CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. 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 Discrete Algs for Mobile Wireless Sys

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. • We’ll study several different strategies: • Constrained flooding • Compass routing • Greedy perimeter routing • Compact routing with Euclidean metrics • Opportunistic routing • First: [Ko, Vaidya] Constrained flooding Discrete Algs for Mobile Wireless Sys

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, don’t flood everywhere---use geographical information to bound the flooding area. Discrete Algs for Mobile Wireless Sys

5. Target Multicast region Sender 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 needn’t 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. Discrete Algs for Mobile Wireless Sys

6. 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. Discrete Algs for Mobile Wireless Sys

7. 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 sender’s location and the entire multicast region. • Large forwarding zones introduce more flooding overhead, but may improve completeness. Discrete Algs for Mobile Wireless Sys

8. Target Multicast region Sender 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. Discrete Algs for Mobile Wireless Sys

9. 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 coordinatescm 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 i’s 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 atcm, radius = dist(sender, cm). Discrete Algs for Mobile Wireless Sys

10. Sender cm Geocasting Algorithm 2 • Message keeps getting closer to cm. • Does not flood through larger rectangle, but may flood around the multicast region. Discrete Algs for Mobile Wireless Sys

11. 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. • Don’t 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? Discrete Algs for Mobile Wireless Sys

12. 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 don’t 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. Discrete Algs for Mobile Wireless Sys

13. 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 D’s 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). Discrete Algs for Mobile Wireless Sys

14. 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 D’s position info more frequently, to reduce size of future multicast regions for route discovery for D. • LAR Algorithm 2: • Try to get closer to D’s 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 D’s current location, so doesn’t require clock synchronization. Discrete Algs for Mobile Wireless Sys

15. 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 don’t 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 target’s position seems like a good idea. • Moral: Geographical information can help in message routing, even if it isn’t accurate and up-to-date. Discrete Algs for Mobile Wireless Sys

16. 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 planargraphs 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. Discrete Algs for Mobile Wireless Sys

17. 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? Discrete Algs for Mobile Wireless Sys

18. 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. Discrete Algs for Mobile Wireless Sys

19. 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 Discrete Algs for Mobile Wireless Sys

20. 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 Discrete Algs for Mobile Wireless Sys

21. Compass Routing I s • Not every planar geometric graph supports compass routing. • Example: t Discrete Algs for Mobile Wireless Sys

22. 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. Discrete Algs for Mobile Wireless Sys

23. Delaunay Triangulation • Theorem: A unique Delaunay triangulation can always be constructed, provided that no four points are “cocircular”. • Example if 4 points are cocircular: Discrete Algs for Mobile Wireless Sys

24. Delaunay Triangulation • Be careful though: not every triangulation will be a Delaunay triangulation: Discrete Algs for Mobile Wireless Sys

25. s x t   y 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) < 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  <  then d(x,t) < d(s,t). • If    then d(y,t) < d(s,t). Discrete Algs for Mobile Wireless Sys

26. s s x x t t     y y 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) < d(s,t). • Suppose that   , so y is chosen; must show that d(y,t) < 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) < d(s’,t) = d(s,t). s’ Discrete Algs for Mobile Wireless Sys

27. t s 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. Discrete Algs for Mobile Wireless Sys

28. t s 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. Discrete Algs for Mobile Wireless Sys

29. t s 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 needn’t be convex. • Some edges could stick out. Discrete Algs for Mobile Wireless Sys

30. 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) < 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 Discrete Algs for Mobile Wireless Sys

31. Face Routing Example t s Discrete Algs for Mobile Wireless Sys

32. 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? Discrete Algs for Mobile Wireless Sys

33. 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? Discrete Algs for Mobile Wireless Sys

34. 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. Discrete Algs for Mobile Wireless Sys

35. 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. Discrete Algs for Mobile Wireless Sys

36. 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… Discrete Algs for Mobile Wireless Sys

37. 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 Discrete Algs for Mobile Wireless Sys

38. Planarizing the Graph • But face routing is defined only for planar graphs! • Mobile network unit disk graphs aren’t 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) Discrete Algs for Mobile Wireless Sys

39. w v u 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 < d(u, v). • That is, the interior of the “lune” is empty. Discrete Algs for Mobile Wireless Sys

40. RNG(V) • Example [KK]: • RNG(V) is planar (no crossing edges) - prove later Discrete Algs for Mobile Wireless Sys

41. 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). Discrete Algs for Mobile Wireless Sys

42. 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 there’s another (neighboring) node w with max(d(u,w), d(w,v) ) < 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. Discrete Algs for Mobile Wireless Sys

43. 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? Discrete Algs for Mobile Wireless Sys

44. w v u 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). Discrete Algs for Mobile Wireless Sys

45. GG(V) • Example [KK]: • original graph Gabriel Graph Relative Neigh. Graph Discrete Algs for Mobile Wireless Sys

46. c b d a 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 < /2. Impossible. • Corollary: RNG(V) is planar. Discrete Algs for Mobile Wireless Sys

47. 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. Discrete Algs for Mobile Wireless Sys

48. 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 there’s 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. Discrete Algs for Mobile Wireless Sys

49. 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) < d(x,t); then resume in greedy mode. Discrete Algs for Mobile Wireless Sys

50. 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! Discrete Algs for Mobile Wireless Sys