Controller and Estimator

for Dynamic Networks

Amos Korman Shay Kutten

Technion

MotivationMany known algorithms are static. However, in most realistic contexts, and especially distributed contexts,

(the Internet, peer to peer networks etc ).setting is dynamic:

Add node

Remove edge

Add edge

Remove node

Motivation – cont.

Therefore, for a distributed scheme to be useful, it should be capable of reflecting up-to date information in dynamic setting,

which may require occasional updates.

A

B

C

D

A removed

Basic update problems

Size-estimation : some center node

maintains an approximation of # nodes.

Name assignment : maintain at each node u, a unique short identity id(u). )Typically O)log n( bits, n is current # nodes(.

Dynamic models

For simplicity, in this talk, we assume the Serialized model : a topological changeoccurs only after all updates concerningprevious topology changes have occurred.

In fact, the protocols work also under the

Controlled model [Afek et at.], in which topology changesmay occur concurrently, as long as we candelay for arbitrary )but finite( time periods.The Controlled model, may be useful in

overlay networks

Related workAfek, Awerbuch, Plotkin, and Saks showed (J. of ACM) how to solve the size-estimation and name-assignment problems on growing trees using O(log2n) amortized message complexity ,

per topology change. They assumed thatthe tree can only grow and

only by allowing leaves to join.

To solve the problems, they use a machinery called (M,W)-CONTROLLER

An (M,W)-controller

Requests arrive )from environment( to nodes. Each request is eventually either granted

a permit or rejected .

If a request is to perform a topology change is granted a permit then the change occurs .

uRequest

v

signal control protocolMessages are sent to update

nodes

permit or reject

An (M,W)-controller : Requirements:

Safety:

At most M permits are given.

Liveness:

If the controller gives a reject then

at least M-W permits were given

(W is the waste)

MM-W

Controller knows how to stop when the #of permits is between M and M-W

)in case w=0, the controller stops after precisely M permits were given(.

Trivial controller

Whenever a vertex u asks for a request ,

a signal is sent to the root .

In turn, the root returns a permit to u ,

unless is has already given M permits.

If the root has already given M permits ,

it returns reject to u.

Problem: message complexity Ω(Mn).

ROOT

M permits

request

reduction from size-estimation and name assignment to controller

(n/2,n/4-)controller with O)π( amortized message complexity

size estimation and name assignment protocols with O)π( amortized message complexity.

(Even if the number of topology changes is not bounded )using iterations( [Afek et. Al] .)

E

The (M,W)-controller of [AAPS]

Can operate on a growing tree allowing

only leaves to join the tree.

Has O)n·log2n·log ) ( message complexity .

(n is the final number of nodes)

Therefore, if W=M/2 then their controller can solve the size estimation and name assignment problems with O)log2n(

amortized message complexity .

W+1M

New Extended (M,W)-controller

In this paper, we give an

extended )M,W(-controller operating

under a more general model allowing

both additions and deletions of

both leaves and internal nodes.

Same amortized

message complexity: O)log2n log) ((. W+1M

Size estimation and name assignment in extended

dynamic model

Constant size estimation with

amortized message complexity=O)log2n(.

Mainiatining unique identities

using log n+O)1( bits per identity and O)log2n( amortized message complexity.

Remark The behavior of node v in the controller of AAPS depends strongly on the depth of v which does not change in their scenario.

Therefore it is not clear how to adapt the previous controller

to the more general dynamic setting.

ROOT

Extended (M,W)-controller

ROOT

M permits root sends packages of different sizes containing permits.Total # permits sent:no more than M.large package

small package

Safety

The root does not send more than M permits .

If it has sent M permits then

it broadcasts a reject message to all nodes.

Message complexity resulting from this

`reject’ broadcast is O(n).

Extended (M,W)-controller

ROOT

M permits

0

root sends packages of different sizes containing permits.

i

Level i package contains preciselyρ2i permits

Level 0 package contains between

1 and ρ permits

ROOT

requestrequest (to add a child)

0

One permit from P is given to the request. Subsequently: a) size(P)=size(P)-1, b) a child is added.

If size(P)=0, P is canceled.

P

The algorithm

ROOT

requestrequest)to delete the node(

0

One permit from P is given to request. Subsequently: a) size(P)=size(P)-1, b) all packages move to parent.c) the node is deleted.

P

Pi

ROOT

request

0Looking for a level-0 package

at distance between0 and 2Ψ.

Issue permit

u

If no level 0 package at u

ROOT

request

iLooking for a level-i package

at distance between2i Ψ and 2i+1 Ψ

u

request

2Ψ

22Ψ

23Ψ

24Ψroot

Look for level-0

Look for level-1

Look for level-2

Look for level-3

U

request

2Ψ

22Ψ

23Ψ

24Ψroot

Look for level-0

Look for level-1

Look for level-2

Look for level-3

U

3If not find, then a package of the

appropriate size is issued at the root)unless it issued already M permits(

request

2Ψ

22Ψ

23Ψ

24Ψ

3

Move & split

request

2Ψ

22Ψ

23Ψ

24Ψ

3

1

0

2

0

No other level-2 package

CorrectnessSafety: The root does issue more that M permits.

Liveness : If a request is rejected, and at most W are stuck in packages then # granted requests is at least M-W .

. ROOT

M permits were sentAt most W are stuck

At least M-W were given

The waste is at most W

ROOT

Level i package contains preciselyρ2i permits i

j

i

request

2Ψ

22Ψ

23Ψ

24Ψ

3

1

0

2

0

No other level-2 package

Domain

Domain invariants1 (Domain of level-i package is of size ~2iΨ

2 (Domains of two level-i package are disjoint.

i

Therefore, # of level-i packages is at most n/ 2iΨ

i

i

What happens to a domain when a topology change

occurs?i

DomainWhen a node leaves a domain it is

still considered as part of the domain

i

DomainWhen a node joins a domain it is considered as part of the domain

and the bottom node leave the domain

#of wasted tokens #of wasted permits in all level-i packages is

n (ρ/Ψ).

We fix ρ and Ψ so that ρ/Ψ= W/(n log n).

Therefore wasted permits in level-i packages ≤ W/log n.

Altogether, wasted permits is at most W.

request

2Ψ

22Ψ

23Ψ

24Ψ

3

Move & split Search for package

communication:Need only bound move of packages

Communication – cont.

Fix level i. A permit belongs to at most

one level i package.

request

3

1

0

2

0

At most M/(size(i))=M/2iρ level-i packages ever exist.Each level-i package travels to distance O(2iΨ).

Total messages incurred by level-i packages ≤ O(M(Ψ/ρ)) = O(n·log n·(M/W)).

Summing over all levels: # messages is O(n·log2n·(M/W)).

Using iterations, reduce to O(n·log2 n·log(M/(W+1))).

Communication –cont.

conclusion

The field of dynamic distributed algorithms brings many challenging and important problems. )In particular, transform known static schemes to dynamic ones.(

We managed to solve the size estimation and dynamic name assignment problems using O)log2n( amortized massage complexity.

Can we do better?

THANK YOU