Geographical Routing

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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
• maintain (end-to-end) routing states for each
  destination
• thus overhead proportional to network size

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GeoRouting Greedy Distance
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• Find neighbors who are the closest to destination
  D
• To make progress, the chosen neighbor should be
  closer to destination

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Geographical Routing

• Each node only needs to keep state for its
  neighbors
• Beaconing mechanism
• each node broadcasts its MAC and position
• to minimize costs piggybacking

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Greedy Routing Not Always Works
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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

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

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

w
x
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Removing Cross Edges
z
v
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u
w

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

x
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How to Make a Graph Planar?

• Convert a connectivity graph to planar
  non-crossing graph by removing bad edges
• make sure the original graph will not be
  disconnected
• 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
• radio range of 250 m
• Bonus remove redundant, competing path ? less
  collision

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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
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e
e
c
c
a
a
f
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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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Geographic Routing
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Pros and Cons of GPSR

• Pros
• low routing state and control traffic
• scalable
• handles mobility well
• Cons
• planarized graph is hard to guarantee under
  mobility
• location might not be available everywhere (we
  will see the problem next week)
• geographic distance does not correlate well with
  network proximity

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Why Geographic Routing Without Location?

• Location is hard to get
• GPS takes power, doesnt work indoors, difficult
  to incorporate in small sensors
• the network localization problem is difficult
• True location may not be useful if there are
  obstacles

In the connectivity space, B is closer to
destination!!
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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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Perimeter Nodes Are Known (True Positions)
3200 nodes 64 perimeter nodes on the boundary
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Perimeter Nodes Are Known (10 iterations)
Internal nodes initialized as the center of the
square
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Perimeter Nodes Are Known (100 iterations)
Internal nodes initialized as the center of the
square
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Perimeter Nodes Are Known (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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Backup Slides
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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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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?