CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1
CPSC 689 Discrete Algorithms for Mobile and
Wireless Systems

• Spring 2009
• Prof. Jennifer Welch

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Lecture 19

• Topic
• Location-Based Routing
• Sources
• Ko, Vaidya. Geocasting.
• Ko, Vaidya. Location-aided routing (LAR).
• Kranakis, Singh, Urrutia. Compass routing.
• Karp, Kung. GPSR Greedy perimeter stateless
  routing.
• Bose, Morin, et al. Routing with guaranteed
  delivery.
• Kuhn, Wattenhofer, Zhang, Zollinger. Geometric
  ad hoc routing.
• Barriere, Fraignaud, Narayanan. Robust
  position-based routing.
• MIT 6.885 Fall 2008 slides

3
Location-based Routing

• Deliver message to a particular geographical
  region, in 2D or sometimes 3D (Geocast).
• Use in combination with a location service, to
  implement point-to-point message delivery.
• Algorithms use geographical location information.
• Well study several different strategies
• Constrained flooding
• Compass routing
• Greedy perimeter routing
• Compact routing with Euclidean metrics
• Opportunistic routing
• First Ko, Vaidya Constrained flooding

4
Constrained Flooding

• Two closely-related papers Ko, Vaidya, with
  similar algorithms, for two slightly different
  problems
• Geocast
• Route discovery for a moving device.
• Assumptions
• 2D only.
• Nodes have exact (or almost exact) knowledge of
  their own geographical locations.
• Based on GPS or a localization algorithm.
• Known constraints on motion patterns of mobile
  devices, which make it possible to guess
  approximately where they are.
• Basic idea is to flood a geocast message, or RREQ
  message throughout the network, to try to reach
  the target geographical area.
• However, dont flood everywhere---use
  geographical information to bound the flooding
  area.

5
Geocast with Constrained Flooding

• Could solve the problem using network-wide
  flooding, but inefficient.
• Ko, Vaidya solutions involve flooding,
  constrained by geography.
• Target multicast region neednt be one-hop
  radius, so some flooding may be needed at the end
  anyway.
• Not clear that flooding along the way is
  best---e.g., might try to pass just one copy.
• Flooding may work well if many messages get lost,
  obstacles arise, etc.

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Completeness Metric

• Which they call accuracy.
• Fraction of the nodes that were in the multicast
  region at the time of the sending event, that
  receive the message.

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Forwarding Zones

• Main idea Constrain flooding using forwarding
  zones, which limit where the message should be
  rebroadcast.
• Their algorithms define particular forwarding
  zones.
• Forwarding zones represent locations that are on
  the way from the sender to the multicast region.
• Should include the senders location and the
  entire multicast region.
• Large forwarding zones introduce more flooding
  overhead, but may improve completeness.

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Geocasting Algorithm 1

• Identical to general flooding, except that only
  nodes in the forwarding zone rebroadcast.
• Forwarding zone is the smallest rectangle that
• Includes both the sender and the multicast
  region, and
• Has sides parallel to x and y axes.
• Animation shows only the first message arriving
  at each node.
• Flooding means many nodes receive the message
  several times.

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Geocasting Algorithm 2

• Again, only nodes in the forwarding zone
  retransmit.
• But now forwarding zone is determined
  on-the-fly.
• Forwarding zone includes entire multicast region.
• Along the way
• Sender sends its own coordinates, and the
  coordinates cm of the center of the multicast
  region.
• Everyone who retransmits sends its own
  coordinates, plus cm.
• When node j receives a packet from node i, j
  decides that it is in the forwarding zone exactly
  if its distance to cm is smaller (technically,
  not too much larger) than is distance to cm.
• Thus, actual forwarding decisions are not based
  on predetermined region, but on one-hop
  improvement.
• Overall region of possible forwarding is the
  multicast region, plus the entire circle centered
  at cm, radius dist(sender, cm).

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Geocasting Algorithm 2

• Message keeps getting closer to cm.
• Does not flood through larger rectangle, but may
  flood around the multicast region.

