dominating set for a graph G = (V, E) is a subset D of V, such that every vertex not in D is adjacent to at least one member in D. An Independent Set is minimal if no vertex can be removed from D without violating the independent set property. Definition (Maximal Independent Set) A Independent Set for a graph G = (V, E) is a subset I of V such that for every two vertices in I there is no edge in E connecting them. A Maximal Independent Set is an Independent Set that is not a subset of any other Independent Set. Ex Prove that any Maximal Independent Set for a given graph G = (V, E) is also a Minimal Dominating Set for G.

7 Ex 1 c) Ex Proof that in an Independent set defined over a given Unit Disk Graph UDG(V ), any node contains at most 5 Independent Set nodes in its Unit Disk.

8 Ex 1 c) (cont d) Ex Proof that in an Independent set defined over a given Unit Disk Graph UDG(V ), any node contains at most 5 Independent Set nodes in its Unit Disk. Proof: Consider x, u, v, s.t. u, v N 1 (x) and u, v are I.S. nodes Must hold that uxv > π/3, otherwise u, v would share an edge in UDG(V ) (Law of Cosines) For k > π/3, 2π/k < 6 holds Hence, there are at most 5 I.S. nodes in a node s neighborhood

9 Greedy Forwarding

10 Ex 2 a) Ex Prove that progress- and distance-based Greedy Forwarding algorithms which only use edges in forwarding direction, cannot guarantee packet delivery.

11 Ex 2 a) Ex Prove that progress- and distance-based Greedy Forwarding algorithms which only use edges in forwarding direction, cannot guarantee packet delivery.

12 Ex 2 b) Ex Explain the differences of the forwarding areas Sector, Releaux triangle, and Circle in Beaconless routing. Why is the forwarding area limited to one of these shapes, instead of using the complete positive progress area? Give an example that explains the problem.

13 Ex 3 Prepare an example that illustrates the different routing paths obtained when executing the algorithms Greedy-Face-Greedy, GOAFR, FACE using before crossing, and FACE using after crossing. The example should be designed such that the routing paths differ in at least a single edge.

14 Ex 3 (cont d) Greedy-Face-Greedy

15 Ex 3 (cont d) Greedy-Face-Greedy

16 Ex 3 (cont d) Greedy Other Adaptive Face Routing (GOAFR)

17 Ex 3 (cont d) Greedy Other Adaptive Face Routing (GOAFR)

18 Ex 3 (cont d) Face using before crossing

19 Ex 3 (cont d) Face using before crossing

20 Ex 3 (cont d) Face using after crossing

21 Ex 3 (cont d) Face using after crossing