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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

This lecture discusses location services for mobile and wireless systems, including maintaining location information, MOVE and FIND operations, and implementing location services over a hierarchical directory service.

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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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

  2. Lecture 17 • Topic: • Location services • Sources: • Awerbuch & Peleg • Li, Janotti, DeCouto, Karger, & Morris • Abraham, Dolev, & Malkhi • MIT 6.885 Fall 2008 slides • CPSC 689 Spring 2006 slides by Sai Siddarth Discrete Algs for Mobile Wireless Sys

  3. Location Services • Maintain location information about mobile devices. • Generally support operations: • MOVE(i) • A mobile device i may move from one location to another in a network. • FIND(i) • Anyone anywhere in the network can try to locate any mobile device i anywhere in the network. • Typically used to obtain an address, which can be used, perhaps repeatedly, for communicating with the target node i. Discrete Algs for Mobile Wireless Sys

  4. Location Service Example are the location servers of Discrete Algs for Mobile Wireless Sys

  5. Location Service Example are the location servers of wants to communicate with . It communicates with to get ‘s information. Discrete Algs for Mobile Wireless Sys

  6. Location Service Example are the location servers of Discrete Algs for Mobile Wireless Sys

  7. Location Services • Three papers: • [Awerbuch, Peleg 91]: A classical paper, presenting a location service for mobile users traveling around a fixed, wired network. • [Li et al. 00]: The GRID paper, presenting a location service (GLS) for mobile ad hoc networks, good behavior in relatively static cases. • [Abraham, Dolev, Malkhi 04]: LLS, improved handling of node mobility. Discrete Algs for Mobile Wireless Sys

  8. [Awerbuch, Peleg] Overview • A location service for mobile devices (users) traveling around a fixed network, modeled as an undirected graph. • Operations supported: • MOVE(,s,t): • User  jumps from node (location) s to node t in the network. • Not necessarily along graph edges. • FIND(,v): • User at node v in the network tries to get a message to user . • Q: How could these results apply in wireless settings? • Goals: • Make both kinds of operations fast and communication-efficient (polylog in the number of nodes in the network). • Keep only polylog information per user at each node. • Main idea: Implement the location service over a hierarchical directory service. Discrete Algs for Mobile Wireless Sys

  9. Directory Services • Single-level directory provides operations: • INSERT(, s): Insert , associated with location s. • DELETE(, s): Delete association of  with location s. • FIND(, v): Starting from v, try to find an associated location for ; may return a location or may fail. • Hierarchical directory service consists of a series of directory services, each associated with successively larger “radii” (successive doubling). • Directory service associated with a certain radius m is guaranteed to return an answer for a FIND for any node that is within distance m of the location v at which the FIND is invoked. Discrete Algs for Mobile Wireless Sys

  10. Directory Services and Location Services • Implementing location service over hierarchical directory service: • Directories don’t always maintain current locations for users: “low-level” directories (small radii) are kept more current, while higher-level directories may lag further behind. • If a directory entry for user  is old, it points to a previous location v for , which has a forwarding pointer to a more recent location for . • By following forwarding pointers, FIND query reaches . Discrete Algs for Mobile Wireless Sys

  11. Directory Services and Location Services • More specifically: • If a level i directory entry for  indicates node v, then v has a forwarding pointer to the same node w that the level i-1 directory contains for  . • That should be closer to . • Length of chain of pointers kept low (logarithmic bound). • Implementing the hierarchical directory service: • By a distributed read/write quorum system algorithm. • Assume all MOVE and FIND operations are sequential; give some ideas about extensions to allow concurrency. Discrete Algs for Mobile Wireless Sys

  12. h i g j a f e b c d The Network Model • Undirected graph G = (V,E), n nodes. • Bidirectional communication channels. • Edges have weights, used to define a path cost measure (sum of weights). • For now, assume each edge has weight 1---so just consider hop counts. • dist(u,v) = length of shortest path (hops). • D, network diameter • Nodes may communicate directly only with neighbors in the graph. • Sometimes, nodes communicate with non-neighboring nodes: assume “efficient” routing is available (approximately shortest path). Discrete Algs for Mobile Wireless Sys

