Online Multi-Path Routing in a Maze

1
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
Position based Routing

• Target geographic position instead of network
  address
• Idea Iteratively choose neighbor closest to the
  target(Greedy-Strategie)

• Advantages
• local decisisions
• no routing tables
• scalable

(4,2)
13,5
(2,5)
s
t
(13,5)
(5,7)
(0,8)
(3,9)
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Position based Routing

• Prerequisits
• All nodes know their positions (e.g. GPS)
• Position of all neighbors are known (Beacon
  Messages)
• Target position is known (Location Service)

(4,2)
13,5
(2,5)
s
t
(13,5)
(5,7)
(0,8)
(3,9)
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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

NFP
t
s
MFR
Transmission radius
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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)

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Position based Routing

• 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

Advance circle
Right hand rule
s
t
?
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Lower bounds und alternatives

• Lower bound for position based routing Kuhn et
  al. 2002

• Alternative strategyy flooding
• Time O(d)
• Traffic O(d2)
• Position based single-path routingstrategies
• Time and traffic O(d2)
• Is Flooding more efficient?
• Worst case analysis not useful

s
t
Time ?(d2)
Time Hops, Traffic Messages
d length of shortest path (distance)
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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

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Finding Cells for a Unit-Disk Graph

• Cell size 1/3
• transmission-distance1
• 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

x
u
v
w
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Lower bounds and comparative analysiss

• Lower bound for Online Navigation Lumelsky,
  Stepanov 1987

• ?(d p) ? lower bound for traffic (online)
• Instead of worst-case-analysisCompare 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
s
t
p
Path length ?(d p)
d length of shortest path p Perimeter of the
barriers
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The Network Model

• Grid network with faulty nodes
• Faulty blocks barriers
• Barriers are unkown (a priori),decisions need to
  be madeonline
• Comparative analysis
• Competitive time-ratio
• Comparatives traffic-ratio

Perimeter
Start
? Time ( Hops)
Target
? Messages
Barrier
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Single-Path versus Flooding
No Barriers (pltd)
Maze (pd2)
A
B
Start
Start
d length ofthe shortest path
Target
Target
Is there a strategy, as fast as flooding and with
as low traffic as single-path ... for all
scenarios ?
Perimeter
A
B
Time O(d p) ? Rt O(d)
Single-Path (sequential)
Traffic O(d)
Traffic O(d2) ? RTr O(d)
Flooding (parallel)
Time O(d)
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Lucas AlgorithmLucas 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

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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(d2)

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Continuous Ring Search

• Modification of Expanding Ring Search
• Source starts flooding
• but with a delay of s 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(d2)

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The JITE Algorithmus

• Message efficient parallel BFS (breadth first
  search)
• using Continuous Ring Search
• Just-In-Time Exploration (JITE) and Construktion
  of search path insteadflooding
• Search paths surround barriers
• Slow Searchslow BFS on a sparse grid
• Fast ExplorationConstruction of the sparsegrid
  near to the shoreline

Start
Barrier
Target
Shoreline
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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
E
E
?
?
E
E
?
?
E
E
?
?
?
E
?
E
?
?
E
?
E
?
Shoreline
?
E
?
E
?
?
E
E
E
E
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Fast Exploration (1)

• Frame traversal(Right hand rule)
• Time limit If the traversal takes too long then
  the fram is divided into smaller frames

• Construction of a path network for the BFS
• Partition into Frames
• Frame borders provide an approximation of the
  shortest path tree

?
entrypoint
Detour
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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

allowed detour g/?(t)
g
E
E
E
E
?
E
?
E
E
?
E
E
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Frame Exploration

• A frame can be explored in parallel from
  different sides (entry points)
• All messages stop after at most 2gg/?(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
• Wake up
• Tell all frame border nodes about the exploration
  in progress
• Find a coordinator
• Count
• Coordinator sends counting messages
• Stop
• Frame has been explored (in time)
• Close
• Stop exploration within frame
• Notify
• Shoreline enters frame Start exploration in
  neighbor frames

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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 log2 d)
• O(p log d) is the size of the path network/frame
  network
• further logarithmic factor for allowed detours

Time Traffic maxRt, RTr
Greedy (Single-Path) O(dp) O(dp) O(d)
Flooding O(d) O(d2) O(d)
JITE O(d) O(d p log2 d) O(log2 d)
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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

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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