We noticed that the trouble is due to face routing. Can we ignore the real coordinates and use virtual coordinates for routing?

Transcription

1 Geographical routing in practice We noticed that the trouble is due to face routing. Is greedy routing robust to localization noise? Can we ignore the real coordinates and use virtual coordinates for routing? 1

2 Approach I: Rubber band representation 2

3 Rubber band drawing of a graph All edges are rubber bands. Nail down some nodes S in the plane, let the graph go. Theorem: the algorithm converges to a unique state rubber band representation extending S. Peterson graph with one pentagon nailed down. 3

4 Rubber band drawing of a graph The rubber band algorithm minimizes the total energy: Claim: E(x) is convex. When any x i goes to infinity, E(x) goes to infinity. So we have a unique global minimum. Peterson graph with one pentagon nailed down. 4

5 Rubber band drawing of a graph How does the rubber band representation look like? E(x)/ x i =0. neighbors The rubber band connecting i and j pulls i with force x j - x i. The total force acting on x i is 0. The graph is at equilibrium. Peterson graph with one pentagon nailed down. 5

6 Rubber band drawing of a graph 1. Every free node is at the center of gravity of its neighbors. 2. no reflex vertices. Peterson graph with one pentagon nailed down. 6

7 More examples 7

8 Rubber band algorithm Recall the mass-spring model. First we assume nodes on the boundary know their location. Fix the nodes on the outer boundary. Iterative algorithm: Every node moves to the center of gravity of its neighbors. Until no node moves more than distance δ. 8

9 A network with 3200 nodes Greedy routing success rate: 0.989, avg path length

10 Perimeter nodes are known (10 iterations) 10

11 Perimeter nodes are known (100 iterations) 11

12 Perimeter nodes are known (1000 iterations) Greedy routing success rate: 0.993, avg path length

13 Resiliency of the rubber band approach Greedy routing success rate: 0.981, avg path length

14 Resiliency of the rubber band approach Greedy routing success rate: 0.99, avg path length

15 How to fix perimeter nodes? Need nodes on the perimeter to stretch out the net. First assume we know nodes on the perimeter, but not the locations. 1. Each perimeter sends hello messages. 2. All the nodes record hop counts to each perimeter node. 3. The hop count between every pair of perimeter node is broadcast to all perimeter nodes. (quite expensive) 4. Embed perimeter nodes in the plane, say by any localization algorithm. 15

16 Perimeter nodes 1. The embedding only gives relative positions: include 2 bootstrapping beacons in the embedding of perimeters. Use the center of gravity as origin. 1 st bootstrap node defines the positive x-axis. 2 nd bootstrap node defines the positive y-axis. 2. Non-perimeter nodes actually have the distances to all perimeter nodes, and embed themselves. Gives good initial positions for the rubber band algorithm. 16

17 How to find perimeter nodes? The bootstrapping nodes send hello messages to everyone. The node which is the farthest among all its 2-hop neighbors will identify itself as a perimeter node. 17

18 Success rate of greedy routing Success rate on virtual coordinates is comparable with true coordinates, when the sensors are dense and uniform. 18

19 Weird Shapes 19

20 Obstacles Success rate on virtual coordinates degrades when there are a lot of obstacles, but better than true coordinates. 20

21 Conclusions Geographical forwarding is quite robust to localization errors, or reasonable virtual coordinates. Geographical forwarding can easily scale to tens of thousands of nodes with acceptable overhead. For dense uniform sensor layout, we can eliminate the need for face routing altogether. Rubber band virtual coordinates respect the connectivity better than the true coordinates. 21

22 Next Rubber band approach is kind of ad-hoc. Next: In-depth study of routing around holes. Greedy embedding? 22

23 Given a graph, find an embedding s.t. greedy routing works Greedy embedding of a graph 23

24 Greedy embedding Given a graph G, find an embedding of the vertices in R d, s.t. for each pair of nodes s, t, there is a neighbor of s closer to t than s itself. t s 24

