1. Ad-Hoc Networks Beyond Unit Disk Graphs
Fabian Kuhn
Roger Wattenhofer
Aaron Zollinger

2. Overview Introduction Flooding Greedy + Flooding Geometric Routing for
Graph Models for Mobile Ad-Hoc Networks
Quasi Unit Disk Graphs
Related Work
Volatile Memory Routing
Flooding
Lower Bound for Message Complexity
Topology for Optimal Flooding
Greedy + Flooding
Volatile Memory Routing Algorithms
Combining Greedy and Flooding
Geometric Routing for
How to obtain a planar graph
Optimality of AFR/GOAFR
DIALM-POMC 2003

3. Mobile Ad-Hoc Networks
Mobile Devices communicating via radio
Network without centralized control (base station)
We consider the abstraction level of graphs
DIALM-POMC 2003

4. Graph Models for Ad-Hoc Networks
Simple Models
Unit Disk Graph is most widely applied model:
Underlying assumption: All nodes are in R2, have exactly the same transmission range (normalized to one), and there are no obstacles.
Far from reality BUT: There are numerous theoretical results.
Realistic Models
We need more general graph models
However, arbitrary graphs are too general to obtain strong results for routing, etc.
We need something between UDG and arbitrary graphs:
general enough to model reality as close as possible
restrictive enough to allow useful theoretical results
DIALM-POMC 2003

5. Quasi Unit Disk Graph Definition Unit Disk Graph:
Edge between u and v if |u-v|·1
No edge between u and v if |u-v|>1
Definition Quasi Unit Disk Graph:
Edge between u and v if |u-v|·d
May have an edge if d<|u-v|<1
DIALM-POMC 2003

6. Related Work
The Quasi Unit Disk Graph model is not new: Barrière, Fraigniaud, and Narayanan have shown that correct geometric routing is possible if Dial-M 2001 and Wireless Networks Journal Vol. 3(2) 2003
Other generalizations of the unit disk graph have been proposed, e.g. (r,s)-civilized graphs by Krumke, Marathe, and Ravi Dial-M 1998 and Wireless Networks Journal Vol. 7(6) 2001
DIALM-POMC 2003

7. Volatile Memory Routing
We want to consider routing without routing tables
We need to allow nodes to temporarily store some information
Volatile Memory Routing Algorithm: For each message, each node is allowed to temporarily store O(log n) bits. (temporary = while the message has not reached the destination)
DIALM-POMC 2003

8. Overview Introduction Flooding Greedy + Flooding Geometric Routing for
Graph Models for Mobile Ad-Hoc Networks
Quasi Unit Disk Graphs
Related Work
Volatile Memory Routing
Flooding
Lower Bound for Message Complexity
Topology for Optimal Flooding
Greedy + Flooding
Volatile Memory Routing Algorithms
Combining Greedy and Flooding
Geometric Routing for
How to obtain a planar graph
Optimality of AFR/GOAFR
DIALM-POMC 2003

9. Message Complexity Lower Bound
Lower Bound Graph is a Quasi Unit Disk Graph (parameter d)
To find destination t, all vertical chains (all nodes) have to be visited
Length of one chain: c
Optimal path has length O(c)
There are O(c2/d2) nodes ! O(c2/d2) messages
DIALM-POMC 2003

10. Flooding Flooding on the Quasi UDG gives unbounded message complexity
We need a subgraph on which flooding is efficient (a kind of topology control)
Desired properties:
Nodes form a dominating set
O(A/d2) nodes per area A
O(A/d2) edges per area A
Optional: spanner
DIALM-POMC 2003

11. Topology Control I
Construct a Minimal Independent Set (MIS) ! dominating set and O(A/d2) nodes per area A
If we make all 2- and 3-hop connections, we have a spanner, but too many nodes and edges
Solution: Choose only a subset of the 2- and 3-hop connections (“virtual edges” of length · 3)
DIALM-POMC 2003

