GMPLS

Label

Space

Minimization

through

Hypergraph

Layouts

Abstract:

All-Optical

Label

Switching

(AOLS)

is

a

new

technology

that

performs

packet

forwarding

without

any

optical-electrical-optical

conversions.

In

this

report,

we

study

the

problem

of

routing

a

set

of

requests

in

AOLS

networks

using

GMPLS

technology,

with

the

aim

of

minimizing

the

number

of

labels

required

to

ensure

the

forwarding.

We

rst

formalize

the

problem

by

associating

to

each

routing

strategy

a

logical

hypergraph,

called

a

hypergraph

layout,

whose

hyperarcs

are

dipaths

of

the

physical

graph,

called

tunnels

in

GMPLS

terminology.

We

dene

a

cost

function

for

the

hypergraph

layout,

depending

on

its

total

length

plus

its

total

hop

count.

Minimizing

the

cost

of

the

design

of

an

AOLS

network

can

then

be

expressed

as

nding

a

minimum

cost

hypergraph

layout.

We

prove

hardness

results

for

the

problem,

namely

for

general

directed

networks

we

prove

that

it

is

NP-hard

to

nd

a

C

log

n-approximation,

where

C

is

a

positive

constant

and

n

is

the

number

of

nodes

of

the

network.

For

symmetric

directed

networks,

we

prove

that

the

problem

is

APX-hard.These

hardness

results

hold

even

if

the

trac

instance

is

a

partial

broadcast.

On

the

other

hand,

we

provide

approximation

algorithms,

in

particular

an

O(log

n)-approximation

for

symmetric

directed

networks.

Finally,

we

focus

on

the

case

where

the

physical

network

is

a

directed

path,

providing

a

polynomial-time

dynamic

programming

algorithm

for

a

xed

number

k

of

sources

running

in

O(n

time.Key-words:

GMPLS,

optical

networks,

label

stacking,

hypergraph

layout,

approximation

algorithms,

dynamic

programming.

All-Optical

Label

Switching

(AOLS)

is

an

approach

to

route

packets

transparently

and

all-optically,

thus

allowing

a

speed-up

of

the

forwarding.

This

very

promising

technology

for

the

future

Internet

applications

also

brings

new

constraints

and

new

problems.

Indeed,

since

the

forwarding

functions

are

implemented

directly

at

the

optical

domain,

a

specic

correlator

(device)

is

needed

for

each

optical

label

processed

in

the

node.

Therefore,

it

is

of

major

importance

to

reduce

the

number

of

employed

correlators

in

every

node,

implying

a

reduction

in

the

number

of

labels

(as

referred

in

the

rest

of

the

report).

Due

to

its

exibility

as

a

control

plane

and

to

the

fact

that

it

handles

trac

forwarding,

the

Generic

MultiProtocol

Label

Switching

(GMPLS)

is

the

most

promising

protocol

to

be

applied

in

AOLS-driven

networks.In

GMPLS,

trac

is

forwarded

through

logical

connections

called

Label

Switched

Paths

(LSPs).

When

GMPLS

is

used

with

packet-based

network,

packets

are

associated

to

LSPs

by

means

of

a

label,

or

tag,

placed

on

top

of

the

header

of

the

packet.

In

this

way,

routers

-called

Label

Switched

Routers

(LSRs)

-can

distinguish

and

forward

packets.The

GMPLS

standards

allow

packets

to

carry

a

set

of

labels

in

their

header,

conforming

a

stack

of

labels.

Even

though

a

packet

may

contain

more

than

one

label,

LSRs

must

only

read

the

rst

(or

top)

label

in

the

stack

in

order

to

take

forwarding

decisions.

This

helps

to

reduce

both

the

number

of

labels

that

need

to

be

maintained

on

the

core

LSRs

and

the

complexity

of

managing

data

forwarding

across

the

backbone.Stacking

labels

and

label

processing,

in

general,

are

standardized

by

the

following

set

of

operations

that

an

LSR

can

perform

over

a

given

stack

of

labels:SWAP:

replace

the

label

at

the

top

by

a

new

one,

PUSH:

replace

the

label

at

the

top

by

a

new

one

and

then

push

one

or

more

onto

the

stack,

and

POP:

remove

the

label

at

top

in

the

label

stack.The

labels

stored

in

the

forwarding

table

are

signicant

only

locally

at

the

node

and

swapped

all

along

the

LSP

(see

Figure

Solutions

deployed

by

GMPLS

for

reducing

the

number

of

labels

are

label

merging

(not

discussed

here)

and

label

stacking

.With

label

stacking,

when

two

or

more

LSPs

follow

the

same

set

of

links,

they

can

be

routed

together

inside

a

higher-level

LSP,

henceforth

a

tunnel.

In

order

to

setup

a

tunnel,

multiple

labels

are

placed

in

the

packet's

header.

Figure

represents

the

general

operations

needed

to

congure

a

tunnel

with

the

use

of

label

stacking.

At

the

entrance

of

the

tunnel,

λ

PUSH

are

performed

in

order

to

route

the

λ

ows

or

units

of

trac

through

the

tunnel.

Then,

only

one

operation

(either

a

SWAP

or

a

POP

at

the

end

of

the

tunnel)

is

performed

in

all

the

nodes

along

the

tunnel,

regardless

of

λ.

In

this

gure,

a

stack

of

size

is

used

to

route

the

λ

units

of

trac

in

one

tunnel

from

node

A

to

node

E.

The

top

label

l

is

swapped

and

replaced

at

each

hop:

by

l

at

node

B,

by

l

at

node

C,

and

is

nally

popped

at

node

D.

tunnel

at

node

E

can

end

or

follow

dierent

paths

according

to

their

bottom

label

k

i

for

all

i

λ}

in

the

stack.A

consequence

of

the

way

in

which

the

GMPLS

operations

can

be

congured

at

LSRs

is

the

following:

trac

can

enter

in

any

node

of

a

tunnel

but

can

exit

in

only

one

point,

the

last

node

of

the

tunnel.

In

other

words,

when

some

trac

is

carried

by

a

tunnel,

it

follows

the

tunnel

until

its

end.Since

the

number

of

labels

used

for

GMPLS

forwarding

aects

the

cost

of

the

AOLS

architecture,

in

this

report

we

mainly

focus

on

the

minimization

of

the

number

of

labels

used.

In

our

previous

example,

the

total

cost

c(t)

of

the

tunnel

t

from

node

A

to

node

E

in

terms

of

number

of

labels

is

c(t)

λ

(t)

where

λ

is

the

number

of

units

of

trac

forwarded

through

this

tunnel

and

(t)

is

its

length

in

terms

of

number

of

hops

(which

is

on

this

example).

We

will

formally

dene

the

cost

function

of

the

problem

in

Section

work

and

our

contribution

The

label

minimization

problem

in

GMPLS

networks

has

been

widely

studied

in

the

literature

during

the

last

few

years

All

these

articles

focus

mainly

on

proposing

and

analyzing

heuristics

to

the

problem,

but

there

is

a

lack

of

theoretical

results,

like

computational

complexity

or

bounds

on

the

approximation

ratio

of

the

proposed

algorithms.Gupta,

Kumar,

and

Rastogi

study

the

trade-o

between

the

stack

depth

and

the

label

size

in

tag

switching

protocols

(see

They

calculate

lower

bounds

for

dierent

problem

instances.

Amongst

these

problems,

they

propose

solutions

using

at

most

n

labels

when

the

network

is

a

path

and

2∆

n

when

it

is

a

tree,

where

n

is

the

number

of

nodes

and

the

maximum

degree

of

the

topology.

However,

the

proposed

solutions

lay

on

the

assumption

of

non-practical

label

distribution

protocols

for

MPLS.In

this

article

we

provide

the

rst

theoretical

framework

for

the

label

minimization

problem

in

general

GMPLS

networks

considering

the

constraints

imposed

by

real

distribution

protocol,

e.g.,

RSVP-TE.

We

translate

the

problem

into

nding

a

set

of

dipaths

in

a

directed

hypergraph.

With

this

new

formulation,

it

turns

out

that

the

problem

is

very

similar

to

classical

Virtual

Path

Layout

(VPL)

problems

originating

from

ATM

networks.

We

provide

hardness

results

and

approximation

algorithms

for

the

problem

in

paths,

trees,

and

general

graphs.

The

approximation

algorithms

strongly

rely

on

the

already

known

algorithms

for

VPL

problems.

Finally,

we

focus

on

the

path

topology,

providing

a

dynamic

programming

approach

that

runs

in

polynomial

time

for

any

bounded

number

of

sources.Organization

of

the

report

In

Section

we

formally

state

the

problem

in

terms

of

hypergraph

layouts

and

x

the

notation

to

be

used

throughout

the

article.

In

Section

prove

that

for

general

directed

networks

it

is

NP-hard

to

nd

a

C

log

n-approximation,

where

C

is

a

a

positive

constant

and

n

is

the

number

of

nodes

of

the

network.

For

symmetric

directed

networks,

we

prove

that

the

problem

is

APX-hard

and

therefore

it

does

not

accept

a

PTAS

unless

P=NP.

In

Section

we

provide

approximation

algorithms

to

the

problem

for

both

general

and

symmetric

networks,

and

discuss

the

gap

with

the

hardness

results.In

Section

we

focus

on

the

directed

path

topology

and

present

a

dynamic

programming

approach

solving

the

problem

in

polynomial

time

when

the

number

of

sources

is

xed.Finally,

Section

is

devoted

to

conclusions

and

further

research.Some

parts

of

this

report

have

been

presented

in

conferences

PTAS

denotes

the

class

of

problems

admitting

a

polynomial

time

approximation

scheme

that

is

guaranteed

to

nd

a

solution

whose

cost

is

within

a

ε

factor

of

the

optimum

cost,

for

any

ε

When

the

solution

is

only

guaranteed

within

a

constant

factor

of

the

optimum

cost,

then

the

problem

belongs

toAPX.

An

APX-hard

problem

does

not

accept

a

PTAS,

unless

P=NP

(see

for

instance

The

logical

network

design

problem

that

we

address

can

be

roughly

described

as

follows:

we

are

given

a

digraph

(directed

graph)

G

together

with

a

set

of

trac

demands

(or

requests)

between

couples

of

vertices

in

G,

and

we

must

nd

a

set

of

tunnels

of

minimum

cost

allowing

and

a

routing

upon

this

set

of

tunnels

for

all

the

trac

requests.

Note

that

usually

communication

networks

are

symmetric

digraphs

(i.e.

when

operators

set

a

link

in

one

direction,

they

also

set

the

opposite

link).

So

it

is

interesting

to

study

the

symmetric

case,

which

turns

out

to

be

computationally

easier

than

the

general

directed

case.

Let

us

now

precise

each

one

of

the

above

terms.A

tunnel

is

simply

a

directed

path

(or

dipath)

in

G,

and

due

to

the

technological

constraints

discussed

in

Section

trac

can

enter

anywhere

in

the

tunnel

but

must

leave

only

at

the

end

of

the

tunnel.

