1. Robustness of wireless ad hoc network topologies
Motivation and Definitions
Model and previous work
NP-hardness
Approximation algorithm
Conclusions

2. Motivation and Definitions
Wireless networks are characterized by several constraints, such as bandwidth, latency, power and mobility.
Graphs are frequently used to model network communication, providing a context for studying these types of problems.
One of standard approaches is the use of special structures in graphs that provides various criteria to solve network problems.
This structure should be robust, i.e. resistible to faults.

3. Motivation and Definitions
The main objective of the research is:
“Given some network topology, we aim to identify a robust structure inside of the network”
In our case the robust structure will be diameter-bounded tree

4. Ad-Hoc Networks
A mobile ad hoc network (MANET) consists of mobile computing devices with radio transmission and reception capabilities
MANET topology is dynamic in nature due to node movements, variations in the radio propagation conditions, and battery power constraints
MANET are established in order to solve specific problems

5. Ad-Hoc Networks
Although MANET has been mainly used in military applications, they are being increasingly used for civilian applications as well: virtual class rooms, wireless local area networks, emergency and law enforcement

6. Broadcast and Multicast in a MANET
Simultaneous broadcast or multicast communication require a number of sources or some core subgraph.
Such a sources/core subgraph could be:
The median nodes for minimal transport
The centers node for minimal delay
Robust connected backbone, such as bounded-hop subtree.

7. Model
Network is modeled by complete graph G(V,E,W), where each edge ei has weight wi.
The weight is proportional to the robustness of the edge.
Given an integer h, we aim to find maximal weight spanning subtree of G with hop-diameter of at most R.

8. Previous work Core works Gupta&Srimani ‘03: use median
Bing-Hong et al.’06: use subtree with minimal number of leaves
Becker et al.’07: use paths under sum criterion

9. Previous work Spanning tree as the backbone works:
Konemann et al.’04: min-hop spanning tree of bounded diameter B: O(√logB n)-approximation.
Bar-Ilan et al.’01: O(log n) – approximation for bounded-diameter minimum spanning tree, R=4,5
Kortsarz&Peleg’97: the same for larger values of R
Alfandari&Paschos’99: 5/4-approximation for h=2 and costs= 1,2.
Hassin&Levin’03: O(1)-approximation for bounded hop paths between pairs of nodes.

10. NP-hardness Reduction to the Exact Cover by 3-Sets
Given a set X with |X| = 3q, and a collection Y of 3-element subsets of X. Does Y contain an exact cover for X, i.e., a sub-collection Y’ of Y such that every element of X occurs in exactly one member of Y’?
Known as NP-hard!

11. NP-hardness 2 1 1 1 1 1 1 2 1 2

12. NP-hardness
R=4, weight of black edge 0, weight of all edges 1 except edges connecting Y subsets (2).
Not possible to reach a node of X from a node of Y that is not directly connected to C.
There is an exact cover for X by Y if and only if there is a spanning tree of weight 2(|Y|-|X|/3)+|X|/3+2 and hop-diameter 4.

13. The algorithm Find Maximum Spanning Tree Kruskal solution.
Check the hop-diameter by BFS. If less or equal than R, we are done.
Otherwise, start to shortcut.

14. Complete Graph (small weights are not shown)
Maximum Spanning Tree
Complete Graph (small weights are not shown)
0.25 1
0.5
0.33 A B C D E F G H I J
0.25 B C
0.25
0.25 1 A E
0.25 F 1
0.5 D
0.33
0.1
0.2 G
0.2
0.15
0.33
0.25 I H
0.33
0.5 J

15. The hop-diameter of the given tree is 8. But we need less.
The algorithm
The hop-diameter of the given tree is 8. But we need less.
0.25 1
0.5
0.33 A B C D E F G H I J
Choose E (arbitrary) as the root and tag the nodes according to their distance from E

16. The algorithm Do R/2 stages: At stage j, 0 <=j <=R/2-1
remove from the tree T
all the edges being tagged
i*R/2 + j, for 0<=i<=2(|V|-j)/R
and connect the nodes tagged i *R/2 + j + 1
directly to the root . 3 2 1 4

17. Example, R=4, |V|=10 Stage 0: Remove edges tagged 0, 2
Connect the nodes tagged
1,3 directly to the root . 3 2 1 4

18. Example, R=4, |V|=10 Stage 1: Remove edges tagged 1, 3
Connect the nodes tagged
2,4 directly to the root . 3 2 1 4

19. The algorithm
From the obtained trees choose the tree having maximal weight. Call it T’. It has diameter at most R. 3 2 1 4 3 2 1 4

20. The analysis What about the weight of T’?
T’ has the maximal weight from R/2 obtained trees (and, therefore, is better than the average)
Every edge of T has been replaced no more than once. Thus,

21. The analysis Combining things together:
Or in other words: T’ provides 1-2/R approximation for optimal solution.
The total running time is O(|V|log|V|).

22. Conclusions
The first provable approximation factor algorithm for maximum spanning tree of bounded diameter is given. The dependency is on given hop-diameter.
Open question: choices of different root may lead to better solutions?
Open question: better approximation factor is possible?
Open question: what happens with triangle inequality?