12. Topology Control II Each “virtual edge” is completely covered
Place grid over the nodes
(cell size = 6)
Add another grid shifted
by (3,3)
Add another grid shifted
by (3,0)
Add another grid shifted
by (0,3)
Each “virtual edge” is
completely covered
by a cell of at least
one of the grids
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13. Topology Control III
In each cell, we calculate a spanner of the nodes (MIS) and “virtual edges” lying completely inside the cell ! Applying a randomized construction of Linial and Saks (SODA 91) yields a O(log(1/d2))-spanner with O(1/d2) virtual edges.
Combining all local spanners gives a O(log(1/d2))-spanner with O(A/d2) “virtual edges” per area A. ! Backbone Graph
DIALM-POMC 2003

14. Flooding on the Backbone Graph
Flooding/Echo with exponentially growing TTL on the Backbone Graph gives:
O(log(1/d2)c) time and O(c2/d2) message complexity in the synchronous model
O(log(1/d2)log3(c/d)c) time and O(log3(c/d)c2/d2) message complexity in the asynchronous model (using a synchronizer described by Awerbuch and Peleg, FOCS 90)
Geometric Flooding/Echo uses disks with exponentially growing radius instead of TTL:
O(c2/d2) time and message complexity (synchronous and asynchronous)
DIALM-POMC 2003

15. Overview Introduction Flooding Greedy + Flooding Geometric Routing for
Graph Models for Mobile Ad-Hoc Networks
Quasi Unit Disk Graphs
Related Work
Volatile Memory Routing
Flooding
Lower Bound for Message Complexity
Topology for Optimal Flooding
Greedy + Flooding
Geometric (Volatile Memory) Routing Algorithms
Combining Greedy and Flooding
Geometric Routing for
How to obtain a planar graph
Optimality of AFR/GOAFR
DIALM-POMC 2003

16. Geometric Routing
A.k.a. location-based, position-based, geographic, etc.
Each node knows its own position and position of neighbors
Source knows the position of the destination
No routing tables in the nodes, all routing information is in the message!
Volatile Geometric Routing:
Geometric Routing + O(log n) bits per message in each node (while message is on the way from s to t)
DIALM-POMC 2003

17. Geometric Routing
??? t s s
DIALM-POMC 2003

18. Greedy Routing Each node forwards message to “best” neighbor t s
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19. ? Greedy Routing Each node forwards message to “best” neighbor
But greedy routing may fail: message may get stuck in a “dead end”
Needed: Correct geometric routing algorithm t ? s
DIALM-POMC 2003

20. Greedy + Flooding
Straight-forward idea to make greedy routing correct (i.e. always find the destination) ! combine greedy and flooding
We want to keep worst-case optimality
Apply geometric flooding with exponentially increasing radius and the right criterion to fall back from flooding to greedy
DIALM-POMC 2003

21. Greedy + Flooding, Fall Back Criterion
Flooding phase with radii r0, r1, … where ri=r02i
Flooding starts at node u, node vi is best (closest to destination t) node for radius ri
Go back to greedy if |u-t| - |vi-t| ¸ q¢ri (q is a predefined constant)
Message and time complexity: O(c2/d2)
Simulations on UDG suggest that the algorithm is efficient in the average case
DIALM-POMC 2003

22. Overview Introduction Flooding Greedy + Flooding Geometric Routing for
Graph Models for Mobile Ad-Hoc Networks
Quasi Unit Disk Graphs
Related Work
Volatile Memory Routing
Flooding
Lower Bound for Message Complexity
Topology for Optimal Flooding
Greedy + Flooding
Geometric (Volatile Memory) Routing Algorithms
Combining Greedy and Flooding
Geometric Routing for
How to obtain a planar graph
Optimality of AFR/GOAFR
DIALM-POMC 2003

23. ???
Questions? Comments?
DIALM-POMC 2003