To

dene

the

problem

formally

we

need

the

following

notation:•

G

(V,

E)

is

the

underlying

digraph

(which

can

be

symmetric

or

not)

with

|V

n.•

r

i,j

is

the

request

from

node

i

V

to

node

j

V

with

multiplicity

m

i,j

(the

trac

being

dierentiated,

m

i,j

represents

the

number

of

dierent

ows

from

i

to

j

and

j

is

not

necessarily

the

nal

destination

of

the

request

r

i,j

R

is

the

set

of

all

requests.•

P

(G)

is

the

set

of

all

simple

dipaths

in

G.•

t

stands

for

a

tunnel,

and

T

is

the

set

of

tunnels,

that

is

t

T

P

(G).•

is

a

length

function

on

the

arcs,

that

is

E

N

A

tunnel

t

has

length

(t)

e∈t

(e)

and

carries

w(t)

ows,

or

as

referred

in

the

rest

of

the

report,

w(t)

units

of

trac.Note

that,

a

priori,

w(t)

depends

on

the

routing

policy.

The

cost

c(t)

of

a

tunnel

t

is

then

c(t)

w(t)

(t)

and

the

cost

of

a

set

of

tunnels

T

isEach

tunnel

can

be

modeled

as

a

directed

hyperarc

on

the

vertex

set

of

G.

This

observation

naturally

leads

to

the

denition

of

a

hypergraph

layout.Denition

(Hypergraph

layout).

Given

a

graph

G

and

a

set

Given

a

hypergraph

layout

H(T

let

L(r

i,j

be

the

number

of

hyperarcs

that

request

r

i,j

uses,

and

let

d

H

(i,

j)

be

the

distance

from

vertex

i

to

vertex

j

in

H(T

Then

the

term

t∈T

w(t)

of

Equation

can

be

rewritten

as

ri,j

∈R

L(r

i,j

m

i,j

and,

since

L(r

i,j

d

H

(i,

j),

we

conclude

that

in

an

optimal

solution

the

routing

necessarily

uses

shortest

dipaths

in

the

hypergraph

layout.

It

follows

that

the

cost

function

of

Equation

can

be

rewrittenThe

cost

of

a

solution

is

of

bicriteria

nature.

The

rst

part

is

the

cost

of

the

hypergraph

structure;

we

call

it

the

total

length

of

the

layout.

The

second

part

is

the

total

distance

that

the

trac

travels

in

the

hypergraph;

we

call

it

the

total

hop

count.

Both

cost

function

parts

are

very

much

conicting.

On

the

one

hand,

to

minimize

the

hop

count,

it

is

enough

to

take

a

tunnel

connecting

any

source

to

any

destination.

On

the

other

hand,

to

minimize

the

total

length

of

the

layout,

it

is

enough

to

consider

a

minimum

arc-weighted

hypergraph

H

such

that

for

each

request

r

i,j

R,

vertices

i

and

j

lie

on

the

same

connected

component

of

H.

Summarizing,

the

problem

can

be

stated

as

follows.Minimum

Cost

Hypergraph

Layout:

Given

a

digraph

G

with

a

length

function

and

a

set

R

of

trac

requests,

nd

a

feasible

hypergraph

layout

of

minimum

cost,

where

the

cost

of

a

hypergraph

layout

is

dened

as

in

Equation

G

is

a

symmetric

digraph,

the

problem

is

denoted

Minimum

Cost

Symmetric

Hypergraph

Layout.

We

note

that

the

cost

function

of

Equation

can

be

naturally

generalizedwhere

α

and

β

are

positive

constants

and

p(t)

is

a

general

cost

function

p

:The

cost

function

of

Equation

corresponds

to

p(t)

c(t)

(t)

of

a

solution

for

the

example

of

Figure

Consider

the

path

[s,

with

one

source

s,

m

units

of

trac

destined

to

node

at

distance

from

s

nodes

between

s

and

and

m

units

of

trac

destined

to

node

at

distance

from

s.

See

Figure

for

an

illustration.

The

optimal

solution

depends

on

the

values

i

and

m

s,i

Indeed,

two

solutions

have

to

be

examined.

In

the

rst

solution,

a

specic

tunnel

(s,

i)

is

congured

for

each

destination

i,

giving

two

tunnels

(s,

and

(s,

with

a

total

cost:

(m

m

+m

The

second

solution

is

composed

of

the

two

tunnels

(s,

and

The

requests

destined

to

will

rst

use

the

tunnel

(s,

and

then

the

tunnel

We

state

now

a

lemma

to

be

exhaustively

used

in

the

sequel,

asserting

that

there

always

exists

an

optimal

solution

such

that

all

the

trac

units

of

each

request

are

routed

via

a

unique

set

of

consecutive

tunnels,

or

dipath.

Lemma

In

any

network,

there

exists

an

optimal

solution

to

the

Minimum

Cost

Hypergraph

Layout

problem

such

that

all

the

trac

units

of

each

request

are

routed

via

a

unique

dipath.Proof.

Let

r

i,j

be

a

request

from

node

i

to

node

j,

and

let

H

be

an

optimal

solution.Suppose

that

the

units

of

trac

arriving

at

j

from

i

are

routed

in

H

via

p

dierent

dipaths

P

P

p

Let

λ

m

m

p,

be

the

number

of

trac

units

forwarded

by

P

m

Let

h

m

(h

standing

for

hops),

m

p,

be

the

number

of

consecutive

tunnels

in

the

dipath

P

m

Let

the

order

of

the

dipaths

be

such

that

P

is

a

dipath

with

the

minimum

number

of

consecutive

tunnels

h

.Then,

for

any

other

dipath

P

m

with

m

reroute

the

λ

m

requests

routed

via

P

m

via

P

We

obtain

a

new

feasible

solution

H

whose

cost

isIndeed,

the

cost

of

each

tunnel

used

in

P

m

is

decreased

by

λ

m

plus

possibly,

if

some

tunnel

t

of

P

m

becomes

empty,

by

(t)

On

the

other

hand,

the

cost

of

each

tunnel

of

P

is

increased

only

by

λ

m

as

the

tunnel

already

exists.

Therefore,

as

h

h

m

we

get

c(H

c(H),

with

strict

inequality

if

h

h

m

(if

the

dipath

P

m

is

strictly

longer

than

P

or

if,

in

the

rerouting,

some

tunnels

of

length

more

than

become

empty.

So

H

is

also

an

optimal

solution.Relation

with

VPL

problems

The

layout

design

problem

dened

above

is

quite

similar

to

well

studied

VPL

problems

in

ATM

networks,

where

one

imposes

a

constraint

on

the

logical

structure

and

then

wishes

to

minimize

either

the

maximum

distance

or

the

average

distance

traveled

by

the

trac.

Concerning

hardness

and

approximation,

we

shall

see

in

the

sequel

of

the

article

that

the

problem

we

study

inherits

most

of

the

characteristics

of

the

classical

VPL

problems

studied

since

the

It

is

not

surprising

that,

even

if

new

technologies

like

GMPLS

are

proposed

to

cope

with

the

increasing

bandwidth

of

communication

networks,

the

computational

complexity

of

the

problems

associated

to

these

technologies

remains

essentially

the

same.Nevertheless,

there

are

two

crucial

dierences

between

the

GMPLS

problem

that

we

study

and

the

classical

VPL

version

of

ATM

networks.

Indeed,

we

have

seen

that

the

GMPLS

logical

network

design

problem

can

be

translated

into

nding

a

set

of

dipaths

in

a

directed

hypergraph,

whereas

the

existing

models

for

VPL

problems

deal

with

digraphs

without

multiple

arcs.

This

feature

will

be

exploited

in

the

dynamic

programming

approach

for

the

path

presented

in

Section

The

second

dierence

is

that

the

cost

function

we

consider

takes

into

account

the

sum

of

the

length

and

the

hop

count

costs,

whereas

usually

in

VPL

problems

the

aim

is

to

minimize

the

maximum

value

of

either

the

length

or

the

hop

count

in

the

network.

Finally,

it

is

important

to

note

that,

if

there

is

a

single

source

in

the

GMPLS

version

(or,

more

generally,

if

the

trac

instance

is

such

that

in

an

optimal

solution

each

hyperarc

has

exactly

vertices),

then

the

problem

is

equivalent

to

a

VPL

problem.

Hardness

ResultsIn

this

section

we

give

hardness

results

for

the

Minimum

Cost

Hypergraph

Layout

problem.

We

distinguish

two

cases

according

to

whether

the

underlying

network

is

symmetric

or

not.

We

focus

on

these

cases

in

Sections

and

that

a

broadcast

is

a

request

scheme

with

all

the

requests

from

a

vertex

u

i

to

all

vertices

u

j

for

j

n,

j

i.

A

request

scheme

is

a

partial

broadcast

if

some

requests

from

u

i

to

u

j

are

missing.General

case

Theorem

The

Minimum

Cost

Hypergraph

Layout

problem

cannot

be

approximated

within

a

factor

C

log

n

for

some

constant

C

even

if

the

instance

is

a

partial

broadcast,

unless

P

NP.Proof.

The

reduction

is

from

the

Minimum

Set

Cover

problem:

given

a

nite

set

S

with

p

elements

a

j

j

p,

and

a

collection

C

of

subsets

of

S,

the

aim

is

to

nd

a

subcollection

C

of

C

of

minimum

cardinality

that

covers

all

the

elements

of

S.To

a

Set

Cover

instance

with

sets

S

S

S

k

with

S

i

{a

a

a

p

we

associate

the

following

digraph:•

We

start

with

a

distinguished

node

s.•

To

each

set

S

i

i

k,

we

associate

a

node

v

i

and

a

directed

path

of

length

L

(L

is

a

parameter

to

be

specied

later)

from

s

to

v

i

To

each

element

a

j

j

p,

we

associate

a

vertex

u

j

and

we

add

the

arcs•

The

request

set

is

a

partial

broadcast

from

s

to

u

j

j

p,

each

request

with

multiplicity

construction

is

illustrated

in

Figure

Observe

that

the

number

of

vertices

of

the

Minimum

Cost

Hypergraph

Layout

instance

is

n

k

(L

p

Let

OP

T

be

the

optimal

cost

to

the

Minimum

Cost

Hypergraph

Layout

instance,

and

let

OP

T

SC

be

the

optimal

cost

to

the

Minimum

Set

Cover

instance.Note

that

any

cover

dened

by

I

k}

induces

an

Hypergraph

Layout

obtained

as

follows:

we

consider

a

tunnel

of

length

L

connecting

node

s

to

each

node

v

i

i

I

corresponding

to

a

set

taken

in

the

cover.

Then

for

each

u

j

we

set

a

tunnel

from

some

v

i

to

u

j

for

i

I

such

that

a

j

S

i

Such

a

solution

has

cost

In

particular,

considering

an

optimal

solution

to

the

Minimum

Set

Cover

instance

we

getConversely,

consider

a

feasible

layout

H

of

cost

S.

For

each

u

j

the

dipaths

from

s

to

u

j

contain

some

v

i

joined

to

u

j

Let

I

be

the

set

of

indices

i

of

the

v

i

obtained

in

such

a

way.

Then

the

sets

S

i

for

i

I,

cover

a

a

p

For

each

i

I,

consider

a

particular

u

j

joined

by

v

i

The

cost

of

the

tunnels

to

route

the

trac

from

s

to

u

j

is

exactly

L

Indeed,

if

this

routing

uses

h

tunnels,

the

total

length

is

L

h

and

the

total

hop

count

is

h

as

m

s,uj

reach

the

p

|I|

vertices

u

j

not

already

considered,

we

may

reuse

some

tunnels.

This

increases

for

each

node

the

total

cost

at

least

(one

more

tunnel

and

one

more

request).