25 Questions to ask We want to find a virtual coordinates such that greedy routing always works. Does there exist such a greedy embedding in R 2? in R 3? in Euclidean metric? Hyperbolic space? If it exists, how to compute? 25

26 Greedy embedding does not always exist K 1,6 does not have a greedy embedding in R 2 26

27 A lemma Lemma: each node t must have an edge to its closest (in terms of Euclidean distance) node u. Otherwise, u has no neighbor that is closer to t than itself. 27

28 Proof K 1,6 does not have a greedy embedding in R 2 Proof: 1. One of the angles is less than π/3. 2. One of ab 2 and ab 3, say, ab 2, is longer than b 2 b Then b 2 does not have edge with its closest point b 3. 28

29 A conjecture Corollary: K k, 5k+1 does not have a greedy embedding in R 2. Conjecture: Any planar 3-connected graph has a greedy embedding R 2. Hint: this is tight. K 2,11 is planar but not 3-connected. K 3.16 is 3-connected but not planar. (it has K 3.3 minor). Planar 3-connected graph has a greedy embedding in R 3 29

30 Polyhedral routing Theorem: Any 3-connected planar graph has a greedy embedding e in R 3, where the distance function is defined as d(u, v) = - e(u) e(v). Proof: 1. Any 3-connected planar graph is the edge graph of a 3D convex polytope, with edges tangent to a sphere. [Steinitz 1922]. 2. Each vertex has a supporting hyperplane with the normal being the 3D coordinate of the vertex. 30

31 Polyhedral routing Proof: For any s, t, there is a neighbor v of s, d(v,t)<d(s,t). 1. d(s,t)-d(v,t)=[e(v)-e(s)] e(t)>0. 2. Now suppose such neighbor v does not exist, then s is a reflex vertex, with all the neighbors pointing away from t. v 3. This contradicts with the convexity of the polytope. t s 31

32 Discussions Papadimitriou s conjecture: Any planar 3-connected graph has a greedy embedding R 2. has been proved! The theorem only gives a sufficient condition, not necessary. K 3.3 has a greedy embedding. A graph with a Hamiltonian cycle has a greedy embedding on a line. Given a graph, can we tell whether it has a greedy embedding in R 2? Is this problem hard? (Recall that many such embedding problems are hard ) More understanding of greedy embedding in R 2, R 3 32

33 Follow-up work Dhandapani proved that any triangulation admits a greedy embedding (SODA 08). Leighton and Moitra proved the conjecture (FOCS 08). Independently, Angelini et al. also proved it (Graph Drawing 08). Goodrich and D. Strash improved the coordinates to be of size O(log n) (under submission). We briefly introduce the main idea. 33

34 Leighton and Moitra All 3-connected planar graph contain a spanning Christmas Cactus graph. All Christmas Cactus graphs admit a greedy embedding in the plane. 34

35 Leighton and Moitra A cactus graph is connected, each edge is in at most one simple cycle. A Christmas Cactus graph is a cactus graph for which the removal of any node disconnects into at most 2 pieces. 35

36 A Christmas Cactus 36

37 Example 37

38 Connection to graph labeling Given a graph, find a labeling of the nodes such that one can compute the (approximate) shortest path distance between any two vertices from their labels only. Tradeoff between approximation ratio and the label size. For shortest path distance, the maximum label size is Θ(n) for general graph, O(n 1/2 ) (Ω(n 1/3 )) for planar graphs, and Θ(log 2 n) for trees. General graph: a scheme with label size O(kn 1/k ) and approximation ratio 2k-1. Google distance labeling for the literature. 38

39 Approach II: Embed a spanning tree in polar coordinate system 39

40 Embed a tree in polar coordinate system Start from any node as root, flood to find the shortest path tree. Assign polar ranges to each node in the tree. The range of a node is divided among its children. The size of the range is proportional to the size of its subtree. Order the subtrees that align with the sensor connectivity. 40

