1. Directed Virtual Path Layouts In ATM networks Jean-Claude Bermond Nausica Marlin David Peleg Stepane Perennes Presented by Boris Mudrik

2. 2 Contents  1. Introduction 1. Introduction 1.1 Definitions Problem  2. The Model 2. The Model Examples  3. The Cycle C n 3. The Cycle C n 3.1 General Case 3.2 Case c =1  4. The Path P n 4. The Path P n

3. 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. 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. 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. 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 p 1,p 2,…,p k of VPs such that p i begins at vertex u, p k ends at vertex v and the beginning of p i+1 coincides with the end of p i, 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. 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. 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. 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  e  E, c(e)=c 0. 2. The Model

10. 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 directed path (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. 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.)

12. 12  For any digraph F, d F (x,y) denotes the distance from x to y in F, and D F denotes diameter of F (maximum d F (x,y) for all x,y ).  The virtual diameter,, of the digraph G with respect to the capacity c, is the minimum of D H over all the c -DVPL H of G. 2. The Model (cont.)

13. 13 2. The Model (example) 1 23 4 5 6 P(5,2)=(5,6),(6,1),(1,2) l(4,5)=l(2,3)=2 l(6,3)=l(5,1)=1 G=(V,E,c)  e  E, 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)} D G =d G (1,6)=5 H=(V,E’) E’={(1,3), (2,3), (3,4), (3,5), (4,6), (5,1), (5,2), (6,3)} D H =d H (4,1)=d H (4,2)=4

14. 14 In figure, G consists of the symmetric directed cycle C n. 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 C n is 1. 2. The Model (example)

15. 15 3. The Cycle C n In this section the physical digraph G is C n, the symmetric directed cycle of length n. We choose arbitrarily a direction on C n. For concreteness, consider as positive, or forward (resp., negative or backward) the clockwise (resp., counterclockwise) direction. We assume that  e  E, 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. 16 3. The Cycle C n (cont.) It turns out that our bounds can be expressed as functions of  = c  + c . It is then convenient to define ub C (n,  ) (resp., lb C (n,  ) ) as an upper bound (resp., lower bound) for valid if c satisfies c  + c  = . By the definition, lb C (n,  )   ub C (n,  ).

17. 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.

18. 18 Lemma 3.1 Proof.  Let H be an optimal c -DVPL of C n.  Let [x 1,y 1 ] + be the dipath consisting of all the vertices of C n between x 1 and y 1 in the positive direction.  Let d + (x 1, y 1 ) denote the number of arcs in [x 1, y 1 ] +.

19. 19 Lemma 3.1 (cont.)  We say that [x 1, y 1 ] + is covered by H if (the VP corresponding to) some virtual arc e ’ contains [x 1, y 1 ] +. Abusively we say that [x 1, y 1 ] + is covered by e ’.

20. 20 Lemma 3.1 (cont.)  First we prove that if [x 1, y 1 ] + is covered by e ’ then For this, we shorten the cycle by identifying all the nodes in [y 1, x 1 ] + with x 1, obtaining a cycle C ’ of length d + (x 1, y 1 ). Virtual arcs are transformed like in figure.

21. 21 Lemma 3.1 (cont.)  A virtual arc from x  [x 1,y 1 ] + to y  [x 1,y 1 ] + is left unchanged.  A virtual arc from x  [x 1,y 1 ] + to y  [y 1,x 1 ] + is transformed into the arc (x,x 1 ).  Note that the virtual arc containing the positive arcs of [x 1,y 1 ] + is transformed into a loop.  We also remove loops or multiple virtual dipaths in order to get a simple DVPL on C ’.

22. 22 Lemma 3.1 (cont.)  This transformation does not increase the load of any arc.  The virtual arc e ’ that contained [x 1,y 1 ] + 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 + (x 1, y 1 ) ) with c ’ + + c ’  = 1, and diameter at most D H. It follows that (1)

23. 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   D H + 1, and therefore there exist vertices u and v such that [u,v] + is covered and with (2)

24. 24 Lemma 3.1 (cont.)  Let  From (2) we have(2)  And from (1) it follows that(1)

25. 25 Lemma 3.2 Proof.  Let us construct a c -DVPL on C n.  W.l.o.g. suppose that c +  c , so c +  0.  Let p  N +, we proceed as follows.

26. 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 ’  = c  1, and c ’  = c , that is c ’  + c ’  = 1.

27. 27 Lemma 3.2 (cont.)  By construction, the diameter D S 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)  p  1 and d(x,S)  p  1.  Hence

28. 28 Proposition 3.3 Proof.  First we consider the lower bound.  We prove by induction on  that

29. 29 Proposition 3.3 (cont.)  For the initial case we have lb C (n,1) = n – 1  n/2.  Now to go from  – 1 to  we use lemma3.1 witch states thatlemma3.1   Hence lb C (n,1) = n – 1  n/2 and the proof is completed.

