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Ad-Hoc Networks Beyond Unit Disk Graphs. Fabian Kuhn Roger Wattenhofer Aaron Zollinger. Overview. Introduction 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 - PowerPoint PPT Presentation
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Ad-Hoc Networks Beyond Unit Disk Graphs
Fabian KuhnRoger Wattenhofer
Aaron Zollinger
DIALM-POMC 2003 2
Overview
• Introduction– 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 realityBUT: 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• No edge between u and v if |u-v|>1• 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 RaviDial-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– 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
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
DIALM-POMC 2003 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– 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
DIALM-POMC 2003 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– 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
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