41 Embed a tree in polar coordinate system Order the subtrees that align with the sensor connectivity. Three reference nodes flood the network. Each node knows the hop count to each reference. Each node embed itself with respect to the references. (trilateration with hop counts) A node s position is defined as the center of mass of all the nodes in its subtree. This will provide an angular ordering of all the children. 41

42 Routing on a tree Route to the common ancestor of the source and destination. Check whether the destination range is included in the range of the current node. If not, go to the parent. Otherwise go to the corresponding child. Root is the bottleneck. Path may be long. 42

43 Routing on a tree Be a little smarter: store a local routing table that keeps the ranges of up to k-hop neighbors. find shortcuts. Virtual Polar Coordinate Routing: check the neighborhood, find the node that is closer to the destination. greedy forwarding in polar coordinates. If the upper/lower bound is closer to the destination. 43

44 Load balancing Root is still the bottleneck even for smart routing. Shortest path routing, still not the most load balanced routing 44

45 Routing on spanning trees in theory and in practice For any graph G there is a spanning tree T, s.t. the average stretch of the shortest paths on T, compared with G, is O((lognloglogn) 2 ). 45

46 General framework Find a substructure such that we can define virtual coordinates. The virtual coordinates guarantee delivery. Greedy routing in the connectivity graph. 46

47 Virtual ring routing Some slides from 47

48 VRR: the virtual ring topology-independent node identifiers, e.g., MAC address 910 8F6 90E 8F0 8E2 FFF 0 each node maintains a virtual neighbor set (vset) nodes organized into virtual ring by increasing identifier value 48

49 VRR: routing paths 8F6 8F6 physical network topology nodes only maintain paths to virtual neighbors: vset-paths are typically multi-hop vset-paths are maintained proactively 49

50 VRR: forwarding table 14A endpointa endpointb nexta nextb pathid 8F6 90E ME F F6 10E ME 10 14A 140 F42 10E 2 8F6 F42 ME F42 FF forwarding table for node 8F6 F42 8F6 10E vset-paths recorded in forwarding tables along path forwarding table contains 140 vset-paths between node and vset members vset-paths between other nodes that go through node paths to physical neighbors 50

51 VRR: forwarding forward message destined to x by picking endpoint e numerically closest to x forwarding message to next hop towards e deliver message to node with id closest to x how does this work? can find x because nodes are connected in a ring low stretch because of additional forwarding state many alternate paths to route around failures 51

52 VRR: example routing physical network Topology 52

53 VRR: example routing there may be some stretch physical network Topology 53

54 Node joining broadcast hellos to find physical neighbors send setup request to 16E (itself) through proxy (19A) A 8F6 Network Topology 16E 54

55 Node joining 16E sends setup requests to nodes in received vset 164 sends setup to 16E with its vset 16E adds node to vset when it receives setup A 8F6 171 Network Topology 16E 55

56 Size of routing table Assume the nodes are randomly placed, each vpath 56

57 Simulation experiments in ns-2 ran experiments with b MAC varied network size, mobility, session lifetime compared with DSDV, DSR, and AODV VRR performed well in all experiments high delivery ratios even with fast movement significantly lower delays with route instability 57

58 Delivery ratio: fast movement Delivery ra atio DSDV DSR AODV VRR Number of nodes 58

59 Delay: fast movement Delay (sec conds) DSDV DSR AODV VRR Number of nodes 59

60 Sensor network sensor network testbed 67 mica2dot motes in UCB building comparison with BVR delivery ratio with mote failures 60

61 Sensor network: mote failures Number of nodes Number of nodes VRR delivery ratio BVR delivery ratio Time (mins) Percentage of pac ckets delivered 61

62 Wireless office testbed 30 machines running windows communicate using a throughput comparison with LQSR using ttcp 62

63 Wireless office testbed: throughput Bandwidth (Mbps) MR-LQSR VRR Machine 63

64 Why a virtual ring? Alternatively, use an Euler tour to define coordinates on the sensor nodes. An Euler tour is a cycle that visits every vertex. Can be constructed by a depth-first tour on a spanning tree. Also use shortcuts for greedy routing. 64