Directed Virtual Path LayoutsIn ATM networks

Jean-Claude Bermond Nausica Marlin

David PelegStepane Perennes

Presented by Boris Mudrik

2

Contents

1. Introduction

1.1 Definitions

Problem

2. The Model

Examples

3. The Cycle Cn

3.1 General Case

3.2 Case c =1

4. The Path Pn

3

1. Introduction

The transfer of data in ATM is based on packets of fixed length, termed cells.

Each cell is routed independently, based on two routing fields at the cell header, called virtual channel

identifier (VCI) and virtual path identifier (VPI).

This method effectively creates two types of predetermined simple routes in the network, namely, routes which are based on VPIs (called virtual paths or VPs) and routes based on VCIs and VPIs (called virtual channels or VCs).

4

1. Introduction (cont.)

VCs are used for connecting network users.

VPs are used for simplifying network management -

routing of VCs in particular.

Route of a VC may be viewed as a concatenation of

complete VPs.

A major problem in this framework is the one of

defining the set of VPs in such a way that some good

properties are achieved.

5

1. Introduction (cont.)

More formally, given a communication network, the VPs form a virtual directed graph (digraph) on the top of the physical one, with the same set of vertices but with a different set of arcs.

Specifically, a VP from u to v is represented by an arc from u to v in the virtual digraph.

This virtual digraph provides a Directed Virtual Path

Layout (DVPL) for the physical graph.

Each VC can be viewed as a simple dipath in the virtual digraph.

6

1.1 Definitions

A virtual path (VP) is a simple path in the graph (network)

A virtual channel (VC) of length k, connecting vertices u

and v, is a sequence p1,p2,…,pk of VPs such that pi

begins at vertex u, pk ends at vertex v and the

beginning of pi+1 coincides with the end of pi, for i < k.

A virtual path layout in the network is a collection of VPs, such that every pair of vertices is connected by a VC composed of VPs

7

1.1 Definitions

A hop count in the VPL is the maximum number of VPs

among a virtual channels, connecting every pair of

vertices. A capacity of an arc is the maximum number of VPs

can share this arc.

A load of an arc is the number of VPs sharing this arc (assuming that capacity of each VP is 1).

8

Problem

In this article, we consider the following problem:

Given a capacity on each physical arc, minimize the

diameter of an admissible virtual graph (a virtual

digraph that doesn't load an arc more than its

capacity)

9

The physical network is presented by a strongly connected directed graph (digraph) G=(V,E,c).

|V|=n. The vertex set V represents the networks switches and end-users.

The arc set E represents the set of physical directed arcs.

The parameter c is the capacity function, assigning to each arc e its capacity c(e) (the amount of data arc can carry) .

For simplicity in this paper eE, c(e)=c0.

2. The Model

10

The network formed by the VPs is represented by a strongly connected digraph H=(V,E’) and a function P assigning to each arc e’=(x,y)E’ a simple directedpath (dipath) P(e’) connecting x to y in G.

In our terminology, the pair (H,P) is a virtual digraph on G.

An arc of H is a virtual arc.

The dipath P(e’) in G associated with a virtual arc e’ is a virtual dipath (VP).

2. The Model (cont.)

11

The load of an arc e of G is the number of virtual dipaths containing the arc e, that is,

A virtual digraph (H,P) on G satisfying the requirement

is referred to as a c-admissible Directed Virtual Paths Layout of G, shortly denoted c-DVPL of G.

The aim is to design c-DVPL of G with minimum hop-count, i.e, to find a virtual digraph with minimum diameter.

2. The Model (cont.)

eP e Eeel

eceE, le

12

For any digraph F, dF(x,y) denotes the distance from x

to y in F, and DF denotes diameter of F (maximum

dF(x,y) for all x,y).

The virtual diameter, , of the digraph G with

respect to the capacity c, is the minimum of DH over

all the c-DVPL H of G.

2. The Model (cont.)

G,cD~

13

2. The Model (example)

1

2 3

4

5

6

P(5,2)=(5,6),(6,1),(1,2)

l(4,5)=l(2,3)=2l(6,3)=l(5,1)=1

G=(V,E,c)

eE, c(e)=2

V={1,2,3,4,5,6}

E={(1,2), (1,5), (2,3), (3,4), (4,5), (5,6), (5,1), (6,1), (6,3)}

DG=dG(1,6)=5

H=(V,E’)

E’={(1,3), (2,3), (3,4), (3,5), (4,6), (5,1), (5,2), (6,3)}

DH=dH(4,1)=dH(4,2)=4

14

In figure, G consists of the symmetric directed cycle Cn.

The virtual graph H consists of arcs (i,i+1) in the

clockwise direction and arcs (ip, (i-1)p) in the opposite

direction (assuming that p divides n). The load of every

arc of Cn is 1.

