Papers - Stony Brook Universityjgao/CSE595-spring09/lecture7.pdf · Papers • [Papadimitriou04 ......

2/19/09 Jie Gao, CSE595-spring09 1

PapersPapers

• [Papadimitriou04] David Ratajczak, Christos Papadimitriou, On a conjecture related to geometric routing, ALGOSENSOR, 2004. Initiates the study of greedy embedding.

• [Leighton08] T. Leighton and A. Moitra. Some results on greedy embeddings in metric spaces. FOCS08. The Ratajczak-Papadimitriou conjecture is proved.

• [Newsome03] James Newsome, Dawn Song, GEM: Graph EMbedding for Routing and Data-Centric Storage in Sensor Networks Without Geographic Information, Sensys'03. Virtual coordinates constructed by using a spanning tree.

• [Caesar06] M. Caesar, M. Castro, E. B. Nightingale, G. O'Shea, A. Rowstron, Virtual Ring Routing: Network Routing Inspired by DHTs, Sigcomm’06. Routing with 1D coordinates.


Greedy embedding of a graphGreedy embedding of a graph

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


Greedy embeddingGreedy embedding

• Given a graph G, find an embedding of the vertices

in Rd, s.t. for each pair of nodes s, t, there is a neighbor of s closer to t than s itself.

s

t


Questions to askQuestions to ask

• We want to find a virtual coordinates such that

greedy routing always works.

• Does there exist such a greedy embedding in R2?

• in R3?

• in Euclidean metric? Hyperbolic space?

• If it exists, how to compute?


Greedy embedding does not always existGreedy embedding does not always exist

• K1,6 does not have a greedy embedding in R2


A lemmaA 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.


ProofProof

• K1,6 does not have a

greedy embedding in R2

Proof:

1. One of the angles is

less than π/3.

2. One of ab2 and ab3, say,

ab2, is longer than b2b3.

3. Then b2 does not have edge with its closest

point b3.


A conjectureA conjecture

• Corollary: Kk, 5k+1 does not have a greedy

embedding in R2.

• Conjecture: Any planar 3-connected graph has a

greedy embedding R2.

• Hint: this is tight.

• K2,11 is planar but not 3-connected.

• K3.16 is 3-connected but not planar. (it has K3.3

minor).

• Planar 3-connected graph has a greedy embedding

in R3


Polyhedral routingPolyhedral routing

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.

Theorem: Any 3-connected planar graph has a greedy

embedding e in R3, where the distance function is

defined as d(u, v) = - e(u)⋅e(v).


Polyhedral routingPolyhedral 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.

3. This contradicts with the

convexity of the polytope.

t

s

v


DiscussionsDiscussions

• Papadimitriou’s conjecture: Any planar 3-connected graph has a greedy embedding R2. has been proved!

• The theorem only gives a sufficient condition, not necessary.

– K3.3 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 R2? Is this problem hard? (Recall that many such embedding problems are hard…)

• More understanding of greedy embedding in R2, R3…


FollowFollow--up workup 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.


Leighton and Leighton and MoitraMoitra

• All 3-connected planar graph contain a spanning

Christmas Cactus graph.

• All Christmas Cactus graphs admit a greedy embedding

in the plane.


Leighton and Leighton and MoitraMoitra

• 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.


A Christmas CactusA Christmas Cactus


ExampleExample

Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

Ankur Moitra () Geometric Graph Theory October 26, 2008 36 / 82

Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

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5b

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7b

1b

2b

3b4b

5b

6b

7b


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

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3b

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7b

1b

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3b4b

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s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

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s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example I

1a

2a

3a

4a

5a

6a

7aR+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t


Example II

4a

1a

2a

3a

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c


Example II

4d

3c

3d

4c

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b

1b

3a

3b

4a

4b

1d

1c1a

2a

3a

4a

1c1d

2c

2d


Example II

4d

3c

3d

4c

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b

1b

3a

3b

4a

4b

1d

1c1a

2a

3a

4a

1c1d

2c

2d


Example II

2a

2b 2d

3b


Example II

3b

2a

2b 2d


Example II

3b

2a

2b 2d


Example II

4a

1a

2a

3a

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

2dt

s2b

2a


Example II

2dt

s2b

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

3c

2a

2b 2d


Example II

3c

2a

2b 2d


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a


General Christmas Cactus Graphs

G



G



G



G



G



G



G



G



G



Connection to graph labelingConnection 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(n1/2) (Ω(n1/3)) for planar

graphs, and Θ(log2n) for trees.