30. 30 Proposition 3.3 (cont.)  Now, we prove the upper bound.  First we show by induction on  that for n = 2a , a  N

31. 31 Proposition 3.3 (cont.)  For  = 1, ub C (n,1)  n-1 is true.  For the inductive step from  – 1 to , we apply lemma 3.2 with p = a, getting lemma 3.2 ub C (n,  )  2(a  1) + ub C (2a  1,   1)  By induction, ub C (2a  1,  1)  2(   1)a – 2(   1) + 1  So we get the expected result.

32. 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 ub C is increasing function on n, we obtain  As, this implies

33. 33 Corollary 3.4  If c + = c  = c then

34. 34 3.2 Case c =1  The upper bound is the one of proposition 3.3.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.

35. 35 Definition 3.5  The forward successor of a vertex x is denoted x +  [x,y] + denotes the dipath from x to y in C n 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 C n associated to the dipath Q in H.

36. 36 Definition 3.6  A circuit-bracelet of size n is a digraph A of order n constructed as follows (see the Figure):Figure The digraph is made of a set of cycles C i, i  I directed in a clockwise manner. For any i, C i and C (i+1) mod |I| share a unique vertex v (i+1) mod |I|. The length of the dipath in C i from v i-1 to v i is denoted p i and is called the positive length of C i. The length of the dipath in C i, from v i to v i-1 is denoted n i and is called the negative length of C i. The successor of v i in C i by w i, and the ancestor of v i+1 in C i by z i.

37. 37 Lemma 3.7  f(n) is the minimal value of D A, where A is any circuit- bracelet of size n. Proof.  If an arc e of C n 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  e  E,  e ’ E ’ s.t. e  P(e ’ ).  This implies (3)  Where w(e ’ ) is the dilation of a VP e ’, i.e. the length of P(e ’ ).

38. 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  e  E, 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. 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). C1C1 w1w1 z1z1 C7C7 v7v7 C6C6 C3C3 z0z0 – w5w5 C5C5 + v6v6 v5v5 z5z5 C2C2 v2v2 v3v3 w 2 = z 2 H = + vovo wowo C0C0 v1v1 C4C4 v4v4 u1u1 u2u2 u3u3 u4u4 u5u5 u1=viu1=vi u2u2 u3u3 u4u4 u 5 =v i+1

40. 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, soproposition 3.3  The size of any circuit in an optimal circuit-bracelet is at most, otherwise the distance from w i to the second neighbor of v i on the bigger cycle C i is more than f(n).

41. 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.

42. 42 Definitions  We prove proposition 3.10 for the special case of regular circuit-bracelet satisfying  i, n i = 1.  The circuits of regular circuit-bracelet all consist of a single arc of type – and p i arcs of type +. Remark that p i is then the length of C i.  Let g(n) denote the minimal value of D A where A is any regular circuit-bracelet of size n. vivi v i+1 pipi

43. 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.

44. 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.figure  Let k  {0,1,…,b–1} and consider dipaths from  In the positive direction the cost is exactly

45. 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.

46. 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

47. 47 Lemma 3.9 (cont.)  (4) implies (4)  Using that fact that  But if, and the latter inequality is true if p  33 and n is large enough.  It is follows that

48. 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 n i +p i  D+1, so that we can find an integer k such that

49. 49 Proposition 3.10 (cont.)  Consider the shortest dipath from v 1 to v k+1 and suppose that it uses the positive direction. So  It follows that  So, the dipath from v k to v 1 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

50. 50 4. The Path P n  In this section the physical digraph G is the n -vertex symmetric directed path P n. 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. 51 Proposition 4.1 Proof.  Let us first prove the lower bound.  Let H be a c -DVPL of P n.  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.

52. 52 Proposition 4.1 (cont.)  First we show that if [x,y] is covered then D H > lb C ( 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).lemma 3.1  We obtain a c ’-DVPL for C d(x,y) with c ’ + +c ’ – =  –2.

53. 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 2D H 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 D H  lb C (m,  – 2).

54. 54 Proposition 4.1 (cont.)  Hence  Using the lower bound on lb C (m,  ) given in proposition 3.3, and maximizing in m, completes the lower bound proof. proposition 3.3

55. 55 Proposition 4.1 (cont.)  To prove the upper bound, we construct a VPL based on the best VPL we know on the cycle C n 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 P n.  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 C n–1, sum of capacities  – 1 plus 1.Fig. 4

56. 56 Figure 4: P n, c=2