2. The Model (example)

15

3. The Cycle Cn

In this section the physical digraph G is Cn, the

symmetric directed cycle of length n. We choose

arbitrarily a direction on Cn. For concreteness, consider

as positive, or forward (resp., negative or backward)

the clockwise (resp., counterclockwise) direction. We

assume that eE, c(e)=c if e is a forward arc and c(e)=c

if e is a backward arc, for some constant nonnegative

integers c, c.

16

3. The Cycle Cn (cont.)

It turns out that our bounds can be expressed as

functions of = c+ c. It is then convenient to define

ubC(n,) (resp., lbC(n,)) as an upper bound (resp., lower

bound) for valid if c satisfies c+ c= .

By the definition, lbC(n,) ubC(n,).

G,cD~

G,cD~

17

3.1 General Case

In this section, we show the following upper and

lower bounds on the virtual diameter for the cycle.

The bounds are both proved by induction from the

next two lemmas.

12

2122

2~

2

111

nn,cCD

nn

18

Lemma 3.1

Proof.

Let H be an optimal c-DVPL of Cn.

Let [x1,y1]+ be the dipath consisting of all the vertices of

Cn between x1 and y1 in the positive direction.

Let d+(x1,y1) denote the number of arcs in [x1,y1]+.

1,,

2maxmin, plb

p

nnlb CNpC

19

Lemma 3.1 (cont.)

We say that [x1,y1]+ is covered by H if (the VP

corresponding to) some virtual arc e’ contains [x1,y1]+.

Abusively we say that [x1,y1]+ is covered by e’.

20

Lemma 3.1 (cont.)

First we prove that if [x1,y1]+ is covered by e’ then

For this, we shorten the cycle by identifying all the

nodes in [y1,x1]+ with x1, obtaining a cycle C’ of length d+

(x1,y1). Virtual arcs are transformed like in figure.

1,, 11 yxdlbD CH

21

Lemma 3.1 (cont.)

A virtual arc from x[x1,y1]+ to y[x1,y1]+ is left

unchanged.

A virtual arc from x[x1,y1]+ to y[y1,x1]+ is transformed

into the arc (x,x1).

Note that the virtual arc containing the positive arcs

of [x1,y1]+ is transformed into a loop.

We also remove loops or multiple virtual dipaths in

order to get a simple DVPL on C’.

22

Lemma 3.1 (cont.)

This transformation does not increase the load of any arc.

The virtual arc e’ that contained [x1,y1]+ disappears,

so the congestion of any positive arc decreases. Our transformation does not increase the virtual

diameter. Consequently, we obtain a c’-DVPL of C’ (a cycle of

length d+(x1,y1)) with c’++c’=1, and diameter at

most DH. It follows that

(1) 1,, 11 yxdlbD CH

23

Lemma 3.1 (cont.)

Now we argue that there exist vertices u and v with large d+(u,v) such that [u,v]+ is covered.

Let P be the shortest dipath in H from 0 to n/2, and assume w.l.o.g. that P contains the arcs of [0,n/2]+.

Let S denote the set of vertices of P between x and y in the positive direction.

Then S DH + 1, and therefore there exist vertices

u and v such that [u,v]+ is covered and with

(2) .

2,

HD

nvud

24

Lemma 3.1 (cont.)

Let

From (2) we have

And from (1) it follows that

covered is vuvudp ,,max

p

nDH 2

1, plbD CH

25

Lemma 3.2

Proof.

Let us construct a c-DVPL on Cn.

W.l.o.g. suppose that c+ c, so c+ 0.

Let pN+, we proceed as follows.

1,12min,

p

nubpnub CNpC

26

Lemma 3.2 (cont.)

Use n virtual arcs (i,i+1)i[0..n-1] of dilation 1 in positive direction.

Let S be the set of vertices ,

and note that vertices of S form a cycle .

Use an optimal c’-DVPL for with c’= c1, and

c’= c, that is c’+ c’= 1.

p

p

npp 1,,2,,0

p

nC

p

nC

27

Lemma 3.2 (cont.)

By construction, the diameter DS of the set S (i.e., the maximal distance of two vertices in S) is at most

For any vertex x, we have d(S,x) p1 and d(x,S) p1. Hence

1,

p

nubC

1,12,

p

nubpDy,SdxSdx,yd CS

28

Proposition 3.3

Proof.

First we consider the lower bound.

We prove by induction on that

12

2122

2~

2

111

nn,cCD

nn

.2

1,

1

nnlbC

29

Proposition 3.3 (cont.)

For the initial case we have lbC(n,1) = n – 1 n/2.

Now to go from – 1 to we use lemma3.1 witch

states that

Hence lbC(n,1) = n – 1 n/2 and the proof is completed.

1

1

2

1,

2maxmin, p

p

nnlb

NpC

.21

21

,2

max1

11

1

1

npnp

pn

for attained

30

Proposition 3.3 (cont.)