So

altogether

the

cost

of

the

solution

is

at

least

S

|I|

(L

|I|)

k

L

|I|

Therefore,

we

have

a

solution

to

the

Minimum

Set

Cover

with

costSuppose

that

C

is

a

constant

such

that

we

can

approximate

in

polynomial

time

the

Minimum

Cost

Hypergraph

Layout

problem

within

a

factor

C

log

n.

That

is,

we

can

nd

a

solution

such

that

S

C

log

n

OP

T

By

the

above

discussion,

we

can

then

nd

in

polynomial

time

a

solution

to

the

Minimum

Set

Cover

instance

with

cost

S

SC

such

thatUsing

Equation

in

Equation

we

obtainLet

now

L

p

We

can

assume

that

the

number

of

sets

of

the

Minimum

Set

Cover

instance

is

bounded

by

a

polynomial

on

the

number

of

elements

Therefore,

n

k

(p

p

p

for

some

xed

constant

C

In

other

words,

log

n

C

log

p.

Using

the

latter

inequality

and

the

fact

that

L

p

in

Equation

we

getRR

version

Oct

the

rightmost

term

of

Equation

tends

to

as

p

tends

to

we

have

obtained

a

polynomial

time

(C

C

log

p)-approximation

algorithm

for

Minimum

Set

Cover.

On

the

other

hand,

Raz

and

Safra

proved

that

Minimum

Set

Cover

is

not

approximable

within

a

factor

C

log

p,

for

some

constant

C

unless

P

NP.

So

for

C

approximating

Minimum

Cost

Hypergraph

Layout

within

a

factor

C

log

n

is

also

NP-hard.When

the

input

graph

G

is

symmetric,

we

can

consider

G

as

an

undirected

graph

where

the

edge

{i,

j}

corresponds

to

the

two

arcs

(i,

j)

and

(j,

i).Theorem

The

Minimum

Cost

Symmetric

Hypergraph

Layout

problem

is

APXhard

even

if

the

instance

is

a

partial

broadcast.

Therefore,

it

does

not

accept

a

PTAS

unless

P=NP.Proof.

The

reduction

is

from

Minimum

Steiner

Tree

problem:

given

an

edge-weighted

graph

G

(V,

E)

and

a

subset

S

V

nd

a

connected

subgraph

with

minimum

edge-weight

containing

all

the

vertices

in

S.

We

can

assume,

by

subdividing

edges,

that

all

edge-weights

are

equal

to

This

problem

is

known

to

be

APX-hard

hence

it

does

not

accept

a

PTAS

unless

P

NP.

Given

an

instance

(G

(V,

E),

S

V

of

Minimum

Steiner

Tree

on

n

vertices,

we

build

an

instance

of

Minimum

Cost

Symmetric

Hypergraph

Layout

as

follows.

We

take

as

underlying

digraph

G

and

as

request

set

a

partial

broadcast

from

some

vertex

s

in

S

to

all

the

other

vertices

in

S,

all

with

multiplicity

We

set

all

the

edge

lengths

to

L

L

being

a

parameter

to

be

specied

later.

Let

OP

T

be

the

optimal

cost

to

the

Minimum

Cost

Symmetric

Hypergraph

Layout

instance,

and

let

OP

T

ST

be

the

optimal

cost

to

the

Minimum

Steiner

Tree

instance,

realized

by

a

subgraph

H.

Since

H

connects

s

to

all

the

other

vertices

in

S,

it

follows

thatConversely,

given

any

solution

to

the

Minimum

Cost

Symmetric

Hypergraph

Layout

instance

with

cost

S

and

realized

with

a

graph

H

S

we

can

nd

a

solution

to

the

Minimum

Steiner

Tree

instance

(just

by

taking

the

edges

used

by

some

tunnel)

with

costSuppose

for

the

sake

of

contradiction

that

there

exists

a

PTAS

for

the

Minimum

Cost

Symmetric

Hypergraph

Layout

problem.

Then,

for

each

ε

we

can

nd

in

polynomial

time

a

solution

S

such

that

S

ε)

OP

T

Then,

we

can

nd

a

solution

to

the

Minimum

Steiner

Tree

instance

such

that

where

we

have

used

Equation

in

the

last

inequality.

Let

now

L

n

Equation

becomes

Figure

The

binary

layout

depicted

above

gives

a

log

n-approximation

algorithm

for

Minimum

Cost

Hypergraph

Layout

on

the

path.

If

the

path

is

undirected,

we

add

the

symmetric

tunnels.In

this

section

we

provide

approximation

algorithms

for

the

Minimum

Cost

Hypergraph

Layout

problem.

For

the

sake

of

presentation,

we

describe

our

algorithms

when

the

network

is

a

path,

a

tree,

and

a

general

graph

in

Sections

and

respectively.

Although

Proposition

is

a

particular

case

of

Proposition

we

describe

in

Section

a

simple

and

intuitive

layout

for

the

path,

which

diers

from

the

approach

of

Section

assume

that

the

path

is

directed.

We

can

suppose

w.l.o.g.

that

n

is

a

power

of

two,

otherwise,

just

add

dummy

vertices.We

dene

the

following

binary

layout,

such

as

illustrated

in

Figure

we

connect

node

to

node

(n

node

(n

to

node

n,

and

we

consider

recursively

the

binary

layout

for

(n

on

the

subdipaths

(n

and

[(n

n].

It

is

clear

that

any

request

can

be

routed

in

this

layout

with

at

most

log

n

hops,

and

that

the

total

length

of

this

layout

is

bounded

above

by

log

n

n]),

where

n])

denotes

the

length

of

the

dipath

going

from

node

to

node

n.

Therefore

the

cost

of

this

layout

is

log

n

ri,j

m

i,j

log

n

n]).We

now

distinguish

two

cases.Consider

rst

the

case

where

the

set

of

requests

covers

all

the

arcs

of

the

path

(an

arc

(u,

u

is

covered

by

a

request

r

i,j

if

i

u

u

j).

The

total

cost

of

a

tunnel

is

at

least

(t)

as

each

tunnel

carries

at

least

one

request.

As

the

set

of

requests

and

so

the

set

of

tunnels

covers

all

the

arcs

of

the

path,

a

lower

bound

for

the

minimum

cost

is

e

(e)

n]).

Another

trivial

lower

bound

is

ri,j

m

i,j

Therefore,

n])

ri,j

m

i,j

is

also

a

lower

bound,

and

so

using

the

binary

layout

in

the

whole

path

yields

a

log(n)-approximation.If

the

set

of

requests

does

not

cover

all

the

arcs,

we

consider

the

span

of

an

instance

as

the

minimum

(in

terms

of

length)

set

of

disjoint

intervals

of

the

path

such

that

any

request

can

be

routed

using

only

one

of

these

intervals.

Each

arc

of

these

intervals

is

covered

by

at

least

one

request

included

in

the

interval,

so

we

can

apply

the

binary

layout

described

above

for

any

interval.

We

obtain

for

an

interval

of

length

n

i

a

log(n

i

)-approximation,

and

thus

a

log(max

i

(n

i

log(n)

approximation

for

our

problem.If

the

path

is

undirected

(or

equivalently,

a

symmetric

directed

path),

we

add

to

the

binary

layout

(dened

analogously

in

the

span

of

the

instance)

all

the

symmetric

tunnels,

c

and

the

diameter

of

a

virtual

topology

allowing

to

route

an

all-to-all"

(in

the

sense

that

each

node

sends

trac

to

all

the

nodes

reachable

from

it)

trac

in

a

general

tree.Theorem

(Bermond

et

al.

In

a

directed

tree

on

n

nodes

such

that

each

node

sends

trac

to

all

the

nodes

reachable

from

it,

for

each

value

of

c

there

exists

a

virtual

topology

allowing

to

route

all

trac

with

diameter

at

most

and

load

at

most

c.

In

addition,

such

a

virtual

topology

can

be

constructed

in

polynomial

time.In

particular,

if

we

set

c

log

Theorem

implies

that

we

can

nd

in

polynomial

time

a

virtual

topology

with

load

O(log

n)

and

diameter

at

most

log

n

n

log

n

log

n

O(log

n).Consider

a

general

directed

tree

and

suppose

rst

that

the

instance

of

the

Minimum

Cost

Hypergraph

Layout

problem

is

such

that

each

node

sends

trac

to

all

the

nodes

reachable

from

it.

Each

arc

must

be

used

by

some

tunnel,

and

so

like

for

paths,

one

lower

bound

is

e∈E

(e)

ri,j

m

i,j

.In

the

layout

described

above,

each

arc

is

used

at

most

log

times,

and

therefore

the

total

length

of

this

layout

is

O(log

n

e∈E

(e)).

Since

the

diameter

is

also

O(log

n),

the

total

hop

count

is

O(log

n·

ri,j

∈R

m

i,j

So

altogether,

we

have

an

O(log

n)-approximation.Suppose

now

that

the

trac

instance

is

a

general

one.

Similarly

to

Section

we

dene

the

span

of

an

instance

as

a

minimum

set

of

subtrees

such

that

any

request

can

be

routed

within

one

of

these

trees.

Then,

we

apply

the

layout

of

Theorem

to

each

connected

component

of

the

span,

obtaining

the

same

approximation

ratio.

Finally,

for

symmetric

trees,

we

just

multiply

the

length

of

the

layout

by

by

adding

the

symmetric

tunnels,

as

we

did

in

Section

Summarizing,

Proposition

When

the

network

is

a

tree,

there

exists

a

polynomial-time

approximation

algorithm

for

the

Minimum

Cost

Hypergraph

Layout

problem

with

an

approximation

ratio

O(log

n).

that

it

is

not

the

case

in

a

general

network.

Namely,

the

problem

seems

much

easier

to

approximate

in

symmetric

networks.Let

us

introduce

the

following

problem,

that

will

be

used

in

the

approximation

algorithms

presented

in

this

section:

in

the

Minimum

Generalized

Steiner

Network

problem,

we

are

given

a

graph

G

(V,

E),

a

weight

function

w

E

N,

a

capacity

function

c

E

N,

and

a

requirement

function

r

V

V

N.

The

objective

is

to

nd

a

Steiner

network

over

G

that

satises

all

the

requirements

and

obeys

all

the

capacities,

i.e.,

a

function

f

E

N

such

that,

for

each

edge

e

E,

f

(e)

c(e)

and,

for

any

pair

of

nodes

i

and

j,

the

number

of

edge-disjoint

paths

between

i

and

j

is

at

least

r(i,

j),

where

for

each

edge

e,

f

(e)

copies

of

e

are

available.

The

aim

is

to

minimize

the

cost

of

the

network,

i.e.,

e∈E

w(e)

f

(e).

If

the

input

graph

G

is

undirected,

the

problem

is

approximable

within

O(log

r

max

where

r

max

is

the

maximum

requirement

and

within

a

constant

factor

when

all

the

requirements

are

equal

The

directed

version

of

the

problem

is

approximable

within

O(n

log

n)

.Symmetric

network

Suppose

rst

that

the

network

is

symmetric.

Given

an

instance

(G,

R)

of

Minimum

Cost

Hypergraph

Layout

in

a

general

symmetric

network,

we

build

an

instance

of

the

associated

Minimum

Generalized

Steiner

Network

problem

as

follows.

We

take

as

underlying

graph

G

itself,

and

we

take

as

weight

function

the

length

function

of

G,

that

is,

w(e)

(e)

for

all

e

E(G).

For

i,

j

V

(G),

we

set

r(i,

j)

whenever

m

i,j

and

r(i,

j)

otherwise.