• General graph: ∃ a scheme with label size O(kn1/k) and approximation ratio 2k-1.

• Google “distance labeling” for the literature.


Approach II:Approach II:

Embed a spanning tree in polar coordinate Embed a spanning tree in polar coordinate

systemsystem


Embed a tree in polar coordinate systemEmbed 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.


Embed a tree in polar coordinate systemEmbed 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.


Routing on a treeRouting 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.


Routing on a treeRouting 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.


Load balancingLoad balancing

• Root is still the bottleneck even for smart routing.

Shortest path routing, still not

the most load balanced routing


Routing on spanning trees Routing on spanning trees –– in theory and in in theory and in

practicepractice

• 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).


General frameworkGeneral framework

• Find a substructure such that we can define virtual

coordinates.

– The virtual coordinates guarantee delivery.

– Greedy routing in the connectivity graph.


Virtual ring routingVirtual ring routing

Some slides from http://research.microsoft.com/en-us/um/people/antr/vrr-

sigcomm06.ppt


VRR: the virtual ringVRR: the virtual ring

8F690E

910

8F08E2

0FFFtopology-independentnode identifiers, e.g.,

MAC address

nodes organized into virtual ringby increasing identifier value

each node maintainsa virtual neighbor set (vset)


VRR: routing pathsVRR: routing paths

physical network topology

8F6

8F6

nodes only maintain paths to virtual neighbors:

• vset-paths are typically multi-hop

• vset-paths are maintained proactively


VRR: forwarding tableVRR: forwarding table

• vset-paths recorded in forwarding tables along path • forwarding table contains

• vset-paths between node and vset members

• vset-paths between other nodes that go through node• paths to physical neighbors

endpointB pathIdnextA nextBendpointA

10E 10

ME F42 318F6 90E910

forwarding table for node 8F6

8F6 ME

14A 140 F42 10E 2

8F6 F42 ME F42 FF

14A

140

F42

10E

8F6


VRR: forwardingVRR: 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


VRR: example routingVRR: example routing

physical network Topology


VRR: example routingVRR: example routing

physical network Topology

there may be

some stretch


Node joiningNode joining

Network Topology

19A

164

8F6

16E

broadcast hellos

to find physical neighbors

send setup request to 16E

(itself) through proxy (19A)


Node joiningNode joining

Network Topology

19A

164

8F6

16E

171

164 sends setup to 16E

with its vset

16E sends setup requests

to nodes in received vset

16E adds node to vset

when it receives setup


Size of routing tableSize of routing table

• Assume the nodes are randomly placed,

each vpath


Simulation experiments in nsSimulation experiments in ns--22

• ran experiments with 802.11b 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


Delivery ratio: fast movementDelivery ratio: fast movement

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200

Number of nodes

Deli

very

rati

o

DSDVDSRAODVVRR


Delay: fast movementDelay: fast movement

0

2

4

6

8

10

12

14

0 50 100 150 200

Number of nodes

De

lay (

sec

on

ds

)

DSDV

DSR

AODV

VRR


Sensor networkSensor network

• sensor network testbed

– 67 mica2dot motes in UCB building

• comparison with BVR

• delivery ratio with mote failures


Sensor network: mote failuresSensor network: mote failures

50

60

70

80

90

100

110

120

130

0 1 2 3 4 5 6 7 8 9 101112131415Time (mins)

Nu

mb

er

of

no

des

80

82

84

86

88

90

92

94

96

98

100

Number of nodes

VRR delivery ratio

BVR delivery ratio

Pe

rce

nta

ge

of

pa

ck

ets

de

liv

ere

d


Wireless office Wireless office testbedtestbed

• 30 machines running windows

• communicate using 802.11a

• throughput comparison with LQSR using ttcp


Wireless office Wireless office testbedtestbed: throughput: throughput

0

2

4

6

8

10

12

Machine

Ba

nd

wid

th (

Mb

ps

)

MR-LQSR

VRR


Why a virtual ring?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.

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