Geographic Routing

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Geographic Routing

• Y. Richard Yang
• 4/7/2009

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Admin.

• Homework 4
• Exam
• Highest 75
• Median 63
• Average 59

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NP-Hardness of Location Computation
Location computation of globally rigid networks
is NP-hard
Proof sketch Assume we have algorithm X that
takes as input a realizable globally rigid
network and outputs its unique positions. We
will find the set-partition of the partitionable
set S 1) scale so that the sum of elements in S
is less than p/2 2) construct wheel network with
di,i12sin(si/2) 3) call X to localize.
sin(s1/2)
1
sin(s1/2)
2
3
s1s4s2s3
s4
s2
rights lefts
s1
4
s3
0
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The Localization Process
Applications Using Location
Location Computation
Localizability
Measurements
5
Geographical Routing
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Motivations

• A need for geographical based distribution
• e.g., sensornets and location-based services
• Reducing state overhead
• the routing protocols we discussed so far may not
  scale to large-scale networks
• routing states proportional to network size
• objective of georouting each node keeps state
  (positions) only of its direct neighbors

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Greedy Geographical Routing

• Given destination D, find neighbor who is the
  closest to D

S
D
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Problem of Greedy Georouting

• May stuck with local minimum
• To make progress (avoiding local min), the chosen
  neighbor should be closer to destination

D
A
B
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Greedy Routing Works Well in Dense Networks
D

• If node density is high, it is a low probability
  event for the region to have no nodes

10
Dealing with Void
Right-hand rule When arriving at node a from
node b, the next edge traversed is the next one
sequentially counterclockwise about a from edge
(a,b)
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Example Right Hand Rule on Convex Subdivision
Convex subdivision a planar graph where every
internal face is a polytope Applying the
right-hand rule first remove the edges crossing
the line from source to destination, and then
apply the right hand rule.
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Right-Hand Rule Does Not Work Well with Cross
Edges
z
v
D
u

• s originates a packet to u
• Right-hand rule results in the long detour
  s-u-z-w-u-s

w
s
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Removing Cross Edges
z
v
D
u
w

• Remove (w,z) from the graph
• Right-hand rule results in the route s-u-z-v-D

s
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Removing Cross-Edges -gt Planar Graph

• Convert a connectivity graph to planar
  non-crossing graph by removing bad edges
• Requirements
• make sure the original graph will not be
  disconnected
• local, distributed algorithm

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Two Types of Planar Graphs

• Relative Neighborhood Graph (RNG)
• Gabriel Graph (GG)

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Relative Neighborhood Graph

• Edge uv can exist only if there does not exist
  another node w inside the intersection of the
  two circles centered at u and v with radius d(u,
  v)
• i.e., no w such that d(w, u) lt d(u, v) and d(w,
  v) lt d(u, v)
• Or equivalently ?w ? u, v d(u,v)
  maxd(u,w),d(v,w)

not empty ? remove uv
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Gabriel Graph

• An edge (u,v) exists between vertices u and v if
  no other vertex w is present within or on the
  circle whose diameter is uv.
• ?w ? u, v d2(u,v) lt d2(u,w) d2(v,w)

Not empty ? remove uv
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Examples
Full graph
GG subset
RNG subset

• 200 nodes
• randomly placed on a 2000 x 2000 meter region

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Properties of GG and RNG
RNG

• RNG is a sub-graph of GG
• because RNG removes more edges
• GG is a planar graph
• and thus RNG is also planar
• Connectivity
• if the original graph isconnected, RNG is also
  connected

GG
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Connectedness of RNG Graph

• Key observation
• Any edge on the minimumspanning tree of the
  originalgraph is not removed
• Assume (u,v) is such an edge but removed in RNG
  due to w

u
v
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Delaunay Triangulation

• Let disk(u,v,w) be a disk defined by the three
  points u,v,w
• The Delaunay Triangulation (Graph)
• There is a triangle of edges between three nodes
  u,v,w iff the disk(u,v,w) contains no other
  points
• Properties of the Delaunay Triangulation graph
• it is the dual of the Voronoi diagram
• DT graph is planar

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Some Interesting Properties

• Since the MST(V) is connected and the DT(V) is
  planar, all the planar graphs above are connected
  and planar

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Final Algorithm Greedy Perimeter Stateless
Routing (GPSR)

• Maintenance
• all nodes maintain a single-hop neighbor table
• Use RNG or GG to make the graph planar
• Routing
• use greedy forwarding whenever possible
• resort to perimeter routing when greedy
  forwarding fails and record current location Lc
• resume greedy forwarding when we are closer to
  destination than Lc

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Example
d
d
D
D
e
e
c
c
a
a
f
S
S
b
For details about GPSR algorithm, please GPSR.
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Evaluations