Finally,

we

set

c(e)

for

all

e

E(G).

Let

F

be

an

optimal

solution

to

this

Minimum

Generalized

Steiner

Network

instance

(note

that

F

may

be

disconnected),

and

let

(F

e∈E(F

(e).

The

following

easy

observation

will

be

useful:

since

F

is

the

minimum

(in

terms

of

total

edge-length)

subgraph

of

G

such

that

any

couple

source-destination

lies

on

the

same

connected

component,

the

total

length

of

any

solution

to

the

Minimum

Cost

Hypergraph

Layout

problem

is

at

least

(F

Using

the

algorithm

of

we

can

nd

in

polynomial

time

a

Steiner

network

F

with

(F

(F

Since

the

edge

capacities

are

set

to

we

can

assume

that

such

a

Steiner

network

F

is

a

forest.

The

layout

is

then

obtained

by

applying

the

algorithm

described

in

Section

to

each

connected

component

of

F

.It

is

clear

that

the

hop

count

of

this

layout

is

at

most

O(log

n)

times

the

lower

bound

ri,j

∈R

m

i,j

On

the

other

hand,

the

total

length

of

this

layout

is

O(log

n

(F

O(log

n·

(F

Since

the

total

cost

of

any

layout

is

lower-bounded

by

(F

ri,j

m

i,j

the

O(log

n)-approximation

follows.

Summarizing,

Theorem

In

a

general

symmetric

network,

there

exists

a

polynomial-time

approximation

algorithm

for

the

Minimum

Cost

Symmetric

Hypergraph

Layout

problem

with

an

approximation

ratio

O(log

n).

General

directed

network

In

a

not

necessarily

symmetric

network,

if

we

follow

the

same

approach

as

in

the

symmetric

case,

the

assumption

that

the

graph

F

(the

solution

to

the

Minimum

Generalized

Steiner

Network)

is

a

directed

forest

does

not

hold

anymore,

and

therefore

we

cannot

apply

the

layout

of

Section

directly

to

F

.To

overcome

this

problem,

we

proceed

as

follows:

suppose

that

F

is

connected,

otherwise

we

proceed

independently

in

each

connected

component.

We

partition

F

into

strongly

connected

components

F

F

l

(Note

that

this

partition

can

be

found

in

linear

time

Then,

if

we

shrink

each

F

k

to

a

single

vertex,

for

k

l,

we

obtain

a

directed

acyclic

graph

(a

DAG

for

short).

We

remove

arcs

from

this

DAG

until

we

obtain

a

directed

tree

T

such

that

all

the

requests

can

be

routed

using

only

edges

from

T

and

F

k

k

l.For

k

l,

we

select

in

component

F

k

an

arbitrary

distinguished

node

v

k

and

let

T

out

k

and

T

in

k

be

two

directed

spanning

trees

of

F

k

such

that

T

out

k

is

routed

at

v

k

and

such

that

v

k

is

reachable

in

T

in

k

from

any

vertex

in

F

k

(Note

that

T

out

k

and

T

in

k

exist

and

can

be

eciently

computed

since

F

k

is

strongly

connected

The

routing

within

each

of

the

trees

T

and

T

out

k

T

in

k

k

l,

is

carried

out

according

to

the

layout

described

in

Section

whose

diameter

(in

each

tree)

is

at

most

log

n.

Then

our

routing

strategy

is

the

following.

If

m

i,j

and

i,

j

lie

on

the

same

component

F

k

we

send

the

request

from

i

to

v

k

using

the

arcs

of

T

in

k

and

then

from

v

k

to

j

using

the

arcs

of

T

out

k

Otherwise,

if

i

lies

on

a

component

T

k

and

j

lies

on

a

component

T

k

with

k

k

we

send

the

request

from

i

to

v

k

using

the

arcs

of

T

in

k

then

from

v

k

to

v

k

using

the

arcs

of

T

and

nally

from

v

k

to

j

using

the

arcs

of

T

out

k

.Using

the

above

routing

scheme,

each

request

is

routed

either

with

at

most

log

n

hops

(if

source

and

destination

lie

on

the

same

connected

component

of

T

or

at

most

log

n

(otherwise).Recall

that

the

best

approximation

ratio

to

the

Minimum

Generalized

Steiner

Network

problem

is

O(n

log

n)

and

therefore

the

total

length

of

the

obtained

layout

is

at

most

O(n

log

n)

times

the

total

length

of

an

optimal

one.

The

layout

used

in

each

tree

introduces

just

a

multiplicative

term

to

the

total

length

bounded

by

O(log

n)

(see

Section

On

the

other

hand,

by

the

arguments

above

the

total

hop

count

of

this

layout

is

at

most

O(log

n)

times

the

total

hop

count

of

an

optimal

layout.

Summarizing,

we

obtain

an

O(n

log

n)-approximation.Theorem

In

a

general

network,

there

exists

a

polynomial-time

approximation

algorithm

for

the

Minimum

Cost

Symmetric

Hypergraph

Layout

problem

with

an

approximation

ratio

O(n

log

n).

In

this

section

we

focus

on

the

case

when

the

underlying

digraph

is

a

directed

path.

Our

approach

consists

in

a

dynamic

programming

algorithm

that

computes

partial

solutions

induced

on

subdipaths

of

the

original

dipath.We

provide

the

details

for

one

and

two

sources

in

Sections

and

respectively,

and

then

we

show

in

Section

how

to

generalize

the

previous

ideas

to

deal

with

any

xed

number

k

of

sources

on

the

path.First,

let

us

introduce

some

notations

that

will

be

useful

in

the

sequel.Nodes

are

numbered

from

left

to

right

n.

We

denote

by

[i,

j]

the

subdipath

from

node

i

to

node

j

(with

i

j)

and

by

OP

T

(s

s

s

k

[s

n])

the

cost

of

an

optimal

solution

with

k

sources

s

s

s

k

on

the

dipath

[s

n].Notation

α(i).

For

a

given

node

i

and

an

interval

[u,

v],

let

α(i)

be

the

rightmost

endvertex

in

[u,

v]

of

a

tunnel

starting

in

i

(said

otherwise,

(i,

α(i))

is

the

longest

tunnel

issued

from

i

and

ending

in

[u,

v]).Note

that

on

the

path

n],

when

all

the

requests

are

destined

to

node

n,

the

optimal

solution

is

simply

the

tunnel

from

node

i

to

node

n

where

i

is

the

leftmost

source.Case

of

a

single

sourceWe

present

in

this

section

the

algorithm

for

a

single

source

(Gerstel

et

al.

used

a

similar

approach

in

We

also

give

a

closed

formula

of

the

optimal

cost

when

the

requests

and

the

lengths

of

the

arcs

are

uniform

(see

Proposition

one

source,

we

denote

OP

T

(i;

[i,

j])

by

simply

OP

T

[i,

j],

that

is,

the

cost

of

an

optimal

solution

for

the

dipath

[i,

j]

with

a

unique

source

located

at

i

and

sending

to

a

node

u,

i

u

j

a

request

of

multiplicity

m

i,u

m

.Since

the

path

is

directed,

we

assume

w.l.o.g.

that

the

source

is

located

in

the

leftmost

node

of

the

path

(node

The

rst

crucial

observation

is

that

the

structure

of

the

tunnels

in

an

optimal

solution

is

non-crossing,

i.e.,

two

tunnels

can

only

intersect

in

an

optimal

solution

if

one

is

strictly

inside

the

other,

as

stated

in

the

following

lemma.Lemma

Let

the

network

be

a

directed

path,

with

a

unique

source

at

node

i

and

with

requests

to

nodes

in

[i,

j].

The

set

of

tunnels

T

of

an

optimal

solution

for

Minimum

Cost

Hypergraph

Layout

is

such

that,

if

(i,

α(i))

is

the

longest

tunnel

from

i

to

[i,

j]

Proof.

Suppose

there

exists

such

a

tunnel

(k,

l)

(see

Figure

As

α(i)

is

the

rightmost

node,

then

k

i,

otherwise

(i,

l)

would

have

been

longer

than

(i,

α(i)).

Therefore,

the

number

of

consecutive

tunnels

from

i

to

l,

namely

h(i,

l),

satises

h(i,

l)

Consider

the

set

of

tunnels

T

obtained

from

T

by

deleting

the

tunnel

(k,

l)

and

adding,

if

it

does

not

exist,

the

tunnel

(α(i),

l).

Any

request

from

node

i

to

some

node

u

in

[l,

j]

which

was

routed

via

the

tunnel

(k,

l)

is

now

routed

till

l

through

two

consecutive

tunnels

(i,

α(i))

and

(α(i),

l).

It

is

an

admissible

solution

whose

cost

satises:where

λ

l

is

the

number

of

requests

arriving

at

l

or

transiting

via

l.

As

h(i,

l)

andLemma

leads

to

the

following

approach:

consider

the

rightmost

tunnel

originating

from

the

source,

node

and

assume

it

ends

at

node

As

there

is

no

tunnel

crossing

all

the

requests

for

nodes

in

n]

have

to

be

routed

rst

by

tunnel

α

and

so

can

be

considered

as

emitted

by

a

source

at

node

Therefore,

we

can

split

the

problem

into

two

subproblems:

nd

an

optimal

solution

for

the

requests

to

and

an

optimal

solution

for

the

dipath

n]

with

source

at

approach

allows

us

to

compute

the

optimal

solution

for

a

path

with

n

vertices

recursively.Proposition

The

cost

of

an

optimal

solution

OP

T

[i,

j]

for

problem

Minimum

Cost

Hypergraph

Layout

on

the

dipath

[i,

j]

with

source

i

may

be

expressed

as

follows:Proof.

By

Lemma

let

α(i)

be

the

rightmost

node

in

[i,

j]

from

i

in

an

optimal

solution.

Then,

the

cost

of

the

solution

is

the

sum

of

the

cost

of

the

tunnel

(i,

α(i))

equals

//We

consider

all

requests

from

to

some

node

in

the

interval//value

is

the

cost

of

the

solution

if

α(i)

is

the

splitting

point

for

sub-Compute

the

optimal

set

of

tunnels

from

the

table

S;

end

Theorem

Let

the

network

be

a

directed

path

n]

with

a

unique

source

at

node

then

an

optimal

solution

of

the

Minimum

Cost

Hypergraph

Layout

problem

can

be

computed

in

O(n

time

by

Algorithm

Algorithm

proceeds

as

follows.

First,

it

computes

optimal

solutions

for

dipaths

of

min

So,

we

can

compute

the

optimal

solution

using

dynamic

programming

(Algorithm

with

time

complexity

O(n

and

space

complexity

O(n

).The

optimal

algorithm

in

the

example

of

Figure

Let

us

compute

an

optimal

solution

to

the

example

in

Figure

using

Algorithm

where

all

the

lengths

are

equal

to

and

m

m

m

and

m

We

rst

have

to

compute

the

table

OP

T

containing

the

costs

of

the

suboptimal

solutions

for

each

subpath.

Table

Computation

of

the

tables

OP

T

and

S

for

the

optimal

solution

of

the

example

on

Figure

the

nodes

in

brackets

representing

the

splitting

points

of

table

S.

Table

the

set

of

tunnels

composing

the

optimal

solution

can

be

deduced

from

the

splitting

points.

