The general goal

1. Performance-Guaranteed Strongly Connected Dominating Sets in Heterogeneous Wireless Sensor Networks Chunyan Liu∗ , Hejiao Huang∗†‡, Hongwei Du∗ and Xiaohua Jia∗ ∗Harbin Institute of Technology Shenzhen Graduate School, China. †Shenzhen Key Laboratory of Internet Information Collaboration, China. ‡Corresponding author:hjhuang@aliyun.com IEEE INFOCOM 2016 Presentation prepared by Rakefet Menda

2. The general goal find a special Strongly Connected Bidirectional Dominating Set within minimum routing cost for each pair of nodes in directed graphs.

3. Efficient routing CDS- connected dominating set To achieve efficient broadcasting and routing, the size of CDS should be kept as small as possible while the length of routing path does not increase a lot through the nodes in the CDS. However, sometimes, CDS with smaller size may result in some terrible long routing paths.

4. Minimum connected dominating set (virtual backbone- VB) The shortest path through VB from 1 to 10 is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} length - 10

5. Minimum connected dominating set (virtual backbone- VB) connected dominating set The shortest path through VB from 1 to 10 is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} length - 10 If some nodes (10) are added to the CDS, the shortest path from 1 to 10 based on the VB is {1, 10} length - 2

6. Minimum connected dominating set (virtual backbone- VB) connected dominating set The shortest path through VB from 1 to 10 is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} length - 10 If some nodes (10) are added to the CDS, the shortest path from 1 to 10 based on the VB is {1, 10} length - 2 Longer routing paths will cause a lower delivery ratio. If the size of the CDS is too large, too many nodes are involved to forward messages.

7. Problem statement Our main goal is to solve the α-MOC-SCBDS problem- (α Minimum rOuting Cost Strongly Connected Bidirectional Dominating Set) We want to find set S of graph G which is a SCBDS with routing cost less than α*Minimum routing cost. • finding minimum connected dominating set is a np hard problem.

8. Some assumptions and definitions • G = (V, E) Direct graph • S ⊆ V • S is a α-MOC-SCBDS if and only if: • Each node v ∈ V has a transmission range rv ∈ [rmin, rmax]

9. Centralized algorithm Nin(v) - the set of nodes from which node v can receive messages Nout(v) - the set of nodes which can receive messages from v N(v) = Nin(v) ∩ Nout(v) N +(v) = N(v) ∪ {v}

10. Centralized algorithm Nin(v) - the set of nodes from which node v can receive messages Nout(v) - the set of nodes which can receive messages from v N(v) = Nin(v) ∩ Nout(v) N +(v) = N(v) ∪ {v} • First stage - find the Bidirectional Dominating Set (BDS) C of G using a greedy method (Algorithm 1): 1: Step1: Select a node v ∈ V with the maximal transmission range (or minimal node ID) join the C: C = C ∪ {v}. 2: Step2: V = V − N +(v) 3: Step3: repeat Step1 and Step2 until V = ∅ 4: Output: C

11. Centralized algorithm Note that, for any dominated node v, if it is dominated by u, then u ∈ N +(v). Therefore, a bidirectional dominating set is constructed Lemma In a directed disk graph G = (V, E), the size of Dominating Set got by Algorithm 1 is upper bounded by (2ρ + 1)^2 opt where ρ = rmax / rmin and opt is the size of the optimal solution of the Minimum Strongly Connected Bidirectional Dominating Set (MSCBDS) problem.

12. Centralized algorithm • Second stage - connect the BDS as a SCBDS (Algorithm 2) : For any pair nodes u, v ∈ C, if d(u, v) ≤ 4, all nodes on the path join the SCBDS S. Then S forms a SCBDS of G. For any nodes u, v ∈ V \ C, dS(u, v) ≤ αd(u, v), when α = 5.

13. Centralized algorithm Lemma 8. Given a directed graph G = (V, E) and a Bidirectional Dominating Set C of G. ∀u, v ∈ V with d(u, v) = 2, find a shortest path p(u, v) through their dominating nodes and let all nodes on the path join the set S. S is a strongly connected dominate set and S is with the upper bound where ρ = rmax /rmin

14. Distributed Algorithm In the distributed environment, each node can only sense its incoming neighbors but outgoing neighbors. For an arbitrary vertex v ∈ V , let N(v) denote the neighbors of v that are within the coverage of v while v is within the coverage of each node in N(v). It is also that: N(v) = {u|dist(u, v) ≤ min{ru, rv}}.

15. Distributed Algorithm First step - Distributed algorithm to construct a BDS (Algorithm 3) : • Each node in the network is assigned a positive number ID and marked white. • Firstly, each node generates a neighbor list through a handshake of broadcasting and replying. • Next, in each iteration, each white node compares its ID with those of white neighbors. The one with the smallest ID among all its white neighbors is marked black and all its neighbors are marked gray. • Until all nodes are marked either black or gray. Then all black nodes form a Dominating Set.

16. Distributed Algorithm Second step - Distributed algorithm to construct SCBDS with guaranteed routing cost (Algorithm 4) : The nodes are connected in order to form a SCBDS. • Each node is located in the state of dominator or dominated and different ID. The state is assigned according to the result of the Algorithm 3. • Similarly with Algorithm 3, each node generates a neighbor list through a handshake of broadcasting and replying.

17. Distributed Algorithm • Next, each node with the state of Dominator broadcast its ID to all its neighbors to find a shortest path which the length is not larger than 4 to those nodes with the state of Dominator. • Let all nodes on the paths join the set S. • Until all node pairs with the state of Dominator have been traversed while the shortest path length between them is no more than 4. S is a SCBDS

18. Distributed Algorithm • Next, each node with the state of Dominator broadcast its ID to all its neighbors to find a shortest path which the length is not larger than 4 to those nodes with the state of Dominator. • Let all nodes on the paths join the set S. • Until all node pairs with the state of Dominator have been traversed while the shortest path length between them is no more than 4. S is a SCBDS

19. Distributed Algorithm The Algorithm 3 and 4 are the distributed version of Algorithm 1 and 2. Therefore, S constructed in Algorithm 4 forms a SCBDS, and the distributed algorithms have the same performance ratio with the centralized algorithms

20. Distributed Algorithm The Algorithm 3 and 4 are the distributed version of Algorithm 1 and 2. Therefore, S constructed in Algorithm 4 forms a SCBDS, and the distributed algorithms have the same performance ratio with the centralized algorithms. Lemma The distributed algorithms α-MOC-SCBDS-D produces a SCBDS with the upper bound where ρ = rmax /rmin

21. Simulation • In the simulation, we model a wireless sensor network as a set of nodes randomly deploy in a 100×100 Euclidean plane. The number of nodes varies among 100, 150, 200, 250, 300, 350 and 400. • Each node has a fixed transmission range in the range of [rmin, rmax]. rmin changes among 15, 20, 25 and 30. rmax=ρ*rmin where ρ=rmax/rmin varied among 1.25, 1.5, 1.75 and 2.

22. Simulation Compared results: • CDS-BD-D is a approximate algorithm that generates a CDS whose size and average path length are bounded. The author improve CDS-BD-D as SCBDS-BD-D using forward tree and backward tree to work on directed graphs.

23. Simulation Compared results: • CDS-BFS, algorithm under unidirectional graph. SCDS size is bounded, but dominated nodes may not communicate with its dominating nodes. Improve CDS-BFS as SCBDS-BFS through constructing bidirectional dominating set to work on directed graphs.

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28. Questions?