  13. h  i g j a f e b c d The Problem • Each user , at each time, is located at some node Addr() of G. • A location service is a global service providing operations: • MOVE(, s, t): • Invoked at node s = Addr(). • User  moves from node s to node t. • FIND(,v): • Invoked at node v. • Delivers a message submitted (by external client) at node v, to Addr(). Guaranteed. Discrete Algs for Mobile Wireless Sys

  14. h i g j a f e b c d Full Information Strategy • Every node keeps a complete table of where all the users are. • Every time a user moves, immediately communicates with all nodes to update their information. • FIND(,v) operations are cheap---route message directly to user . • MOVE operations are very expensive (e.g., amount of communication). • Useful only in settings where MOVEs are infrequent.  Discrete Algs for Mobile Wireless Sys

  15. Other Strategies • No-information strategy: • No information maintained anywhere. • MOVEs are very cheap. • FINDs are very expensive: involve a global search. • Useful only in settings where FINDs are infrequent (occasional location queries). • Complete chain of forwarding pointers: • When user moves, leave a pointer from the previous node to the new one. • MOVEs cheap, but FIND can involve following a length-n chain of pointers. • They want some intermediate “partial information” strategy, in which both MOVE and FIND operations are cheap, for any sequence of MOVE and FIND operations. Discrete Algs for Mobile Wireless Sys

  16. Cost Measures • MOVEs to nearby locations, and FINDs for nearby users, should cost less than long-distance MOVEs and FINDs. • Measure communication cost, for each FIND and MOVE, in terms of number of hops that messages traverse in the implementation. • (Actually, weighted costs. And charge based on number of bits in the message. Ignore these for now.) • Compare actual costs to ideal costs, where full info is available for free. These are the ideal costs: • FIND(, v): dist(v,Addr()), the distance (hops), from the location v at which the operation is invoked to the actual location of . • MOVE(, s, t): dist(s,t) • Measure is amortized, over a sequence of MOVE and FIND operations. • FIND-stretch: Ratio of the sum of the actual costs of all the FIND operations to the sum of the ideal costs for the same FIND operations. • MOVE-stretch: Similar. Discrete Algs for Mobile Wireless Sys

  17. Main Result • Algorithm that guarantees: • FIND-stretch O(log2(n)) (counting log(n) for each message; with unit cost, may be just O(log(n)). • MOVE-stretch O( log D log n + (log D)2 / (log n) ). • Low communication. Discrete Algs for Mobile Wireless Sys

  18. Hierarchical Directory Service • Hierarchy of regional directories, RDi, i = 1,..., log D. • RDi enables a searcher at any location v to find any user residing within distance 2i of location v. • Each RDi is a single-level global directory data structure. • It maintains (conceptually) some global state, which is a set of (user, address) pairs (, s). • Call s the level i regional address of . • RDi provides operations: • INSERT(, s): Invoked from location s. Associates  with s. • DELETE(, s): Invoked from location s, where  is associated with s. Deletes the association of  with s. • FIND(, v): Starting from v, try to find the location associated with . May return a location or may fail. Guaranteed to succeed if  has an associated location s with dist(v,s) ≤ 2i. Discrete Algs for Mobile Wireless Sys

  19. Implementing Location Service Over a Hierarchical Directory Service • Suppose we have a hierarchical directory service (see later how to implement it, with a distributed algorithm). • How to use it to implement a location service? • Each RDi maintains associations between users and locations (regional addresses). • Ideally, these would be kept up-to-date, so when a MOVE(, s, t) occurs, all directories would be updated to remove (, s) and add (, t). • Too costly. • Instead, store “forwarding pointers” at some nodes of the network. • Pointers of the form forward(). • Outside the directory service. • The node s at which a user  currently resides knows it is there: represented by setting forward() at s to s itself. Discrete Algs for Mobile Wireless Sys