The

optimal

solution

for

the

subpath

has

cost

and

a

splitting

point

Thus,

the

optimal

solution

is

composed

of

a

tunnel

and

of

optimal

solutions

for

the

subpaths

and

The

rst

subsolution

has

a

splitting

point

which

gives

tunnels

The

optimal

solution

for

the

subpath

is

obviously

the

tunnel

Finally,

the

optimal

solution

is

composed

of

tunnels

and

formula

when

the

requests

and

the

lengths

are

uniform

In

the

special

case

where

both

the

multiplicities

of

the

requests

and

the

lengths

of

the

arcs

are

all

we

give

a

closed

formula

of

the

cost

of

an

optimal

solution,

as

stated

in

the

following

proposition.Proposition

Let

the

network

be

a

path

n]

with

n

q

r,

where

r

q

such

that

([i,

i

for

all

i

n

and

with

a

unitary

distribution,

that

is,

for

all

∀i

n],

m

Then

the

cost

of

an

optimal

solution

is

q

(q

(q

Let

us

consider

only

the

solutions

satisfying

the

non-crossing

lemma,

that

we

call

normal

solutions.

A

normal

solution

for

n]

is

obtained

by

taking

for

a

normal

solution

S

for

i]

and

a

normal

solution

S

for

[i

n],

with

the

source

at

i

Therefore,

the

cost

is:

c(S

c(S

n

Associate

to

a

normal

solution

S,

a

binary

tree

T

(S)

on

n

vertices

obtained

as

follows:

T

(S)

consists

of

a

root

joined

to

the

root

of

the

left

subtree

T

(S

associated

to

S

and

to

the

root

of

the

right

subtree

T

(S

associated

to

S

.Let

d(T

(S))

be

the

sum

of

the

distances

of

the

nodes

to

the

root

node

in

T

(S).

Then,

we

prove

by

induction

that

the

cost

of

a

solution

S

is

c(S)

d(T

(S))

|T

(S)|.

That

is

true

for

n

as

T

(S)

consists

of

the

tree

reduced

to

one

vertex

and

the

cost

of

the

solution

is

Table

Optimal

cost

for

the

path

with

a

uniform

distribution.Therefore,

minimizing

the

cost

for

n

is

equivalent

to

minimizing

the

distance

of

the

nodes

to

the

root

in

a

binary

tree

with

n

nodes.

The

optimal

tree

is

very

simple:

it

is

just

a

complete

binary

tree

with

an

extra

non-complete

level.

Since

a

complete

binary

tree

with

depth

q

has

q

vertices,

we

get

when

n

q

r,

r

q

that

the

sum

of

the

distances

of

the

nodes

to

the

root

is

equal

to

i

qr

n

q

(q

(q

Table

gives

the

values

up

to

nodes

on

the

path

with

uniform

distribution,

each

node

at

distance

one

from

its

neighbours.We

use

a

dynamic

program

similar

to

the

one

used

for

the

single

source

case

in

Section

but

slightly

more

complicated.Let

s

and

s

be

the

two

sources

with

s

s

First,

we

give

the

idea

of

the

algorithm,

then

the

recursive

formulae

used

to

compute

OP

T

(s

s

[s

n]),

and

nally

prove

their

validity,

obtaining

an

O(n

algorithm.The

key

idea

of

the

algorithm

is

that

we

can

nd

an

optimal

solution

satisfying

the

following

property

analogous

to

Lemma

If

i

acts

as

a

source

(representing

s

or

s

and

(i,

α(i))

denotes

the

longest

tunnel

from

i,

then

the

trac

of

i,

destined

to

nodes

after

α(i)

is

routed

via

the

tunnel

(i,

α(i)).

So

starting

at

s

α(s

acts

as

a

source

for

the

nodes

in

[α(s

n].Three

cases

may

if

α(s

s

then

we

consider

the

same

problem

with

sources

α(s

(carrying

the

trac

of

s

and

s

on

a

smaller

dipath

[α(s

if

α(s

s

then

we

consider

the

problem

with

only

one

source

s

carrying

the

trac

from

both

sources

s

and

s

For

that,

we

will

have

to

compute

only

two

types

of

values

dened

as

follows:,

the

cost

of

an

optimal

solution

on

subdipath

[i,

u]

with

sources

i

and

s

where

s

i

s

i

u

n

and

i

carries

the

trac

of

s

that

is,

m

i,x

m

for

i

x

OP

T

δ

(i,

j;

[j,

u]),

with

δ

or

the

cost

of

an

optimal

solution

on

subdipath

[j,

u]

with

sources

i

and

j,

where

s

i

j

u

n

and

where

there

exists

a

tunnel

(β(j),

j)

with

β(j)

i.

For

OP

T

(i,

j;

[j,

u]),

i

carries

the

trac

of

s

(m

i,x

m

for

x

[j,

u])

and

j

the

trac

of

s

For

OP

T

(i,

j;

[j,

u]),

i

carries

the

trac

of

s

and

j

the

trac

of

s

Note

that

in

OP

T

the

left

source

i

is

outside

the

interval

of

destination

nodes,

but

we

have

the

possibility

to

use

the

tunnel

(β(j),

j)

where

β(j)

α

−1

(j)

with

a

cost

reduced

to

its

weight

(so

we

use

a

notation

OP

T

).Note

also

that

in

both

cases

if

i

s

or

i

j

the

problem

is

reduced

to

the

case

of

one

source

i

carrying

the

trac

of

both

sources

s

and

s

that

is,

m

i,x

m

m

for

i

x

u.

We

denote

this

special

source

by

i

is

the

optimal

cost

of

a

problem

with

a

single

source

i

carrying

the

trac

from

s

Note

that

if

α(i)

s

the

problem

is

reduced

to

OP

T

(s

[α(i),

u])

where

s

carries

the

trac

from

both

sources.In

both

cases

α(i)

carries

the

trac

of

i,

that

is

of

s

Computation

of

OP

T

δ

(i,

j;

[j,

u]),

with

s

i

j

u,

δ

or

and

where

i

carries

the

trac

of

source

δ,

j

that

of

the

other

source

δ,

and

there

exists

a

tunnel

(β(j),

j)

(in

fact,

β(j)

α

−1

(j)).

where

α(i)

carries

the

trac

of

i

(that

is,

of

the

source

it

represents).b)

if

there

is

no

tunnel

(i,

α(i))

with

j

α(i)

u,

then

the

value

iswhere

OP

T

corresponds

to

a

problem

with

a

unique

source

j

carrying

the

trac

of

both

sources

i

and

j

(that

is,

m

m

for

j

x

u).

Note

that

in

this

case

the

trac

from

i

uses

the

tunnel

[β(j),

j]

and

so

there

is

no

cost

corresponding

to

the

length.First,

let

us

prove

the

following

lemma,

which

is

valid

for

any

pair

of

sources

i,

j,

and

that

will

be

needed

to

prove

the

formulae.Lemma

Suppose

we

have

two

nodes

i

and

j,

with

i

j,

i

carrying

the

trac

of

one

source

and

j

the

trac

of

the

other

source,

and

j

α(i)

u.

There

exists

an

optimal

solution

where

the

trac

of

i

destined

for

the

nodes

in

[α(i),

u]

uses

as

rst

tunnel

(i,

α(i)).Note

that

in

particular

the

optimal

solution

for

OP

T

δ

(i,

j;

[α(i),

u])

is

independent

of

the

tunnels

used

to

bring

trac

to

nodes

x

α(i).

Proof.

Let

us

suppose

that

the

lemma

is

not

true

and

let

i

be

the

rst

value

for

which

i

is

the

source

and

some

trac

to

[α(i),

u]

is

not

carried

by

(i,

α(i)).Note

that

i

is

of

form

α

h

(s

δ

with

δ

or

So

we

consider

an

optimal

solution

satisfying

the

lemma

for

all

α

h

(s

δ

for

h

h

and

if

δ

α

h

(s

As

the

lemma

is

not

true,

this

solution

uses

to

bring

the

trac

of

i

to

some

node

x,

with

α(i)

x

u,

a

tunnel

(k,

l)

with

k

α(i)

l

u.

By

the

minimality

of

i,

k

i.

Otherwise,

k

will

carry

the

trac

of

some

source

and

k

α

h

(s

δ

But

then

α(k)

j

l

brings

a

contradiction

as

thewould

not

be

the

longest

tunnel

from

i.

Therefore,

k

i

and

the

solution

uses

t

tunnels

to

route

the

trac

from

i

to

k.

Reroute

the

trac

from

i

to

x

by

using

rst

the

tunnel

(i,

α(i))

and

then

injecting

the

trac

arrived

in

α(i)

into

the

tunnel

(k,

l),

and

then

follow

the

same

route

as

x

in

the

optimal

solution.

Doing

so,

we

have

increased

the

cost

by

m

i,x

by

using

(i,

α(i))

but

decreased

the

cost

by

at

least

tm

i,x

(perhaps

more

if

some

tunnel

becomes

empty).

So

the

cost

of

the

obtained

solution

is

less

than

or

equal

to

that

of

an

optimal

one,

and

therefore

it

is

optimal

too.Remark.

We

can

see

the

analogy

between

Lemmas

and

as

we

have

no

tunnel

(k,

l)crossing

α(i)

with

k

α(i)

l

except

perhaps

for

k

j.

In

some

cases

the

cost

of

using

directly

a

tunnel

(j,

l),

that

is

m

j,l

(j,

l)

might

be

less

than

that

of

using

the

tunnel

(i,

α(i))

and

a

tunnel

(α(i),

l),

of

value

j,l

(α(i),

l)

us

now

prove

that

the

computation

of

the

above

formulae

is

valid,

rst

for

the

formula

to

compute

OP

T

(i,

s

[i,

u])

and

then

for

the

formula

to

compute

OP

T

δ

(i,

j;

[j,

u]).For

case

a),

a

proof

analog

to

that

of

Lemma

shows

that

in

an

optimal

solution

there

does

not

exist

a

tunnel

(k,

l)

with

i

k

α(i)

l

u.

Indeed,

deleting

the

tunnel

(k,

l)

and

adding,

if

it

does

not

exist,

the

tunnel

(α(i),

l)

we

obtain

a

better

solution.

So

in

case

a)

we

can

separate

the

trac

into

that

destined

for

the

nodes

in

[i,

α(i)

which

is

only

that

of

i

(that

is,

of

s

as

the

source

s

is

outside

the

interval,

plus

the

trac

destined

for

the

nodes

in

[α(i),

u]

which

uses

as

rst

tunnel

(i,

α(i)).

We

have

to

add

the

cost

of

theIn

case

b),

by

Lemma

there

exists

an

optimal

solution

in

which

the

trac

destined

for

the

nodes

in

[α(i),

u]

uses

rst

the

tunnel

(i,

α(i)).

So

the

cost

of

an

optimal

solution

consists

of

the

optimal

cost

for

the

trac

to

the

nodes

in

[i,

For

case

a),

by

Lemma

there

exists

an

optimal

solution

such

that

the

trac

to

[α(i),

u]

uses

as

rst

tunnel

(i,

α(i)).

So

we

can

separate

the

cost

into

the

cost

of

an

optimal

solution

to

send

the

trac

of

i

and

j

to

the

nodes

in

[j,

where

α(i)

now

carries

the

trac

of

i

to

x

such

that

α(i)

x

u,

plus

the

cost

of

the

tunnel

(i,

α(i)).For

case

b)

we

can

inject

the

trac

of

i

into

the

existing

tunnel

(β(j),

j)

for

a

minimum

cost

of

x∈

[j,u]

m

i,x

Then

we

have

the

trac

from

both

sources

to

x,

such

that

j

x

u,

concentrated

in

node

j,

so

we

use

an

optimal

solution

for

one

source.