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Geocasting Evaluation

• Simulation results, comparing their algorithms to
  basic, unconstrained flooding.
• Both algorithms achieve completeness comparable
  to basic flooding, but with much less
  communication overhead.
• Dont compare to non-flooding approaches.
• Q Is the flooding approach an efficient way to
  get a message to the multicast region? Is it
  better to send one, or a limited number of
  copies, to the multicast region, then flood
  through the multicast region? Tradeoffs?

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Route Discovery Using Constrained Flooding

• Location-Aided Routing (LAR).
• Now the problem is to establish a route to a node
  with an unknown location, as in DSR or AODV.
• Use constrained flooding methods, flooding RREQ
  messages instead of data messages.
• Q But how to constrain? Now we dont know the
  target location
• A Guess current location of target node based
  on previous location and known limitations on
  motion patterns (speed, direction, etc.)
• Use standard RREQ, RREP on reverse route.
• Establish routes on-demand, like AODV.
• Sender reinitiates route discovery when active
  route breaks.

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Route Discovery

• Multicast region
• An area that sender S thinks is likely to contain
  the location of the target node D.
• S calculates this based on most recent knowledge
  of Ds location, plus knowledge of (maximum or
  average) speed, direction, etc.
• The more info S has, the smaller the multicast
  region can be.
• Requires synchronized clocks---info originating
  from D about its position at some time must be
  correlated with a time on S's clock.
• Forwarding zone
• As before, used to constrain who forwards the
  message a node forwards exactly if it is in the
  forwarding zone.
• Forwarding zone includes S and the multicast
  region, plus enough other area to allow a path
  between them.
• If route discovery fails, S retries with a larger
  forwarding zone---perhaps the entire plane
  (unconstrained flooding).

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The LAR Algorithms

• LAR Algorithm 1
• Forwarding zone is smallest axis-aligned
  rectangle, as before.
• S includes info about the forwarding zone in the
  RREQ message.
• When D sends a RREP, includes its current
  position, which S could use to help in future
  route discoveries.
• Might want to send Ds position info more
  frequently, to reduce size of future multicast
  regions for route discovery for D.
• LAR Algorithm 2
• Try to get closer to Ds last known position
  (take cm last known position, in previous
  Algorithm 2).
• Use some slack---at each hop see if new distance
  is approximately better than previous distance.
• No estimation of Ds current location, so doesnt
  require clock synchronization.

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LAR Evaluation

• Simulations, comparing to unconstrained flooding
  save communication overhead.
• Improvement is most pronounced with high density,
  large transmission ranges.
• In this case, more route discoveries succeed.
• When route discoveries dont succeed, fall back
  on unconstrained flooding anyway.
• For route discovery, some type of flooding seems
  necessary limiting flooding by using any
  available info about the targets position seems
  like a good idea.
• Moral Geographical information can help in
  message routing, even if it isnt accurate and
  up-to-date.

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Compass Routing Kranakis, Singh, Urrutia

• Network model is an (undirected) geometric graph
• Vertices Points in the 2D-plane.
• Edges Straight-line segments between some of
  the points.
• Consider planar graphs only (no edge crossings).
• Assume source node s, destination node t.
• Problem Pass a message from s to t.
• Now consider algorithms that involve passing one
  copy of the message only, no flooding.
• The question is which path to follow.
• Would like an algorithm that is guaranteed to
  deliver the message (perhaps for some subclass of
  all geometric graphs).
• Ideally, using a close-to-shortest path.

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Compass Routing Relevance

• Use in wireless ad hoc networks?
• By superimposing graph model on broadcast model,
  e.g., unit disk model, or more general neighbor
  discovery.
• Suitable for static networks extend to dynamic?

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Local Algorithms

• Require algorithms to be local
• Message includes
• Coordinates of current node and of destination t.
• Extra storage, for a constant number of node
  identifiers cannot store entire network
  topology.
• Each node knows identity and coordinates of its
  neighbors.
• Node information doesn't change during the
  execution of the algorithm.
• Choose next edge based just on info in message
  and info at current node.
• Q Why not let the node information change
  during the algorithm? E.g., node could remember
  visit history, detect when a message is following
  a cycle.
• A To save space---algorithm could be used
  concurrently for many messages, requiring lots of
  storage.