  20. FIND(,v) Implementation • Using (global) hierarchical directory and (distributed) forwarding pointers. • FIND queries RD1, RD2,... about , until it gets some location s as an answer from some regional directory RDi. • That will eventually happen, because it's guaranteed at the top level, directory Rlog D. • Then FIND goes to node s, sees if user  is there; if not, FIND follows forwarding pointers from node to node until it reaches ’s current location. • Forwarding pointers and RDi directories are not changed by FIND operations, just during MOVEs. Discrete Algs for Mobile Wireless Sys

  21. MOVE (, s, t) Implementation • Modifies some of the RDi entries for  to contain the new location t. • Definitely modifies lowest-level directory, RD1. • Possibly some higher-level directories also. • If any directory is updated, so are all the lower-level ones. • To determine which directories to update, maintain cumulative counts of how far  has moved (in all its MOVEs) since last updating each directory. • When  has moved distance 2i-1 or greater, regional address RDi() is guaranteed to be updated. • Let i denote the highest-level directory that is updated. • Then, if i is not the top-level directory, set a forwarding pointer forward() = t at the node associated with  in RDi+1. Discrete Algs for Mobile Wireless Sys

  22. Reachability Invariant • Expresses an interesting relationship between regional addresses for  stored in the regional directories, and forwarding pointers for : • If i  2, RDi() = s, and RDi-1() = t, then forward() at node s contains t. • Thus, if location s is the regional address of  at some level i of the directory hierarchy, then following the forwarding pointer forward() from s leads to the same location t that would be obtained from level i-1 of the directory hierarchy. • So, following a path of forward() pointers starting from any regional address for  in the directory hierarchy, is the same as stepping through all the successively lower-level regional addresses for forward(). Discrete Algs for Mobile Wireless Sys

  23. Proximity Invariant • Says that directory entries are updated sufficiently often: • For any i, the total distance traveled by user  since the last time RDi() was updated (to its exact location) is ≤ 2i-1 -1. • This use of powers of 2 yields the logarithmic bounds. • Analysis LTTR (not in the paper). Discrete Algs for Mobile Wireless Sys

  24. Implementing a Hierarchical Directory Service • Implement each level separately. • For each level, use a kind of quorum configuration: • Define m-regional-matching, or m-quorum-configuration, where m is a distance (nonnegative real): • Assigns to each node v in the graph, two sets of nodes, Read(v) and Write(v). • If dist(v,w) ≤ m, then Read(v)  Write(w)  . • Thus, if two operations are performed, one involving all the nodes in Read(v), and the other involving all the nodes in Write(w), for nearby nodes v and w, they would encounter at least one node in common. • E.g., if a write-quorum is used to write an update to several copies of a directory, then accessing a read-quorum from a nearby node is guaranteed to get an up-to-date copy of the directory. Discrete Algs for Mobile Wireless Sys

  25. Implementing a Hierarchical Directory Service • Theorem 4.1 says that certain efficient m-regional matchings exist: • Small size read-quorums and write-quorums. • Short distance between v and each member of Read(v) and Write(v). • Measure is normalized by dividing by distance parameter m. • Thus, larger m are allowed to have proportionally larger distances between quorum nodes, without penalty. Discrete Algs for Mobile Wireless Sys

  26. Implementing Regional Directory RDi • Use a 2i-regional matching. • Information being kept in RDi for node v is copied at all nodes in Write(v). • To read the information starting from node w, read from all copies at all nodes in Read(w). • If dist(v,w) ≤ 2i, this is guaranteed to return the last value written in the directory at v. • More specifically, in terms of actual directory operations: • INSERT(, s): Implemented using writes to copies at the nodes in Write(s). • DELETE(, s): Same • FIND(, v): Implemented using reads of the nodes in Read(v). Discrete Algs for Mobile Wireless Sys

  27. Find Example [AP] Discrete Algs for Mobile Wireless Sys

  28. x5 x5 x5 x5 Move (x4 to x5) Example [AP] x5 Discrete Algs for Mobile Wireless Sys