We

can

ll

up

the

tables

by

increasing

u.

Indeed,

suppose

we

have

lled

the

table

until

u

Then

we

ll

up

the

values

of

OP

T

(i,

j;

[j,

u])

starting

at

j

u,

in

which

case

OP

T

δ

(i,

u;

[u,

u])

m

s

δ

,u

Then

we

ll

up

the

values

for

j

u

and

so

on

until

j

s

Note

that

for

a

given

j

we

need

values

with

either

an

interval

ending

in

α(i)

u

or

with

a

j

j.

Then

we

ll

up

OP

T

(i,

s

[i,

u])

by

starting

at

i

s

and

then

decreasing

i,

the

last

value

to

be

computed

being

OP

T

(s

s

[s

u]).At

each

step,

we

need

at

most

O(n)

operations,

so

the

overall

complexity

is

O(n

Generalization

to

an

arbitrary

number

k

of

sourcesIn

the

case

of

k

sources

s

s

s

k

the

computation

is

similar

to

the

case

of

two

sources

except

that

now

we

need

to

compute

k

dierent

types

of

values.

For

h

k,

we

need

to

compute

OP

T

π

(i

i

i

h

s

s

k

[i

h

u]),

where

i

i

i

h

s

s

k

s

i

s

i

s

h

i

where

there

exists

a

tunnel

(β(i

h

i

h

with

β(i

h

i

The

node

i

j

acts

as

a

source

and

carries

the

trac

of

the

source

s

π(j)

with

π,

a

permutation

of

k}.

In

fact,

for

j

h

i

j

s

j

so

the

source

i

j

carries

the

trac

of

s

j

and

therefore

for

j

h

π(j)

j.

Using

a

lemma

analogous

to

Lemma

and

letting

for

a

node

i

and

an

interval

[u,

v],

α(i)

be

such

that

(i,

α(i))

is

the

longest

tunnel

from

i

and

ending

in

[u,

v].

The

nodes

carry

the

same

trac

as

before,

except

α(i

which

carries

the

trac

of

i

that

is

of

s

.The

permutation

π

is

such

that:for

j

h,

as

i

j

i

π

(j)

π(j

if

α(i

s

then

π

(h)

if

s

h+m

α(i

s

then

π

(h

m)

Otherwise,

π

(j)

π(j).If

α(i

s

h+m

for

some

m,

then

the

node

s

h+m

will

carry

the

trac

of

both

sources

s

h+m

and

s

and

we

apply

the

algorithm

with

k

sources.

b)

there

is

no

tunnel

from

i

ending

in

[i

h

u].

Then

the

trac

of

i

can

be

directly

sent

to

i

h

via

the

tunnel

(β(i

h

i

h

and

then

we

have

to

compute

the

optimum

for

k

sources.

In

this

report

we

modeled

a

question

raised

by

label

minimization

in

GMPLS

networks

as

a

hypergraph

layout

problem.

In

the

single

commodity

case

we

showed

the

problem

to

be

closely

related

to

well

studied

VPL

problems.

However,

the

optimization

criteria

(average

hop

count

and

average

load)

that

appear

in

our

problem

are

among

the

less

studied

ones.We

provided

hardness

results

for

the

general

directed

case

and

for

the

symmetric

case,

and

proposed

approximation

algorithms.

More

specically,

we

gave

a

log

n-approximation

on

paths

and

trees,

and

observed

that

in

a

general

network

the

hardness

of

our

problem

is

essentially

equivalent

to

the

hardness

of

nding

generalized

Steiner

networks.

This

is

the

reason

why

closing

the

approximability

gap

of

our

problem

looks

like

a

challenging

problem.In

the

multi-sources

case,

we

presented

a

dynamic

program

on

the

path

that

is

polynomial

when

the

number

of

sources

is

xed.

Namely,

our

algorithm

runs

in

O(n

time

on

a

path

with

n

nodes

and

k

sources.

In

view

of

this

running

time,

it

is

unlikely

that

the

problem

is

NP-hard

on

the

path,

so

nding

a

polynomial

algorithm

for

an

arbitrary

number

of

sources

on

the

path

remains

open.

Likely

extensions

of

the

dynamic

program

to

the

case

of

trees

and

bounded

treewidth

networks

remain

also

to

be

done.

The

complexity

of

the

problem

when

the

routing

is

part

of

the

input

of

the

problem

(that

is,

there

is

a

dipath

associated

to

each

request)

remains

open.

We

want

to

investigate

also

the

possibility

of

a

constant

factor

approximation

for

a

general

graph

when

there

is

a

single

source.

We

suspect

that

the

problem

may

become

polynomial-time

solvable

when

there

is

a

single

source

that

sends

trac

to

all

the

nodes

of

the

network

(note

that

the

reductions

of

Section

do

not

apply

to

this

case),

but

we

have

not

been

able

to

prove

it.

Last,

we

believe

that

more

general

approximation

results

can

be

given

for

low

dimension

Euclidean

metric

graphs

using

the

classical

Arora

paradigm

.INRIA

All-Optical

Label

Switching

(AOLS)

is

an

approach

to

route

packets

transparently

and

all-optically,

thus

allowing

a

speed-up

of

the

forwarding.

This

very

promising

technology

for

the

future

Internet

applications

also

brings

new

constraints

and

new

problems.

Indeed,

since

the

forwarding

functions

are

implemented

directly

at

the

optical

domain,

a

specic

correlator

(device)

is

needed

for

each

optical

label

processed

in

the

node.

Therefore,

it

is

of

major

importance

to

reduce

the

number

of

employed

correlators

in

every

node,

implying

a

reduction

in

the

number

of

labels

(as

referred

in

the

rest

of

the

report).

Due

to

its

exibility

as

a

control

plane

and

to

the

fact

that

it

handles

trac

forwarding,

the

Generic

MultiProtocol

Label

Switching

(GMPLS)

is

the

most

promising

protocol

to

be

applied

in

AOLS-driven

networks..

In

GMPLS,

trac

is

forwarded

through

logical

connections

called

Label

Switched

Paths

(LSPs).

When

GMPLS

is

used

with

packet-based

network,

packets

are

associated

to

LSPs

by

means

of

a

label,

or

tag,

placed

on

top

of

the

header

of

the

packet.

In

this

way,

routers

-called

Label

Switched

Routers

(LSRs)

-can

distinguish

and

forward

packets..

The

GMPLS

standards

allow

packets

to

carry

a

set

of

labels

in

their

header,

conforming

a

stack

of

labels.

Even

though

a

packet

may

contain

more

than

one

label,

LSRs

must

only

read

the

rst

(or

top)

label

in

the

stack

in

order

to

take

forwarding

decisions.

This

helps

to

reduce

both

the

number

of

labels

that

need

to

be

maintained

on

the

core

LSRs

and

the

complexity

of

managing

data

forwarding

across

the

backbone..

Stacking

labels

and

label

processing,

in

general,

are

standardized

by

the

following

set

of

operations

that

an

LSR

can

perform

over

a

given

stack

of

labels:.

SWAP:

replace

the

label

at

the

top

by

a

new

one,

PUSH:

replace

the

label

at

the

top

by

a

new

one

and

then

push

one

or

more

onto

the

stack,

and

POP:

remove

the

label

at

top

in

the

label

stack..

The

labels

stored

in

the

forwarding

table

are

signicant

only

locally

at

the

node

and

swapped

all

along

the

LSP

(see

Figure

Solutions

deployed

by

GMPLS

for

reducing

the

number

of

labels

are

label

merging

(not

discussed

here)

and

label

stacking

With

label

stacking,

when

two

or

more

LSPs

follow

the

same

set

of

links,

they

can

be

routed

together

inside

a

higher-level

LSP,

henceforth

a

tunnel.

In

order

to

setup

a

tunnel,

multiple

labels

are

placed

in

the

packet's

header.

tunnel

at

node

E

can

end

or

follow

dierent

paths

according

to

their

bottom

label

k

i

for

all

i

λ}

in

the

stack..

A

consequence

of

the

way

in

which

the

GMPLS

operations

can

be

congured

at

LSRs

is

the

following:

trac

can

enter

in

any

node

of

a

tunnel

but

can

exit

in

only

one

point,

the

last

node

of

the

tunnel.

In

other

words,

when

some

trac

is

carried

by

a

tunnel,

it

follows

the

tunnel

until

its

end..

Since

the

number

of

labels

used

for

GMPLS

forwarding

aects

the

cost

of

the

AOLS

architecture,

in

this

report

we

mainly

focus

on

the

minimization

of

the

number

of

labels

used.

In

our

previous

example,

the

total

cost

c(t)

of

the

tunnel

t

from

node

A

to

node

E

in

terms

of

number

of

labels

is

c(t)

λ

(t)

where

λ

is

the

number

of

units

of

trac

forwarded

through

this

tunnel

and

(t)

is

its

length

in

terms

of

number

of

hops

(which

is

on

this

example).

We

will

formally

dene

the

cost

function

of

the

problem

in

Section

Previous

work

and

our

contribution

The

label

minimization

problem

in

GMPLS

networks

has

been

widely

studied

in

the

literature

during

the

last

few

years

All

these

articles

focus

mainly

on

proposing

and

analyzing

heuristics

to

the

problem,

but

there

is

a

lack

of

theoretical

results,

like

computational

complexity

or

bounds

on

the

approximation

ratio

of

the

proposed

algorithms..

Gupta,

Kumar,

and

Rastogi

study

the

trade-o

between

the

stack

depth

and

the

label

size

in

tag

switching

protocols

(see

They

calculate

lower

bounds

for

dierent

problem

instances.

Amongst

these

problems,

they

propose

solutions

using

at

most

n

labels

when

the

network

is

a

path

and

2∆

n

when

it

is

a

tree,

where

n

is

the

number

of

nodes

and

the

maximum

degree

of

the

topology.

However,

the

proposed

solutions

lay

on

the

assumption

of

non-practical

label

distribution

protocols

for

MPLS..

In

this

article

we

provide

the

rst

theoretical

framework

for

the

label

minimization

problem

in

general

GMPLS

networks

considering

the

constraints

imposed

by

real

distribution

protocol,

e.g.,

RSVP-TE.

We

translate

the

problem

into

nding

a

set

of

dipaths

in

a

directed

hypergraph.

With

this

new

formulation,

it

turns

out

that

the

problem

is

very

similar

to

classical

Virtual

Path

Layout

(VPL)

problems

originating

from

ATM

networks.

We

provide

hardness

results

and

approximation

algorithms

for

the

problem

in

paths,

trees,

and

general

graphs.

The

approximation

algorithms

strongly

rely

on

the

already

known

algorithms

for

VPL

problems.

Finally,

we

focus

on

the

path

topology,

providing

a

dynamic

programming

approach

that

runs

in

polynomial

time

for

any

bounded

number

of

sources..

Organization

of

the

report

In

Section

we

formally

state

the

problem

in

terms

of

hypergraph

layouts

and

x

the

notation

to

be

used

throughout

the

article.

In

Section

we

prove

that

for

general

directed

networks

it

is

NP-hard

to

nd

a

C

log

n-approximation,

where

C

is

a

a

positive

constant

and

n

is

the

number

of

nodes

of

the

network.

