MAP: Medial Axis Based Geometric Routing in Sensor Networks

1. MAP: Medial Axis Based Geometric Routing in Sensor Networks MobiCom’05 Jehoshua Bruck, Jie Gao, Anxiao(Andrew) Jiang Ku Dara

2. Contents • Introduction • Medial axis • Medial Axis based naming and routing Protocol (MAP) • In continuous region in the Euclidean plane • In discrete sensor field • Simulation • Summary MAP

3. Introduction(1/2) • Design of Routing algorithm • Routing is elementary in all communication networks • is tightly coupled with auxiliary infrastructure that abstracts the network connectivity • Stable link & powerful nodes (Internet) : routing table • Fragile links & constantly changing topologies &nodes with less resouceful h/w (ad-hoc mobile wireless networks) : flooding Too energy-expensive for sensor networks Flooding for route discovery Light infrastructure of sensor networks for efficient and localized routing Medial Axis MAP

4. Introduction(2/2) • Medial Axis • Set of points with at least two closest neighbors on the boundaries of the shape • ‘Skeleton’ of a region • Capture both geometric and topological features by using the connectivity information • MAP • Medial axis based naming and routing protocol as a routing infrastructure • Depend only connectivity graph • Consist of 2 protocols • Madial Axis Construction Protocol(MACP) • Medial Axis based Routing Protocol(MARP) MAP

5. Medial Axis - Definition • Given a bounded region R, boundary • A is the collection of points with two or more closest points in • A cord is a line segment on the medial axis and its closest points on • A point on the medial axis with 3 or more closest points on is called a medial vertex • Canonical cell : the medial axis, 2 chords, MAP

6. Naming w.r.t. medial axis • Point p is named by the chord x(p)y(p) it stays on (x(p), y(p), d(p)) • x(p) is a point on the medial axis • y(p) is the closest point of x(p) on ∂R • d(p) is height. i.e. relative distance from x(p): |px(p)|/|x(p)y(p)| Theorem: Every point is given a unique name MAP

7. Routing between canonical cells(1/2) • The naming system naturally builds a Cartesian coordinate system • x-longitude curve --The chord with medial point x • h-latitude curve --The collection of points with the height h (0 ≤ h ≤ 1) • The canonical cells are glued together by the medial axis. • With the knowledge of the medial axis – route from cells to cells by checking only local neighbor information MAP

8. C2 C2 C1 C1 Routing between canonical cells(1/2) • Two canonical cells adjacent to the same medial vertex may not share a chord • Build rotary systems around medial vertices • Polar coordinate system: (|ap|/r, ), r is the maximum radius of a ball centered at a medial vertex a MAP

9. Routing scheme(1/2) • Routing is done in 2 steps • Check the medial axis graph, find a route connecting the corresponding points on the medial axis as guidance • Realize the route by local gradient descending, in either the Cartesian coordinate system inside a canonical piece, or a polar coordinate system around a medial vertex MAP

10. Routing scheme(2/2) • Routing is done in 2 steps • Check the medial axis graph, find a route connecting the corresponding points on the medial axis as guidance • Realize the route by local gradient descending, in either the Cartesian coordinate system inside a canonical piece, or a polar coordinate system around a medial vertex MAP

11. MAP in discrete networks – naming(1/5) • Detect boundaries of the sensor field • Find sample nodes on boundaries • By manual identification, or automatic detection [Fekete’04, funke’05] MAP

12. MAP in discrete networks – naming(2/5) • Detect boundaries ( the curve construction problem) • Use local flooding to connect nearby boundary nodes • Include nodes on the shortest path between them as boundary nodes MAP

13. MAP in discrete networks – naming(3/5) • Construct the media axis graph • Detect medial nodes (the sensors with 2 or more closest boundary nodes) by restricted flooding • Flooding message: Sensor’s ID, boundary, hop count MAP

14. MAP in discrete networks – naming(4/5) • Construct the medial axis graph • Connect medial nodes into a graph and clean it up • Remove very short branches Broadcast this simple graph to all sensors MAP

15. MAP in discrete networks – naming(5/5) • Assign names to sensors for discrete networks • Replace chords by approximate shortest path trees • “Medial axis with dangling trees” • Shortest path forest rooted at the medial axis • Nodes are assigned names w.r.t. where it lies in the tree All the computation is simple and local MAP

16. Medial axis based routing • Medial Axis based Routing Protocol • Find the shortest path in the medial axis graph A • Route in parallel to the shortest path • Route along the shortest path trees rooted at that medial point to reach the destination q • Guaranteed delivery • If there is no better choice, route toward the medial axis • Maintain balanced load • Try to route in parallel with the medial axis as much as possible to avoid overloading nodes near the medial axis MAP

17. Simulation (1/2) • Outdoor sensor field: Campus(650mX620m) 5735 nodes The simple medial axis graph: 18nodes, 27edges MAP

18. Simulation (2/2) Routing path comparison Load balance comparison MAP destination source GPSR Normalized standard deviation of traffic load on sensors MAP

19. Summary • MAP • Topology-enabled naming and routing schemes that based purely on link connectivity information • Advantage • Takes only connectivity graph as input • Infrastructure is lightweight • Routing is efficient and local MAP