  29. Extensions to Allow Concurrency • Some concurrency isn’t a problem: • Concurrency between different users doesn’t matter. • Can assume MOVEs for the same user  are sequential: can coordinate at ’s location. • Concurrent FINDs for  don't interfere. • The real problem: • When a FIND for  proceeds concurrently with a MOVE of . • Would like to guarantee, under reasonable conditions, that the FIND will succeed in tracking down the moving user. Discrete Algs for Mobile Wireless Sys

  30. Extensions to Allow Concurrency • No definitive solution, but some ideas: • Slow down 's moves to ensure that concurrent FINDs can catch it. • Require stronger conditions for quorum systems, so that a Read set is guaranteed to intersect all “relevant” Write sets, for nodes within the distance that could be covered by concurrent MOVEs. • Coordinate using synchronized clocks. Discrete Algs for Mobile Wireless Sys

  31. The GRID Location Service (GLS) [LJCKM] • For mobile ad hoc networks. • Node addresses are 2D geographical coordinates, not graph nodes. • GLS supports operations: • FIND(y): • A mobile node x anywhere in 2D space may submit a FIND(y) request, asking for the address (coordinates) of any other mobile node y. • GLS should respond to x with reasonably up-to-date coordinates for y. • MOVE(y) Discrete Algs for Mobile Wireless Sys

  32. The GRID Location Service (GLS) • GLS can be combined with a geographical routing algorithm, which sends a message to the vicinity of a target geographical location: • x looks up the address of y using GLS, then use geographical routing to send an actual data message to y. • Also, the GLS implementation uses geographical routing as part of the processing of FIND operations. Discrete Algs for Mobile Wireless Sys

  33. Key Ideas in GLS • Three structures: • Static, hierarchical subdivision of the plane into nested squares. • Static assignment of unique ids to mobile nodes, from a circularly-ordered set of node ids. • Dynamic assignment of a table consisting of particular (node-id, address) pairs to each mobile node. • The pairs in the tables should contain relatively up-to-date coordinates; must be maintained in the face of mobility. • Selection of which pairs reside in each table depends on current locations of mobile nodes in the plane, with respect to the hierarchical subdivision. Discrete Algs for Mobile Wireless Sys

  34. Key Ideas • The selection of which pairs reside in each table is adapted from a consistent hashing technique of Karger and others. • Here, the selection is designed to: • Load-balance, keep tables at different nodes of similar sizes. • Support efficient FIND operation, which follows a greedy strategy, repeatedly moving to the node whose id is “closest” to that of the target, among those whose coordinates appear in the local table. Discrete Algs for Mobile Wireless Sys

  35. Targeted Complexity • Overall table organization is designed to ensure that the number of steps performed during each FIND is proportional to the log of the Euclidean distance from source to target. • Moreover: • The number of hops for the first such step (using geographical routing) is proportional to the distance between the source and target. • The number of hops for each successive step is approximately half of the number for the previous step. • Thus, the total number of total hops used by a FIND is proportional to the geographical distance between source and target. • Queries for locations of nearby nodes are satisfied with correspondingly local communication. • Experiments show: • Scalability, tolerance to failures, recoveries, motion. • Main emphasis is on scalability (to a large metropolitan area). • Analyze correctness and efficiency (in static case). Discrete Algs for Mobile Wireless Sys

  36. Node Identifiers • Mobile nodes have unique ids, chosen from a discrete circularly-ordered set, e.g., integers mod k, for some k. • Consider the ids as increasing in clockwise direction around a circle. • The node “closest” to node with id x in some nonempty set X of ids: • Means the one in X whose id is first encountered, starting from x and moving in the clockwise direction. • If x is in X, the result is x itself. Discrete Algs for Mobile Wireless Sys

  37. Example of Identifiers • Blue nodes are in X • Node closest to 6 is 7 • Node closest to 8 is 1 • Node closest to 4 is 4 0 8 1 7 2 6 3 5 4 Discrete Algs for Mobile Wireless Sys