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Impossibility Claim

• Remember, constraints for a local algorithm
• Claim In ordinary planar graphs, without
  geometric coordinates, this problem is not
  solvable in general with a local algorithm (with
  deterministic rules).
• Proof LTTR

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Compass Routing I

• From any node, always forward message over the
  adjacent edge that points most closely in the
  direction of the target t (closest slope).
• Example

s
t
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Compass Routing I
s

• Not every planar geometric graph supports compass
  routing.
• Example

t
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Delaunay Triangulation

• Delaunay triangulation of a set P of n points in
  the plane
• Take the convex hull (smallest convex set)
  containing all the points in P.
• Divide this into a set of triangles whose
  vertices are exactly the points in P.
• So that, for each triangle, the circle passing
  through its vertices has no other point of P in
  its interior.

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Delaunay Triangulation

• Theorem A unique Delaunay triangulation can
  always be constructed, provided that no four
  points are cocircular.
• Example if 4 points are cocircular

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Delaunay Triangulation

• Be careful though not every triangulation will
  be a Delaunay triangulation

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Compass Routing on Delaunay Triangulation Graphs

• Theorem 2.1 A Delaunay triangulation of any set
  P of points supports compass routing.
• Proof Show that, if compass routing traverses
  edge (s,v) then d(v,t) lt d(s,t), strictly
  decreasing.
• Since there are only finitely many vertices, this
  is enough to show the message reaches t.
• Argument is based on simple geometric case
  analysis, e.g., suppose s wants to reach t, can
  travel to either x or y.
• Claim
• If ? lt ? then d(x,t) lt d(s,t).
• If ? ? ? then d(y,t) lt d(s,t).

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Compass Routing, Delaunay

• Draw circle(s, x, y) t is outside.
• Draw center line to t.
• WLOG, s is above the center line, and so d(x,t) lt
  d(s,t).
• Suppose that ? ? ?, so y is chosen must show
  that d(y,t) lt d(s,t) in this case.
• Reflect s to get s.
• Use the fact that ? ? ? (and some inequalities
  involving angles) to conclude that y is
  counterclockwise of s.
• Then d(y,t) lt d(s,t) d(s,t).

s
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Compass Routing II (Face Routing)

• First consider graphs that are constructed out of
  convex polygons, then remove the restriction.
• Graphs of convex polygons,
  
  where entire collection is
  
  also a convex polygon.
• Line from s to t passes through a sequence of
  convex polygons.
• Normally, passes through successive edges.
• Could pass through vertices, or along edges.
• These two cases are ignored in the paper, though
  probably the algorithm can be extended to handle
  them.

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Compass Routing II (Face Routing)

• In the algorithm, the message moves
  systematically around the successive polygons in
  this sequence, until it reaches t (guaranteed).
• Can do this with the limited available
  information
• Could alternate moving clockwise and
  counterclockwise around the faces.
• The message contains s and t.
• Can check whether a traversed edge crosses edge
  st.

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Arbitrary Planar Geometric Graphs

• Such a graph induces a partitioning of the plane
  into polygonal regions, but now
• The faces needn't be convex.
• Entire collection neednt be convex.
• Some edges could stick out.

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Face Routing Algorithm

• p s
• repeat
• let f be face of G with p on boundary that
  intersects pt
• for each edge (u,v) of f do
• if uv intersects pt at p' and d(p',t) lt
  d(p,t) then
• p p'
• endif
• endfor
• traverse f until reaching the edge (u,v)
  containing p
• // traverse f counterclockwise / clockwise if
  f is the outer face
• until p t

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Face Routing Example
t
s
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Evaluation

• Guaranteed message arrival.
• Number of edge traversals bounded by 4 E
• Each edge can be traversed as part of two faces.
• Double again because algorithm traverses all the
  way around each face before deciding.
• Restriction to planar graphs
• Algorithm given for planar graphs only.
• But mobile ad hoc network communication graphs
  aren't necessarily planar---typically have many
  crossed edges.
• How to apply the results to non-planar graphs?