For

symmetric

directed

networks,

we

prove

that

the

problem

is

APX-hard

and

therefore

it

does

not

accept

a

PTAS

unless

P=NP.

In

Section

we

provide

approximation

algorithms

to

the

problem

for

both

general

and

symmetric

networks,

and

discuss

the

gap

with

the

hardness

results..

In

Section

we

focus

on

the

directed

path

topology

and

present

a

dynamic

programming

approach

solving

the

problem

in

polynomial

time

when

the

number

of

sources

is

xed..

Finally,

Section

is

devoted

to

conclusions

and

further

research..

Some

parts

of

this

report

have

been

presented

in

conferences

PTAS

denotes

the

class

of

problems

admitting

a

polynomial

time

approximation

scheme

that

is

guaranteed

to

nd

a

solution

whose

cost

is

within

a

ε

factor

of

the

optimum

cost,

for

any

ε

When

the

solution

is

only

guaranteed

within

a

constant

factor

of

the

optimum

cost,

then

the

problem

belongs

to.

APX.

An

APX-hard

problem

does

not

accept

a

PTAS,

unless

P=NP

(see

for

instance

The

logical

network

design

problem

that

we

address

can

be

roughly

described

as

follows:

we

are

given

a

digraph

(directed

graph)

G

together

with

a

set

of

trac

demands

(or

requests)

between

couples

of

vertices

in

G,

and

we

must

nd

a

set

of

tunnels

of

minimum

cost

allowing

and

a

routing

upon

this

set

of

tunnels

for

all

the

trac

requests.

Note

that

usually

communication

networks

are

symmetric

digraphs

(i.e.

when

operators

set

a

link

in

one

direction,

they

also

set

the

opposite

link).

So

it

is

interesting

to

study

the

symmetric

case,

which

turns

out

to

be

computationally

easier

than

the

general

directed

case.

Let

us

now

precise

each

one

of

the

above

terms..

A

tunnel

is

simply

a

directed

path

(or

dipath)

in

G,

and

due

to

the

technological

constraints

discussed

in

Section

trac

can

enter

anywhere

in

the

tunnel

but

must

leave

only

at

the

end

of

the

tunnel.

To

dene

the

problem

formally

we

need

the

following

notation:.

G

(V,

E)

is

the

underlying

digraph

(which

can

be

symmetric

or

not)

with

|V

n..

r

i,j

is

the

request

from

node

i

V

to

node

j

V

with

multiplicity

m

i,j

(the

trac

being

dierentiated,

m

i,j

represents

the

number

of

dierent

ows

from

i

to

j

and

j

is

not

necessarily

the

nal

destination

of

the

request

r

i,j

R

is

the

set

of

all

requests..

P

(G)

is

the

set

of

all

simple

dipaths

in

G..

t

stands

for

a

tunnel,

and

T

is

the

set

of

tunnels,

that

is

t

T

P

(G)..

is

a

length

function

on

the

arcs,

that

is

E

N

A

tunnel

t

has

length

(t)

e∈t

(e)

and

carries

w(t)

ows,

or

as

referred

in

the

rest

of

the

report,

w(t)

units

of

trac..

Note

that,

a

priori,

w(t)

depends

on

the

routing

policy.

The

cost

c(t)

of

a

tunnel

t

is

then

c(t)

w(t)

(t)

and

the

cost

of

a

set

of

tunnels

T

is.

Each

tunnel

can

be

modeled

as

a

directed

hyperarc

on

the

vertex

set

of

G.

This

observation

naturally

leads

to

the

denition

of

a

hypergraph

layout..

Denition

(Hypergraph

layout).

Given

a

graph

G

and

a

set

T

P

(G),

H(T

is

the

directed

hypergraph

with

V

(H(T

V

(G),

and

where

for

each

tunnel

t

T

P

(G)

there

is

a

directed

hyperarc

in

H(T

connecting

any

vertex

of

t

to

the

end

of

t.

to

j.

The

problem

can

then

be

simply

expressed

as

nding

a

feasible

hypergraph

layout

of

minimum

cost.

Let

us

now

rewrite

the

cost

function

of

Equation

Given

a

hypergraph

layout

H(T

let

L(r

i,j

be

the

number

of

hyperarcs

that

request

r

i,j

uses,

and

let

d

H

(i,

j)

be

the

distance

from

vertex

i

to

vertex

j

in

H(T

Then

the

term

t∈T

w(t)

of

Equation

can

be

rewritten

as

ri,j

∈R

L(r

i,j

m

i,j

and,

since

L(r

i,j

d

H

(i,

j),

we

conclude

that

in

an

optimal

solution

the

routing

necessarily

uses

shortest

dipaths

in

the

hypergraph

layout.

It

follows

that

the

cost

function

of

Equation

can

be

rewritten.

The

cost

of

a

solution

is

of

bicriteria

nature.

The

rst

part

is

the

cost

of

the

hypergraph

structure;

we

call

it

the

total

length

of

the

layout.

The

second

part

is

the

total

distance

that

the

trac

travels

in

the

hypergraph;

we

call

it

the

total

hop

count.

Both

cost

function

parts

are

very

much

conicting.

On

the

one

hand,

to

minimize

the

hop

count,

it

is

enough

to

take

a

tunnel

connecting

any

source

to

any

destination.

On

the

other

hand,

to

minimize

the

total

length

of

the

layout,

it

is

enough

to

consider

a

minimum

arc-weighted

hypergraph

H

such

that

for

each

request

r

i,j

R,

vertices

i

and

j

lie

on

the

same

connected

component

of

H.

Summarizing,

the

problem

can

be

stated

as

follows..

Minimum

Cost

Hypergraph

Layout:

Given

a

digraph

G

with

a

length

function

and

a

set

R

of

trac

requests,

nd

a

feasible

hypergraph

layout

of

minimum

cost,

where

the

cost

of

a

hypergraph

layout

is

dened

as

in

Equation

If

G

is

a

symmetric

digraph,

the

problem

is

denoted

Minimum

Cost

Symmetric

Hypergraph

Layout.

We

note

that

the

cost

function

of

Equation

can

be

naturally

generalized.

where

α

and

β

are

positive

constants

and

p(t)

is

a

general

cost

function

p

The

cost

function

of

Equation

corresponds

to

p(t)

c(t)

(t)

Computation

of

a

solution

for

the

example

of

Figure

Consider

the

path

[s,

with

one

source

s,

m

units

of

trac

destined

to

node

at

distance

from

s

nodes

between

s

and

and

m

units

of

trac

destined

to

node

at

distance

from

s.

See

Figure

for

an

illustration.

The

optimal

solution

depends

on

the

values

i

and

m

s,i

Indeed,

two

solutions

have

to

be

examined.

In

the

rst

solution,

a

specic

tunnel

(s,

i)

is

congured

for

each

destination

i,

giving

two

tunnels

(s,

and

(s,

with

a

total

cost:.

The

second

solution

is

composed

of

the

two

tunnels

(s,

and

The

requests

destined

to

will

rst

use

the

tunnel

(s,

and

then

the

tunnel

We

state

now

a

lemma

to

be

exhaustively

used

in

the

sequel,

asserting

that

there

always

exists

an

optimal

solution

such

that

all

the

trac

units

of

each

request

are

routed

via

a

unique

set

of

consecutive

tunnels,

or

dipath.

Lemma

In

any

network,

there

exists

an

optimal

solution

to

the

Minimum

Cost

Hypergraph

Layout

problem

such

that

all

the

trac

units

of

each

request

are

routed

via

a

unique

dipath..

Proof.

Let

r

i,j

be

a

request

from

node

i

to

node

j,

and

let

H

be

an

optimal

solution..

Suppose

that

the

units

of

trac

arriving

at

j

from

i

are

routed

in

H

via

p

dierent

dipaths

P

P

p

Let

λ

m

m

p,

be

the

number

of

trac

units

forwarded

by

P

m

Let

h

m

(h

standing

for

hops),

m

p,

be

the

number

of

consecutive

tunnels

in

the

dipath

P

m

Let

the

order

of

the

dipaths

be

such

that

P

is

a

dipath

with

the

minimum

number

of

consecutive

tunnels

h

Then,

for

any

other

dipath

P

m

with

m

reroute

the

λ

m

requests

routed

via

P

m

via

P

We

obtain

a

new

feasible

solution

H

whose

cost

is.

Indeed,

the

cost

of

each

tunnel

used

in

P

m

is

decreased

by

λ

m

plus

possibly,

if

some

tunnel

t

of

P

m

becomes

empty,

by

(t)

On

the

other

hand,

the

cost

of

each

tunnel

of

P

is

increased

only

by

λ

m

as

the

tunnel

already

exists.

Therefore,

as

h

h

m

we

get

Relation

with

VPL

problems

The

layout

design

problem

dened

above

is

quite

similar

to

well

studied

VPL

problems

in

ATM

networks,

where

one

imposes

a

constraint

on

the

logical

structure

and

then

wishes

to

minimize

either

the

maximum

distance

or

the

average

distance

traveled

by

the

trac.

Concerning

hardness

and

approximation,

we

shall

see

in

the

sequel

of

the

article

that

the

problem

we

study

inherits

most

of

the

characteristics

of

the

classical

VPL

problems

studied

since

the

It

is

not

surprising

that,

even

if

new

technologies

like

GMPLS

are

proposed

to

cope

with

the

increasing

bandwidth

of

communication

networks,

the

computational

complexity

of

the

problems

associated

to

these

technologies

remains

essentially

the

same..

Nevertheless,

there

are

two

crucial

dierences

between

the

GMPLS

problem

that

we

study

and

the

classical

VPL

version

of

ATM

networks.

Indeed,

we

have

seen

that

the

GMPLS

logical

network

design

problem

can

be

translated

into

nding

a

set

of

dipaths

in

a

directed

hypergraph,

whereas

the

existing

models

for

VPL

problems

deal

with

digraphs

without

multiple

arcs.

This

feature

will

be

exploited

in

the

dynamic

programming

approach

for

the

path

presented

in

Section

The

second

dierence

is

that

the

cost

function

we

consider

takes

into

account

the

sum

of

the

length

and

the

hop

count

costs,

whereas

usually

in

VPL

problems

the

aim

is

to

minimize

the

maximum

value

of

either

the

length

or

the

hop

count

in

the

network.

Finally,

it

is

important

to

note

that,

if

there

is

a

single

source

in

the

GMPLS

version

(or,

more

generally,

if

the

trac

instance

is

such

that

in

an

optimal

solution

each

hyperarc

has

exactly

vertices),

then

the

problem

is

equivalent

to

a

VPL

problem.


Problems for a Philosophy of Software Engineering Abstract On the basis of an earlier contribution to the philosophy of computer science by Amnon Eden, this essay discusses to what extent Eden's 'paradigms' of computer science can be transferred or applied to software engineering. This discussion implies an analysis of how software engineering and computer science are related to each other. The essay concludes that software engineering can neither be fully subsumed by computer science, nor vice versa. Consequently, also the philosophies of computer science and software engineering-though related to each other-are not identical branches of a general philosophy of science. This also implies that not all of Eden's earlier arguments can be directly mapped from the domain of computer science into the domain of software science. After the discussion of this main topic, the essay also points to some further problems and open issues for future studies in the philosophy of software science and engineering.

