1. 1 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Online Multi-Path Routing in a Maze Christian Schindelhauer joint work with Stefan Rührup Workshop of Flexible Network Design Bertinoro, 1.-6.10.2006 to appear at ISAAC 2006

2. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 2  Target: geographic position instead of network address  Idea: Iteratively choose neighbor closest to the target (Greedy-Strategie) Position based Routing (2,5) (13,5) (5,7) (4,2) (3,9) 13,5 (0,8) s t  Advantages: –local decisisions –no routing tables –scalable

3. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 3 Prerequisits: –All nodes know their positions (e.g. GPS) –Position of all neighbors are known (Beacon Messages) –Target position is known (Location Service) Position based Routing (2,5) (13,5) (5,7) (4,2) (3,9) 13,5 (0,8) s t

4. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 4 First Works (1)  Routing in Packet Radio Networks  Greedy-Strategies: –MFR: Most Forwarding within Radius [Takagi, Kleinrock 1984] –NFP: Nearest with Forwarding Progress [Hou, Li 1986] s t MFR NFP Transmission radius

5. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 5 First Works (2)  Cartesian Routing [Finn 1987]  Routing with geographic Coordinates  n-hop Cartesian regular: Every node has a node in its n-hop- neighborhood which is closer to an arbitrary target  Greedy-Routing and Limited Flooding (restricted to n Hops)

6. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 6 X  Problems: Greedy routing may end in local minima  No neighbors closer to the target available  Recovery-strategy necessary (e.g. GPSR [Karp, Kung 2000] )  Example: Position based Routing s t ? Advance circle Right hand rule

7. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 7 Lower bounds und alternatives  Lower bound for position based routing [Kuhn et al. 2002] : s t  Alternative strategyy: flooding –Time: O(d) –Traffic: O(d 2 )  Position based single-path routing strategies –Time and traffic: O(d 2 )  Is Flooding more efficient?  Worst case analysis not useful Time: Ω(d 2 ) Time = #Hops, Traffic = #Messages d = length of shortest path (distance)

8. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 8 Grid networks and Unit- Disk Networks  Online routing in grid network with faulty nodes is equivalent to position based routing in wireless ad-hoc networks  Implicit geographic clustering  Partitioning of the plane into cells, empty regions = barriers  Distributed protocol for construction and routing

9. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 9 Finding Cells for a Unit- Disk Graph  Cell size ≤ 1/3 –transmission-distance=1  Cell is NOT a barrier if –it is inside of a circle around a node with radius 1/2 –if an edge (u,v) with |u,v| ≤ 1 touches this cell  Cell clustering –Gateways (and leader)  Two-hop communication gives a complete local view of the cell network v x w u

10. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 10 Lower bounds and comparative analysiss  Ω(d + p)  lower bound for traffic (online)  Instead of worst-case-analysis: Compare the algorithm with the best online-algorithm for the class of problems  Characterize the class of problems by the perimeter p and the distance d p Path length: Ω(d + p)  Lower bound for Online Navigation [Lumelsky, Stepanov 1987]: d = length of shortest path p = Perimeter of the barriers d = length of shortest path p = Perimeter of the barriers p s t

11. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 11 The Network Model  Grid network with faulty nodes  Faulty blocks = barriers  Barriers are unkown (a priori), decisions need to be made online  Comparative analysis –Competitive time-ratio –Comparatives traffic-ratio Start Target Perimeter Barrier  Time (# Hops)  Messages

12. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 12 Time: O(d + p)  R t = O(d) Time: O(d) Traffic: O(d 2 )  R Tr = O(d) Traffic: O(d) Single-Path versus Flooding Single-Path (sequential) Flooding (parallel) Start Target d = length of the shortest path A A B B No Barriers (p<d)Maze (p=d 2 ) A A B B Target Perimeter O(d) max{R t,R Tr } O(d) Is there a strategy, as fast as flooding and with as low traffic as single-path... for all scenarios ?

13. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 13 Lucas Algorithm [Lucas 88] 1: repeat 2: Follow the straight line connecting source and target. 3: if a barrier is hit then 4: Start a complete right-hand traversal around the barrier and remember all points where the straight line is crossed. 5: Go to the crossing point that is nearest to the target. 6: end if 7: until target is reached Time: d + 3/2 p Traffic: d + 3/2 p

14. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 14 Expanding Ring Search [Johnson, Maltz 96]  Start flooding with restricted search depth  Repeat flooding while doubling the search depth until the destination is reached  Time: O(d)  Traffic: O(d 2 )

15. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 15 Continuous Ring Search  Modification of Expanding Ring Search:  Source starts flooding –but with a delay of σ time steps for each hop  If the target is reached, a notification message is sent back to the source  Then the source starts flooding without slow-down a second time –Second wave is sent out to stop the first wave  Time: O(d)  Traffic: O(d 2 )

16. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 16 The JITE Algorithmus  Message efficient parallel BFS (breadth first search) –using Continuous Ring Search  Just-In-Time Exploration (JITE) and Construktion of search path instead flooding  Search paths surround barriers  Slow Search: slow BFS on a sparse grid  Fast Exploration: Construction of the sparse grid near to the shoreline Target Start Barrier Shoreline

17. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 17 E E E EE E E E E E E E E E E E Slow Search & Fast Exploration  Slow Search visits only explored paths  Fast Exploration is started in the vicinity of the BFS-shoreline  Exploration must be terminated before a frame is reached by the BFS-shoreline Exploration Shoreline

18. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 18 Construction of a path network for the BFS Partition into „Frames“ Frame borders provide an approximation of the shortest path tree Fast Exploration (1) Detour entry point Frame traversal (Right hand rule) Time limit: If the traversal takes too long then the fram is divided into smaller frames

19. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 19 Fast Exploration (2) Problems: Exploration causes traffic  explore only frames in the vicinity of the shoreline Small barriers cause further subdivision (traffic!)  Allow small detours Exploration needs time  Slow down BFS-Shoreline by a constant factor  Size limit for new neighbor frames Multiple entry points  Coordinate exploration g allowed detour: g/  (t) allowed detour: g/  (t) E E E E E E E E E

20. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 20 Frame Exploration  A frame can be explored in parallel from different sides (entry points) –All messages stop after at most 2g+g/  (t) rounds –If a message is stopped then no messages of type 3 or 4 occured after a specific time further subdivision is triggered when the messages of type 3 do not occur in time 1.Wake up: Tell all frame border nodes about the exploration in progress Find a coordinator 2.Count: Coordinator sends counting messages 3.Stop: Frame has been explored (in time) 4.Close: Stop exploration within frame 5.Notify Shoreline enters frame: Start exploration in neighbor frames

21. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 21 Slow Search  Path network/frame network gives a constant factor approximation of the shortest path tree  Constant factor slow down of the BFS-Shoreline  Allowed detours of g/  (t) per g  g-frame. Choose  (t)= log t. For a portion of 1-1/log d of all frames we observe g/  (t) = O(g/log d) (log g = 1..log d)  Target is reached in time O(d) (constant competitive ratio)  Traffic O(d + p log 2 d) –O(p log d) is the size of the path network/frame network –further logarithmic factor for allowed detours TimeTrafficmax{R t, R Tr } Greedy (Single-Path)O(d+p) O(d) FloodingO(d)O(d 2 )O(d) JITEO(d)O(d + p log 2 d)O(log 2 d)

22. University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer Worksho of Flexible Network Design Bertinoro, 1-6.10.2006 Online Multi-Path Routing in a Maze - 22 Summary  New efficient strategy for position based routing  Comparative analysis for time and traffic  Lower bounds, linear trade-off  Single-Path versus Flooding  JITE Algorithm –asymptotical as fast as flooding –small polylogarithmic overhead for traffic  Results applicable for wireless ad-hoc-networks

23. 23 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Thank you Position based Routing Strategies Christian Schindelhauer joint work with Stefan Rührup Workshop of Flexible Network Design Bertinoro, 1.-6.10.2006 to appear at ISAAC 2006