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Summary of Compass Paper

• Compass Routing I Follow edge pointing in the
  direction closest to that of the target.
• Guaranteed arrival, for subclasses of graphs such
  as Delaunay triangulations.
• Similar to Follow the edge whose other endpoint
  is closest to the target. Compare behavior?
• Compass Routing II Face routing.
• Guaranteed arrival.
• No fault-tolerance
• Algorithms pass just one copy of the message,
  subject to loss if nodes fail/move.
• How to make them more fault-tolerant?
• Dynamic case
• What happens when the network topology changes?

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Greedy Perimeter Stateless Routing (GPSR) Karp,
Kung

• Similar results proved (a year earlier) by Bose,
  Morin, Stojmenovic, Urrutia
• Problem Geographical routing, as before.
• Basic ideas
• Forward a single copy of a message.
• Use greedy routing when possible.
• When this fails, default to (a version of) face
  routing.
• Each node keeps only local information.
• Claim the algorithm scales well to large
  networks, and adapts well to changes in topology.

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GPSR

• Assumptions
• 2D
• Everyone knows their own positions (exactly).
• Each message source knows the (exact) position of
  the message target.
• Unit disk model Edges between all nodes that
  are at most distance r apart reliable
  communication on all edges.
• Sometimes (but not usually), they consider
  changing topology.
• Measures
• Communication cost number of hops traversed.
• Message delivery success ratio.
• Per-node state.

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Greedy Forwarding

• Packets marked by originator with target location
  t.
• Forwarding node u makes locally optimal, greedy
  choice of next hop the neighbor geographically
  closest to t, provided that it is closer to t
  than u is.
• Requires nodes to know their neighbors, and their
  neighbors locations
• Nodes exchange (id, position) information with
  neighbors.
• If topology changes are considered, must do this
  periodically, at a frequency based on rate of
  mobility and radio range r
• Periodically, each node transmits its (id,
  position).
• Others update their neighbor info based on the
  messages they receive.
• Use timeout to remove nodes from the neighbor
  set.
• State size depends on density of nodes in the
  network, but not on the total number of nodes.
• But, sometimes greedy doesn't work

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Greedy Forwarding Can Fail

• x has no neighbor closer to t than x is.
• Even if x were allowed to pass the message to its
  neighbor y, y would just pass it back.
• When a void is encountered, switch to face
  routing mode (perimeter mode).
• Could return to greedy mode, if face routing ever
  reaches a node that the regular algorithm would
  consider an improvement.

t
void
y
x
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Planarizing the Graph

• But face routing is defined only for planar
  graphs!
• Mobile network unit disk graphs arent
  necessarily planar.
• So they construct a planar subgraph of the UDG
  (ignore some of the edges).
• Two approaches
• Relative Neighborhood Graph (RNG)
• Gabriel Graph (GG)

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Relative Neighborhood Graph

• Relative Neighborhood Graph RNG(V), for a set V
  of points
• Nodes V.
• Edge (u, v) included iff there is no w ? u, v
  such that d(u, w) and d(w, v) are both lt d(u, v).
• That is, the interior of the lune is empty.

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RNG(V)

• Example KK
• RNG(V) is planar (no crossing edges) - prove later

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Algorithm to Produce an RNG

• Using local rules, identify the set of edges in
  RNG(V) ? UDG(V), where V is the set of mobile
  nodes.
• That is, the edges in the RNG that have length
  r.
• Resulting graph is still planar (of course).
• Theorem If UDG(V) is connected, then so is
  RNG(V) ? UDG(V).

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Algorithm to Produce an RNG

• Distributed algorithm for constructing RNG(V) ?
  UDG(V)
• Start with UDG(V) and remove edges.
• Each node u begins with its UDG edges, and
  removes any UDG edge (u, v) for which theres
  another (neighboring) node w with
• max(d(u,w), d(w,v) ) lt
  d(u,v).
• Note the edges can all be removed in
  parallel---removal condition depends solely on
  distances, and not on what other edges are
  removed.

