Information Dissemination in

Highly Dynamic Graphs

Regina O’DellRoger Wattenhofer

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 2

Motivation

• Highly Dynamic Networks– Mobility

• zebra herd, flock of birds, cars, ...

– Stationary nodes BUT unstable links• Roof-top network

• Sensor network in changing environment

• Flooding– High mobility reactive protocol flooding element– Guaranteed information delivery– Fundamental ingredient

• Routing

• Service discovery

• Sensor network management

• …

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 3

Motivation – Previous Work

• Simulations– Lots of mobility models– Algorithm correctness depends on model?

• “Coarse-grained” mobility– Periods of stability– Link failures between completed route requests

• Worst-case analysis [Awerbuch et al, 2005]

– Assumptions on path stabilities• Graph restrictions

– Unit Disk Graph (UDG)– Quasi-UDG– “bounded-growth” graphs

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 4

Our View

“fine-grained”, theoretical worst-case mobility analysis

We ask:- how mobile can the network,- how limited can the system

be such that flooding is still possible?

anything?

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 5

Outline

• Model

– Network, mobility, algorithm

• Flooding

– Knowing |V|

– Storing IDs

– Finding max ID

• Impossibility Conjecture

• Routing

• Outlook

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 6

Model – “Environmental Challenges”

• Graph Gt = (V,Et), V same,

– Gt connected for all times t

– Time between N(v) changes ¸ T for all nodes v– T = max message transmission time

• Broadcast medium• Negligible local processing time• Asynchronous message transmissions

– x 2 N(v) during entire transmission• Events:

– Message receipt– Neighborhood change (N(v) 0)

Obs: Transmission at v will reach some node after at most 2T time.

arbitrary changes

Node w enters N(v)

w should receive message from v

“lost messages” possible

simultaneously

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 7

terminatecorrect

Goals – “Engineering Constraints”

do not know |V|

O(log |V|)

space

task dependent no more messages sent

no upper bound!

– storage, header– unique small IDs

N(v)

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 8

“3 out of 4” Flooding

terminatecorrect

do not know |V|

O(log |V|)

space

send nothing

send forever

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 9

COUNTERFLOODING

• Assume: know n ¸ |V| (polynomial upper bound)• Algorithm

counter kv = 0

retransmit message when N(v) ;, inc kvwhile kv < 2n

• Proof idea– Border intact normal flooding– Otherwise N(v) at border node v

• Comments– Reaches n nodes in time O(n)– Explicit termination?

• Synchronous: easy!

N

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“3 out of 4” Flooding

terminatecorrect

do not know |V|

O(log |V|)

space

estimate |V|

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 11

LISTFLOODING

• Assume: store & send O(n) IDs• Algorithm

list lv of known nodes, set nv = |lv|

receive lw merge: lv = lv Å lw if |lv| > nv

• nv = |lv|

• COUNTERFLOODING( f(nv),lv )

• Proof idea– Set of flooding nodes increases

– Or: lmax increases

• Comments– Correct in time O(n2) for f(n) = n + 1

• probably in O(n) for f(n) = 2n

f(n) =– n + 1– 2n– …

lmax

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Flooding

terminatecorrect

do not know |V|

O(log |V|)

space

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IDFLOODING

• Idea: find max ID upper bound on |V|• Algorithm

store nv = max ID seen

receive ID w if w > nv

• nv = w

• COUNTERFLOODING( f(nv),nv )

• Proof idea– Same principle as LISTFLOODING

– Max ID of flooding nodes will grow

• Comments– Needs unique polynomial IDs– Intuition: IDs encode information about |V|

IDs strong assumption!

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Flooding – No IDs

• General idea

– Receive new information about graph (|lv| > nv, w > nv)

– Update estimate nv

– Restart COUNTERFLOODING with f(nv)

• Counter example (idea)

• General argument?

• Guessing ID?

– dynamic naming/initialization problem

– randomized

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Routing

• What about routing? possible!– Destination: send ACK– Initiates “termination” phase

• Idea: 2 modescounter nv

FLOOD: inc nv, send message

• every N(v)

• if received n’ > nv update nv

TERM: COUNTERFLOODING(nv)

• if received n’ > nv update nv

• restart COUNTERFLOODING(nv)

• Correct in time O(n)– Actually: “optimal path”

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Outlook

• First step– Theoretical analysis possible– More general mobility model

• Lots of open questions– Impossibility of flooding?– Does randomization help?– Explicit termination?

• Local dynamic synchronizer

– Even more mobility• Nodes join and leave

– Even less mobility• Restricted link changes

• Timing assumptions

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs

DistributedComputing

Group

Questions?Comments?

Thank You!

DIALM-POMC 2005 Flooding in Highly Dynamic Graphs 18

Goals – “Engineering Constraints”

• Requirements:– Correctness (task dependent)– Termination (no messages sent)

• Conditions:– n = |V| unknown, nor any upper bound– O(log n) overhead (storage, message header)

• Unique O(log n)-bit IDs

• Neighborhood table prohibitively expensive

• Idea: – Separate conditions analyze effect

must haves

design, cost, environment, …