Now, we prove the upper bound.

First we show by induction on that for n = 2a, aN

122122

2,1

an

nubC

31

Proposition 3.3 (cont.)

For = 1 , ubC(n,1) n-1 is true.

For the inductive step from – 1 to , we apply

lemma 3.2 with p = a, getting

ubC(n, ) 2(a1) + ubC(2a1, 1)

By induction,

ubC(2a1, 1) 2( 1)a – 2( 1) + 1

So we get the expected result.

32

Proof of Proposition 3.3 (cont.)

For other values of n, the claim is proved as follows.

a is such that n 2a. As ubC is increasing function on n, we obtain

As , this implies

.2

1

n

a Let

122

21221

σ

nσσσan,σub

σ

C

12

1

n

a .12

2,1

nnubC

33

Corollary 3.4

If c+ = c= c then

12

4~

2

2

12

1

c

n

c nc,cCD

n

34

3.2 Case c =1

The upper bound is the one of proposition 3.3. The lower bound proof requires some care so we

first give some definitions.

Let H be an optimal virtual digraph on G with respect

to the capacity 1.

The following definitions are given for the positive

direction, but similar notions apply for the negative

direction as well.

12232

41,~

122

n

nCDOn n

35

Definition 3.5

The forward successor of a vertex x is denoted x+

[x,y]+ denotes the dipath from x to y in Cn in the

positive direction

A path Q=(e’1,…,e’q) from x to y in H is said to be of

type + if [x,y]+W(Q).

Where W(Q) is the route in Cn associated to the

dipath Q in H.

36

Definition 3.6

A circuit-bracelet of size n is a digraph A of order n constructed as follows (see the Figure): The digraph is made of a set of cycles Ci, iI

directed in a clockwise manner. For any i, Ci and C(i+1) mod |I| share a unique vertex

v (i+1) mod |I| .

The length of the dipath in Ci from vi-1 to vi is

denoted pi and is called the positive length of Ci.

The length of the dipath in Ci, from vi to vi-1 is

denoted ni and is called the negative length of Ci.

The successor of vi in Ci by wi, and the ancestor of

vi+1 in Ci by zi.

37

Lemma 3.7

f(n) is the minimal value of DA, where A is any circuit-bracelet of size n.Proof.

If an arc e of Cn is not used by a virtual dipath P(e’)

with e’E’, we add a virtual arc e’ such that P(e’)=(e). This transformation can only decrease the diameter

of H, which is of no consequence since we only seek for a lower bound on the virtual diameter.

Using this manipulation eE, e’E’ s.t. eP(e’). This implies

(3)

Where w(e’) is the dilation of a VP e’, i.e. the length of P(e’).

1,~

nCDnf

.newew type of e type of e

38

Lemma 3.7 (cont.)

Show that if e’=(x,y)E' is an arc of type + of dilation w(e’) 3 then all the arcs of type – between y– and x+ are of dilation 1.

Since eE, c(e)=1, and there is already a virtual arc of type + between x and y, there is no virtual arc of type + ending at any vertex between x+ and y–.

Since H(V,E’) is strongly connected, there is at least one arc ending at each one of this vertices. These arcs are of type –. For the same reasons of capacity and connectivity, these arcs are of dilation 1.

39

A circuit-bracelet

Due to this property it is easy to see that there exists a digraph isomorphism between H and a circuit-bracelet of size n. (see the figure).

C1w1

z1

C7 v7 C6

C3

z0

–

w5

C5 +v6

v5

z5

C2v2 v3

w2 = z2

H =

+

vowo

C0

v1

C4

v4

u1

u2

u3

u4

u5

u1=vi

u2

u3

u4

u5=vi+1

40

Lemma 3.8

and the total number of circuits in an optimal circuit-bracelet is .

Proof.

By the proposition 3.3, there exists a regular circuit-bracelet with diameter at most , so

The size of any circuit in an optimal circuit-bracelet is at most , otherwise the distance from

wi to the second neighbor of vi on the bigger cycle Ci

is more than f(n).

nnf n

122 n

nOnf

nOnf 2

41

Lemma 3.8 (cont.)

Hence there are at least circuits.

Moreover the total number of circuits is less than

, otherwise there exist two vertices at

distance more than f(n).

Thus and the lemma follows.

n

nOnf 2

nnf

42

Definitions

We prove proposition 3.10 for the special case of

regular circuit-bracelet satisfying i, ni = 1.

The circuits of regular circuit-bracelet all consist of a

single arc of type – and pi arcs of type +.

Remark that pi is then the length of Ci.