O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O B O B I O O O O O O B I O O O O O O O B I O B I O O O O O O O O O B I O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Wireless authentication using remote passwords Current wireless authentication mechanisms typically rely on inflexible shared secrets or a heavyweight public-key infrastructure with user-specific digital certificates and, as such, lack general support for environments with dynamic user bases where guest access is frequent. Simple Authentication for the Web (SAW) facilitates dynamic user bases in the context of web site logins by enabling users to authenticate to personal messaging identifiers (e.g., email addresses, IM handles, cell phone numbers). SAW, however, is ill-suited for wireless authentication because, in most cases, it is dependent on client-side Internet connectivity. Wireless Authentication using Remote Passwords (WARP) overcomes this constraint by building a hybrid protocol that combines the principles of SAW authentication with the Secure Remote Password (SRP) protocol. Wireless networks provide an attractive means for network access as they enable convenient roaming and device deployability without the burden of network cables and port accessibility. Although convenient, wireless communications are subject to passive eavesdropping, active injection, and modification attacks. For these reasons, it is important that wireless networks provide secure authentication to prevent unauthorized access to network resources and provide confidentiality and integrity to transmitted information.. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Current wireless authentication mechanisms are usually based on user-specific certificates (PKI), global passphrases, or username/password pairs (see Section 7) . These methods are either too heavy or inflexible and lack general support for environments with dynamic user bases, such as corporations or universities where guest access is frequent.. Wireless Authentication using Remote Passwords (WARP) augments the Secure Remote Password (SRP) [12] protocol using concepts from Simple Authentication for the Web (SAW) [11] . By proving ownership of an authorized personal messaging identifier (e.g., email address, IM handle, cell phone number), WARP enables users to authenticate without pre-established secrets or the heavy cost and inconvenience of user-specific certificates. The burden of wireless access control in dynamic user bases is dramatically reduced as authentication of these globally unique identifiers is performed by trusted third parties. WARP is a convenient and secure wireless authentication mechanism. By preserving the principles of SAW, WARP enables decentralized user authentication and facilitates dynamic user bases in the wireless realm without requiring client-side Internet or IP-connectivity. WARP's use of SRP enables users to authenticate using existing personal messaging account identifiers and passwords, without fear that relying parties or eavesdroppers can compromise their login credentials. Although identity providers assist users in creating authenticated, encrypted sessions with relying parties, the identity providers cannot compromise these sessions.. sSRP could replace the use of email or instant messages in the original SAW protocol for website logins. Using sSRP in this manner increases SAW's ability to thwart active impersonation attacks and eliminate latency issues associated with personal message delivery.. We are currently investigating how to integrate the intuitive delegation between personal messaging identifiers and natural client-side auditing capabilities of SAW into WARP.. Forthcoming revisions to sSRP eliminate the need to use PKI-based approaches, e.g., EAP-TTLS, to prevent passive observation of KSU and KSIDP . These revisions also enable the authentication of relying parties to identity providers. Many authentication mechanisms rely on pre-established shared secrets. Global passphrases used in systems like WPA-PSK [1] can be difficult to distribute. If the shared secret is compromised a new one must be deployed to each device. Guest access can only be achieved by disclosure of the shared secret. It is also difficult to audit access because each user connects using the same passphrase.. Individual user accounts with passwords are employed by challenge/response-based systems such as MSCHAPv2 [14] . Account-level password-based authentication provides reasonable deployability for a static group of users. It also lends itself to auditing because authentication is tied to a set of credentials. However, these systems suffer from the same issues of other password-based authentication mechanisms: password re-use and difficulty in remembering strong passwords. It is also difficult to configure guest access where account creation incurs significant overhead or time.. Recent authentication mechanisms, e.g., EAP-TLS [3] , rely on user-specific digital certificates to provide authentication. However, managing user-specific certificates and securing public/private key-pairs are challenging tasks for even savvy end-users. The sign-up process can be resource intensive, adding additional overhead on administrators who must verify user identity and issue certificates according to stringent company policies. Often times this process must be performed not only for each user, but for each of the user's devices, which increases complexity.. Greenpass [6] leverages EAP-TLS for authentication. EAP-TLS relies on PKI to provide the client and server certificates used in the TLS handshake. Greenpass also provides decentralized delegation of access by allowing delegators to sign a SPKI/SDSI certificate that a guest can then use as their EAP-TLS client-side certificate. Greenpass still involves a CA issuing user-specific certificates to regular users.. Network-in-a-Box (NiaB) [4] enrolls devices in the wireless network by employing location-limited communication channels (e.g., infrared or a USB key) to securely distribute the necessary PKI keys and certificates. Although key distribution is simplified, NiaB still requires an administrator be present to authorize each device during enrollment. This location-limited channel approach creates a physical bottleneck when many devices need to be enrolled within a short time period (i.e., conference wireless access where most delegates arrive within a short time period). Simple Authentication for the Web (SAW) [11] enables decentralized authentication of globally unique personal messaging identifiers (e.g., email addresses, IM handles, cell phone numbers). This approach is ideal for systems with dynamic user bases because no shared secrets (e.g., passwords) are required between clients and relying parties (e.g., web sites). Instead, SAW leverages unmodified personal messaging providers (e.g., email, text and instant message services) to act as identity providers to relying parties.. SAW builds on the same basic technique employed by the "Forgot your password?" link common to many web sites; users must retrieve personal messages (e.g., email, text and instant messages) sent to them by the relying party through their messaging provider. Relying parties depend on providers to deliver messages only to authenticated recipients. As WARP represents a significant departure from SAW, a detailed protocol overview is omitted.. WARP builds on the following principles of a SAW authentication:. 1. Reuse existing identifiers and authenticators.. 2. Tightly couple identifiers and identity providers, e.g., email addresses specify the locations of their respective mail servers.. 3. Authentication requires that users obtain two tokens known to the relying party. The first token is given to the initiator of an authentication, while the second is only obtained after a successful authentication to the identity provider. Software to support wireless authentication using WARP has been developed and will soon be available.. libssrp is a general purpose library written in C that provides the functionality needed to conduct sSRP authentication. It is meant to be used by applications that supply their own transport functionality.. The current version of libssrp relies on OpenSSL [10] for its cryptographic primitives and arbitrary precision integers. libssrp, by default, relies on an SRP-compatible password file populated with salts and verifiers generated from plaintext passwords. An API can alternatively be used to allow flexible retrieval of salts and verifiers. An argument to stay with the default configuration is given in Section 6.. The wpa_supplicant [7] open-source package has been extended to support EAP-WARP. The extension consist of two files: 1) One C source file to provide EAP-WARP support; and 2) A patch file that modifies wpa_supplicant to include the extension. The extension is less than 400 lines of code.. This simple extension provides a layer that extracts sSRP packet data and provides it to libssrp. sSRP packet data returned from libssrp is inserted into an EAP packet that is returned to wpa_supplicant. The extension also exports the EAP MSK, derived from KS, to wpa_supplicant.. FreeRADIUS [9] , an open-source RADIUS server, has also been extended with EAP-WARP support. The extension is around 800 lines of C code and comments, and like the extension for wpa_supplicant, provides extraction and insertion of sSRP messages to and from EAP packets, as well as exporting keying material. The libssrp library is again used to provide the bulk of the functionality.. An incarnation of the sSRP service has been written using the libssrp library and provides sSRP over TCP/IP. The service is written in C, can daemonize, and supports logging to syslog.

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Incremental XPath evaluation We study the problem of incrementally maintaining an XPath query on an XML database under updates. The updates we consider are node insertion, node deletion, and node relabeling. Our main results are that downward XPath queries can be incrementally maintained in time O(depth(D) · poly(Q)) and conjunctive forward XPath queries in time O(depth(D)· log(width(D))·poly(Q)), where D is the size of the database, Q the size of the query, and depth(D) and width(D) are the nesting depth and maximum number of siblings in the database, respectively. The auxiliary data structures for maintenance are linear in D and polynomial in Q in all these cases. The XPath language, proposed by the World Wide Web Consortium (W3C), is essentially a query language for selecting nodes in an XML document. As node-selection is one of the most basic operations on XML documents, XPath lies at the core of most of today's data processing languages for XML. For example, it forms an essential component of languages such as XQuery, XSLT, XML Schema (which uses XPath for defining keys), etc.. The most fundamental algorithmic question concerning XPath is query evaluation. That is, given an XPath query Q and an XML database D, return all elements that are selected by Q in D. The query evaluation problem for various fragments of XPath has been researched quite intensely over the last decade (see [4] for an overview). In this paper, we are interested in the incremental XPath evaluation prob- * Supported by the DFG Grant SCHW678/3-1. † Research Assistant of the Fund for Scientific ResearchFlanders (Belgium). ‡ Supported by a grant of the North-Rhine Westfalian Academy of Sciences.. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the ACM. To copy otherwise, or to republish, to post on servers or to redistribute to lists, requires a fee and/or special permissions from the publisher, ACM. lem. That is, given an XPath query Q and XML data D, we assume that the answer for Q on D is already known. However, when D is updated to D , we want to be able to infer the updated answer for Q on D as quickly as possible. The idea is, of course, to maintain extra information such that the updated answer to Q on D can be computed without having to re-evaluate Q from scratch.. There are many motivations for incremental XPath evaluation. An obvious motivating scenario comes from a database system that is interested in whether a trigger condition is satisfied in a database or not. If the precondition of the trigger is stated by means of an XPath query, the system may be interested in knowing very quickly after an update whether the event of the trigger needs to be carried out. Another scenario comes from exchanging data on the Web. When a community exchanges data, it is often the case that a certain user X is interested in the result of a fixed query on the data of another user Y . The data of Y may change often, while the interests of X remain the same. Of course, it would be beneficial for both parties if the query does not have to be completely recomputed every time Y 's data changes. In such a setting, X may herself have a representation of the current result of her query. So, after an update of Y 's data, it would be relevant for Y to be able to quickly determine the changes that X has to make to her old result, rather than sending her the complete new result, which may be much larger than the update.

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Efficient geometry-based similarity search of 3D spatial databases Searching a database of 3D-volume objects for objects which are similar to a given 3D search object is an important problem which arises in number of database applications -for example, in Medicine and CAD. In this paper, we present a new geometrybased solution to the problem of searching for similar 3D-volume objects. The problem is motivated from a real application in the medical domain where volume similarity is used as a basis for surgery decisions. Our solution for an efficient similarity search on large databases of 3D volume objects is based on a new geometric index structure. The basic idea of our new approach is to use the concept of hierarchical approximations of the 3D objects to speed up the search process. We formally show the correctness of our new approach and introduce two instantiations of our general idea, which are based on cuboid and octree approximations. We finally provide a performance evaluation of our new index structure revealing significant performance improvements over existing approaches. The main contribution of the paper is a new geometrybased index structure which generalizes the well-known R-tree approach for an efficient volume-based similarity search on 3D volumle objects. Our solution is based on the general concept of using both, progressive (MIV) and conservative (MSV) approximations, and the concept of using a hierarchy of approximations. The approximations are used to define a minimum and maximum volume difference measure which1 is formally shown to be correct and allows an efficient pruning of the search space. We developed two instantiations of our geometry-based index structure, which are based on cuboid and octree approximations. The practical relevance and feasibility of our approach is shown by applying our new techniques to the real data from our medical applications. Our experimental evaluation of the two variants of the GSS-tree reveals significant performance improvements over existing approaches. Although the GSS-tree has been developed with our medical application in mind, it is generally applicable in a wide range of other applications.. There are a number of open research directions and a lot of future work to do: Other MIV and MSV approximations need to be explored and more experience needs to be col-

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