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Algorithm to Produce an RNG

• Preserving connectivity
• Eliminating edges that aren't part of the RNG
  can't disconnect a connected UDG
• An edge is removed only if there is some other
  node that is closer to both endpoints.
• Use an inductive argument, based on removing one
  edge at a time, in order from longest to
  shortest. LTTR.
• Consistency issue
• Both endpoints of an edge (u, v) should make the
  same decision about whether to include edge (u,
  v).
• Depends on symmetry of neighbor relationship, and
  accuracy of geographical information.
• This works fine for the unit-disk model, in
  static situations.
• How to extend to weaker or more dynamic models?

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Gabriel Graph GG(V)

• Similar to RNG(V)
• Nodes V.
• Edge (u, v) included iff there is no w ? u, v
  such that
• (d(u,w))2 d(w,v)2 d(u,v)2.
• That is, the closed disk contains no nodes except
  u and v.
• A superset of the edges in RNG(V).

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GG(V)

• Example KK
• original graph Gabriel Graph
  Relative Neigh. Graph

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Gabriel Graph GG(V)

• Theorem GG(V) is planar.
• Proof
• If not, we have a crossing pattern.
• But then both b and d are outside the disk with
  diameter ac, and both a and c are outside the
  disk with diameter bd. Means that all the four
  outer angles are lt ?/2. Impossible.
• Corollary RNG(V) is planar.

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Algorithm to Produce a GG

• Using local rules, identify the set of edges in
  GG(V) ? UDG(V), where V is the set of mobile
  nodes.
• That is, the edges in the GG that have length
  r.
• Resulting graph is planar.
• Theorem If UDG(V) is connected, then so is
  GG(V) ? UDG(V).
• Proof Follows from connectivity of the RNG,
  which is sparser.

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Algorithm to Produce a GG

• Distributed algorithm for constructing GG(V) ?
  UDG(V)
• Start with UDG(V) and remove edges.
• Each node u begins with its UDG edges, and
  removes any UDG edge (u, v) for which theres
  another (neighboring) node w with
• (d(u,w))2 d(w,v)2 d(u,v)2.
  
• Edges can be removed in parallel---condition
  depends solely on distances.
• Again, consistency depends on symmetry of
  neighbor relationship, and accuracy of
  geographical information.

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Putting the Pieces Together

• For the static case, assuming UDG(V) is
  connected.
• Predetermine a planar connected subgraph---RNG(V)
  or GG(V).
• Start traversing the entire graph UDV(V) using
  greedy mode.
• When the message reaches a node x with no
  neighbors closer to t, then start traversing the
  planar subgraph in perimeter mode (using face
  routing).
• Do this until
• Reach t then done.
• Revisit a node then failed.
• Reach a node y with d(y,t) lt d(x,t) then resume
  in greedy mode.

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What if the Topology is Dynamic?

• The UDG depends on the current set of neighbors,
  which changes continually in response to
  mobility, failures, recoveries.
• Must recalculate the planar subgraph in response
  to these changes.
• KK recalculate the subgraph upon every
  acquisition of a new neighbor and every loss of a
  former neighbor.
• But this isn't really enough The choices of
  links to remove in the RNG and GG algorithms
  could change even in the face of minute movements
  that don't change the neighbor relation.
• Needs more work!

51
Evaluation

• Local state Small
• Simulation results KK
• Assumptions
• Mobility, random waypoint model.
• Use pause length to capture degree of mobility.
• Dense networks.
• Recalculate the planar graph only when neighbor
  sets change.
• Results
• Packet delivery success rate High, especially
  with frequent neighbor updates.
• Communication overhead Small, because the
  algorithm is local.
• Path length Delivers vast majority in optimal
  number of hops. But thats because greedy
  routing works almost always in dense networks.
• Effect of network diameter Little effect,
  because most of the processing is local.
• Needed work
• Better handling of asymmetric communication,
  dynamic networks.

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Kuhn, Wattenhofer,

• Face routing algorithm with guaranteed upper
  bound, assuming density bound and bounded
  stretch.
• Worst-case lower bound of d2.
• Simulation results, for combination of face and
  greedy routing.

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RobustBarriere, et al.

• Allows fixed edges, but of lengths that need not
  always fit the UDG assumptions. Face routing
  still works.