Let g(n) denote the minimal value of DA where A is

any regular circuit-bracelet of size n.

vi

vi+1

pi

43

Lemma 3.9

Proof. We assume that n is sufficiently large. Let p be an integer and D the diameter of the

considered circuit-bracelet. Call a circuit big if its size is greater than D/p, small

otherwise. Recall that the size of any circuit is less than D + 2. Let b be the number of big circuits Let s be the number of small circuits.

.122 Onng

44

Lemma 3.9 (cont.)

We have

(4)

Suppose that big circuits are ordered cyclically according to the circuit-bracelet structure:

as shown on the figure. Let k{0,1,…,b–1} and consider dipaths from In the positive direction the cost is exactly

.22 DbsDbp

Dsn and

110,,,

biii CCC

2

,

pkkj

ik jpd

.pkk ii zw

to

45

Lemma 3.9 (cont.)

These circuits are big Hence

So we must use the negative direction.

The length is then

where – the number of vertices in the all small circuits.

p

Dp

ji

22

1 DpDD

p

pdk

if

.3 Diisbppd kpkiik pkk

.321

0

bDbsbpsbbndbk

kk

46

Lemma 3.9 (cont.)

so

Note now that , so

(5)

If the coefficient of s in (5) is positive then the left

factor of (5) is greater than which is

greater than

In turn, the coefficient of s is positive if

.3

2D

b

sppsb

b

n

p

Ds

.32

12

Dpb

p

bp

Dsb

b

n

32

pbb

n

.322 pn

.2

pp

Db

47

Lemma 3.9 (cont.)

(4) implies

Using that fact that But

if , and the latter inequality is true if p

33 and n is large enough.

It is follows that

.2

222

p

D

D

nbbD

p

Dn so and

.8

162,22

p

pnbnD obtain we

pp

D

p

pn

2

8

162

.3622 nng

02

48

2

n

pp

48

Proposition 3.10

Proof.

The upper bound is the one given for the general

case. We conjecture that this bound is tight. It would

be desirable to obtain a simpler argument that could

extend to higher capacities.

Recall that Consider a circuit-bracelet, and

recall that ni+pi D+1, so that we can find an integer

k such that

12232

41,~

122

n

nCDOn n

.nD

.12,1,1

npnDpnki iiki ii with

49

Proposition 3.10 (cont.)

Consider the shortest dipath from v1 to vk+1 and suppose

that it uses the positive direction. So

It follows that

So, the dipath from vk to v1 cannot use the negative

direction, and must use the positive one.

It follows that

Globally,

If we remove this vertices we obtain a regular circuit-bracelet with lesser diameter. It follows that

.,1

Dpki i

.,1

Dnki i

.,1

Dpki i

.2 nDpi n

.1221

122

n

nnnngnf

50

4. The Path Pn

In this section the physical digraph G is the n-vertex

symmetric directed path Pn. Our bounds are valid for

any capacity function c such that positive (resp.,

negative) arcs have capacity c+ (resp., c–) and the

additional requirement c+ 1, c– 1.

Let = c++ c–.

51

Proposition 4.1

Proof.

Let us first prove the lower bound.

Let H be a c-DVPL of Pn.

We say that a sub-path [x,y] is covered by H if the

dipaths from x to y and from y to x are both contained

in (the VP corresponding to) some virtual arc.

422

112,

~2

1

11

1

n

cPDn

n

52

Proposition 4.1 (cont.)

First we show that if [x,y] is covered then

DH > lbC ( d(x,y) , - 2 ) .

Indeed if [x,y] is covered we identify x and y and

collapse the path into a cycle of length d(x,y).

We ignore the virtual paths covering [x,y] (see the

proof of lemma 3.1 for details).

We obtain a c’-DVPL for Cd(x,y) with c’++c’–= –2.

53

Proposition 4.1 (cont.)

Now, consider two shortest dipaths in H, one from 0

to n–1 and the second from n–1 to 0.

There are at most 2DH intermediate points (including

0 and n–1) on these two dipaths.

Hence we can find two consecutive intermediate

vertices x and y, with [x,y] covered, such that

If m = max{d(x,y) | [x,y] is covered}, we have

But due to the covering property DH lbC(m, – 2).

.2

,HD

nyxd

.2mn

DH

54

Proposition 4.1 (cont.)

Hence

Using the lower bound on lbC(m,) given in proposition 3.3, and maximizing in m, completes the lower bound proof.

.

2,,

2max, mlb

m

nmlb CP

55

Proposition 4.1 (cont.)

To prove the upper bound, we construct a VPL based

on the best VPL we know on the cycle Cn with c’+ = c+

and c’– = c– – 1.

In this VPL, no VP passes over vertex 0. So we cut

the cycle at vertex 0 and consider it as the path Pn.

On the negative direction, we add a VP of dilation n

from n – 1 to 0 (See Fig. 4). The added VP is used at

most once in a path on H. The bound is the one for

the cycle Cn–1, sum of capacities – 1 plus 1.

56

Figure 4: Pn, c=2