• 50, 112, and 200 nodes with 802.11 WaveLAN radios
• Maximum velocity of 20 m/s
• 30 CBR traffic flows, originated by 22 sending
  nodes
• Each CBR flows at 2 Kbps, and uses 64-byte
  packets

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Packet Delivery Success Rate
Very dense network 20 neighbors
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Routing Protocol Overhead
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Routing State

• State per router for 200-node
• GPSR node stores state for 26 nodes on average in
  pause time-0

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Path Length
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Worst Case of GeoRouting in Terms of Hop Count

• A worst case scenario
• destination is central node
• source is any node on ring
• any spine can go to middle
• O(c) nodes along ring and O(c) nodes along each
  spine
• Best path length O(c)O(c)
• Geographic routing
• Test O(c) spines of length O(c)
• Cost O(c2) instead of O(c)

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Pros of GPSR

• Provides geographical routing at low routing
  overhead

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Cons of GPSR

• Localization is challenging
• True location may be less effective if there are
  obstacles

In the connectivity space, B is closer to
destination!!
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Geographical Routing without Location
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Clarification

• Two interpretations of the term geographical
  routing
• Routing to geographical locations
• Using the technique of georouting for scalable
  routing (assign each node a coordinate)
• What we discuss is mostly the second usage

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Overview

• Objective assign virtual coordinates to nodes
  so that
• the coordinates are computed efficiently,
• routing works well using the computed coordinates
• Progress in three steps
• the perimeter nodes and their locations are known
• the perimeter nodes are known but not their
  coordinates are not known
• nothing is known
• We cover the first two steps

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The Perimeter Nodes and Their Locations Are Known

• Image a rubber band from each node to each
  (connected) neighbor
• The force of a rubber band is proportional to
  its length, directedto the neighbor

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The Perimeter Nodes and Their Locations Are Known

• The equilibrium is achievedwhen the position p
  of a node is equal to the average of its
  neighbors
• where n is number of neighbors
• Algorithm
• each node sends its position to its neighbors
• a node updates its new position to be the average
  of those of its neighbors

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True Node Positions
3200 nodes 64 perimeter nodes on the boundary
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Virtual Node Positions (10 iterations)
Internal nodes initialized as the center of the
square
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Virtual Node Positions (100 iterations)
Internal nodes initialized as the center of the
square
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Virtual Node Positions (1000 iterations)
Internal nodes initialized as the center of the
square
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Routing Performance

• 32000 packets with random source-destination pairs

Success rate using (distance) greedy routing
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Two More Scenarios
Success rate 0.981 Avg. path length 17.3
Success rate 0.99 Avg. path length 17.1
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Weird Shapes
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The Perimeter Nodes Are Known But Their Locations
Are Not Known

• Assume the distance between two nodes is the
  (minimum) number of hops to go from one to the
  other
• Distances can be derived by flooding the network
• each perimeter node sends a HELLO message with a
  hop counter of 0
• when seeing a message from a perimeter node with
  a lower hop counter, a node increases the counter
  by 1 and forwards it

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Virtual Coordinates by Triangulation

• Each perimeter node solves the minimization
  problem
• Detail

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Convergence and Performance
One iteration success rate 0.992 avg. path
length 17.2 Ten iterations success rate
0.994 avg. path length 17.2
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Improving Resiliency

• If perimeter vector is inaccurate, it results in
  inconsistent information
• Augment with two beaconing nodes which provide
  two coordinate axes
• special perimeter nodes or run leader election to
  select the two nodes
• Origin is chosen as the center of gravity of all
  perimeter nodes
• more robust to lost information

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Selecting Perimeter Nodes

• Rule A node is a perimeter node if
• It is farthest away from the first beacon node
  among all its one-hop neighbors

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Perimeter Node Detection
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Convergence and Performance
Ten iterations success rate 0.996 avg. path
length 17.3
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Summary

• Localization is a fundamental primitive as
  communications
• Locations can be real or virtual
• We saw one virtual coordinate system to help
  with routing
• Multiple other virtual coordinate systems such
  Vivaldi as an Internet coordinate system

Applications Using Location
Location Computation
Localizability
Measurements
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Backup Slides
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Projecting on Circle After First Computation

• Circle center is center of gravity radius is
  average of distance of the perimeter nodes to the
  CG
• Motivation maintain a consistent coordinate
  space

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Virtual Polar Coordinate Space

• Each node is assigned a label
• label is number of hops to root and virtual angle
  range

Discussion how to do routing?
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Build the Tree

• Root node
• starts as root
• broadcasts level 0
• A non-root node
• receives message level n marks parent
• broadcasts message saying level n1
• All subtrees report size back to parent.
• Root does assignment of virtual angles.

Question what about the issues of the described
algorithm?