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

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

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

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

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

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

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

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

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

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

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

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 DIALM-POMC 2003

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

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

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

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

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

Greedy Routing Each node forwards message to “best” neighbor t s DIALM-POMC 2003

? 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

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

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

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

??? Questions? Comments? DIALM-POMC 2003