  38. The Planar Subdivision • Nested squares, of order 1, 2, ..., k. • Order-1 squares are the basic unit squares. • Overall square is an order-k square, consisting of 2k-1 2k-1 order 1 squares. • In general, an order-(i+1) square is composed of four order-i squares. • Assume that every mobile node, at any time, is at some location in the overall square. • In only one square at any time. • Use a tiebreaker for boundaries, or ignore cases where the nodes reside exactly on the boundaries. Discrete Algs for Mobile Wireless Sys

  39. Order 1 Squares Discrete Algs for Mobile Wireless Sys

  40. Order 2 Squares • 4 order n squares make up an order n+1 square Discrete Algs for Mobile Wireless Sys

  41. Order 3 Squares • 4 order n squares make up an order n+1 square Discrete Algs for Mobile Wireless Sys

  42. Constraint • Every node can only be in one square of each size. Discrete Algs for Mobile Wireless Sys

  43. Placement of Location Servers Host Location Server Discrete Algs for Mobile Wireless Sys

  44. The Tables • Describe for static case, where mobile nodes don’t move. • Namely, for each node x at geographical location l, and each node y, put (x, l) in y's table if one of the following hold: • x and y are in the same order-1 square. • For some i, 1 ≤ i ≤ k-1: • x and y are in the same order-(i+1) square but in different order-i squares, and • y's id is the closest to x's id (clockwise order), among nodes in y's order-i square. • That is, there is no y' in the same order-i square as y such that x < y' < y. • Thus, a single (x, l) could appear in: • Tables of all the nodes in the same order-1 square as x, plus • 3(k-1) tables, 3 for each order (each order-i square has 3 “sibling” order-i squares within the same order-(i+1) square). Discrete Algs for Mobile Wireless Sys

  45. The Tables • Each entry in node y’s table corresponds to some level. • Entries for lower levels are for nodes that are closer geographically to y. • Not clear whether y has more entries for higher levels: • The “load” of potential nodes x for which y could have an entry is larger for higher levels. • However, for each potential node x at a higher level, it is less likely that y is the chosen node, since y has many other neighbors to share the load. Discrete Algs for Mobile Wireless Sys

  46. Example 90 38 39 37 45 70 50 91 11 62 5 51 19 1 35 27 63 41 23 72 44 14 2 17 20 87 28 10 83 32 98 55 61 6 21 81 31 43 12 76 84 Discrete Algs for Mobile Wireless Sys

  47. FIND Protocol • Suppose that FIND(t) is invoked at node s. • Then: • Node s sends a query message to the node x whose id is closest to that of t, among those whose coordinates appear in s's local table. • If node w  t receives a query message intended for t, then w sends the query message to the node x whose id is closest to that of t among those whose coordinates appear in w's local table. • If node t receives a query message intended for itself, then t sends (using geographical routing) its own coordinates back to the site s that initiated the FIND. Discrete Algs for Mobile Wireless Sys

  48. FIND Protocol Example 90 38 39 37 45 70 50 91 11 62 5 51 19 1 35 27 63 41 23 72 44 14 2 87 17 28 10 83 32 98 55 61 6 20 21 81 31 43 12 76 84 Discrete Algs for Mobile Wireless Sys

  49. Termination and Complexity Bound for FIND • The key is an invariant that holds after any number j of steps of the FIND algorithm (invoked from s for target t): • After at most j steps of FIND, 1 ≤ j ≤ k, the query is at the closest node to t, from among those in the order-j square containing s. • Thus, the query progresses systematically, finding the closest node to t in successively larger squares containing s. • So when j = k, the query is at the closest node to t in the entire square, which is t itself. • Proof is inductive. (See p. 125; the heart of the paper.) Discrete Algs for Mobile Wireless Sys

  50. Inductive Proof of FIND Behavior: Basis • After at most j = 1 step, the query has reached the closest node to t from among those in the order-1 square containing s. • If s is itself the best, then we're done after 0 steps. • If not, then recall that s knows about everyone in its order-1 square. • So in 1 step, it can certainly route the query to the best one, say u. • Potential problem: suppose that s happened to have something else in its table, say v, that isn't in the same order-1 square, but that is closer to t than u is (in id-space)… Discrete Algs for Mobile Wireless Sys

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