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Controller and Estimator for Dynamic Networks. Amos Korman Shay Kutten Technion. Motivation. Many known algorithms are static . However , in most realistic contexts, and especially distributed contexts, ( the Internet, peer to peer networks etc.) setting is dynamic :. Remove edge.

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controller and estimator for dynamic networks

Controller and Estimator for Dynamic Networks

Amos Korman Shay Kutten

Technion

motivation
Motivation

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

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

Remove edge

Remove node

Add node

Add edge

motivation cont
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.

C

A removed

A

D

B

basic update problems
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
Dynamic models

For simplicity, in this talk, we assume the

Serialized model : a topological change

occurs only after all updates concerning

previous topology changes have occurred.

In fact, the protocols work also under the

Controlled model [Afek et at.],

in which topology changes

may occur concurrently, as long as we can

delay for arbitrary (but finite) time periods.

The Controlled model, may be useful in

overlay networks

related work
Related work

Afek, 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 that

the 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
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.

u

control protocol

Request

signal

Messages are sent to update

nodes

v

permit or reject

an m w controller requirements
An (M,W)-controller : Requirements:

Safety:

At most M permits are given.

Liveness:

If the controller gives a reject then

at leastM-W permits were given

(W is the waste)

slide9

Controller knows how to stop when the

# of permits is between M and M-W

M-W

M

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

trivial controller
Trivial controller

M permits

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

request

reduction from size estimation and name assignment to controller
reduction from size-estimation and name assignment to controller

E

(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]).

the m w controller of aaps
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 withO(log2n) amortized message complexity.

M

W+1

new extended m w controller
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( )).

M

W+1

size estimation and name assignment in extended dynamic model
Size estimation and name assignment in extended dynamic model

Constant size estimation with

amortized messagecomplexity=O(log2n).

Mainiatining unique identities

usinglog n+O(1) bits per identity andO(log2n)amortized messagecomplexity.

remark
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
Extended (M,W)-controller

M permits

root sends packages of different sizes containing permits.

Total # permits sent:

no more thanM.

ROOT

large package

small package

safety
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 controller1
Extended (M,W)-controller

M permits

ROOT

root sends packages of different sizes containing permits.

Level i package contains precisely

ρ2i permits

i

Level 0 package contains between

1 and ρ permits

0

slide19

The algorithm

ROOT

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

0

request

(to add a child)

request

slide20

ROOT

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.

Pi

P

0

request

request

(to delete the node)

slide21

If no level 0 package at u

ROOT

0

Looking fora

level-0 package

at distance

between

0 and 2Ψ.

Issue permit

u

request

slide22

ROOT

Looking fora

level-i package

at distance between

2i Ψand 2i+1 Ψ

i

u

request

slide23

root

24Ψ

Look for level-3

23Ψ

Look for level-2

22Ψ

Look for level-1

Look for level-0

request

U

slide24

root

3

24Ψ

If not find, then a package of the

appropriate size is issued at the root

(unless it issued already M permits)

Look for level-3

23Ψ

Look for level-2

22Ψ

Look for level-1

Look for level-0

request

U

slide25

24Ψ

3

23Ψ

Move & split

22Ψ

request

slide26

24Ψ

3

23Ψ

2

No other level-2 package

22Ψ

1

0

request

0

correctness
Correctness

Safety: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.

.

M permits were sent

ROOT

At most W are stuck

At least M-W were given

the waste is at most w
The waste is at most W

ROOT

Level i package contains precisely

ρ2i permits

j

i

i

slide29

24Ψ

3

23Ψ

No other level-2 package

2

Domain

22Ψ

1

0

request

0

domain invariants
Domain invariants

1) Domain of level-i package is of size ~2iΨ

2) Domains of two level-i packageare disjoint.

i

i

i

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

what happens to a domain when a topology change occurs
What happens to a domain when a topology change occurs?

i

Domain

When a node leaves a domain it is

still considered as part of the domain

i

Domain

When 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 tokens

# of wasted permits in all level-i packages isn (ρ/Ψ).

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

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

Altogether, wasted permits is at mostW.

slide33

24Ψ

3

communication:

Need only bound move of packages

23Ψ

Move & split

Search for package

22Ψ

request

communication cont
Communication – cont.

Fix level i.

Apermitbelongs to at most

one level i package.

3

2

1

0

request

0

communication cont1
Communication –cont.

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))).

conclusion
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?