Wide-Sense Nonblocking Multicast in a Class of Regular Optical Networks

1. Wide-Sense Nonblocking Multicast in a Class of Regular Optical Networks From: C. Zhou and Y. Yang, IEEE Transactions on communications, vol. 50, No. 1, Jan. 2002

2. Abstract • Study multicast communication in a class of optical WDM networks with regular topologies. • Derive the necessary and sufficient conditions on the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for multicast communication under some commonly used routing algorithm.

3. Outline • Introduction • Linear Arrays • Rings (unidirectional, bidirectional) • Meshes and Tori • Hypercubes • Conclusions

4. Definitions • Wavelength-division multiplexing (WDM): multiple wavelength channels from different end-users can be multiplexed on the same fiber. • A connection or a lightpath is an ordered pair of nodes (x,y) corresponding to transmission of a packet from source x to destination y.

5. Definitions • Multicast communication: transmitting information from a single source node to multiple destination nodes. • A multicast assignment is a mapping from a set of source nodes to a maximum set of destination nodes with no overlapping allowed among the destination nodes of different source nodes.

6. Examples of multicast assignments in a 4-node network • There are a total of N connections in any multicast assignment. • An arbitrary multicast communication pattern can be decomposed into several multicast assignments.

7. Nonblocking • Strictly Nonblocking (SNB):For any legitimate connection request, it is always possible to provide a connection path without disturbing existing connections. • Wide-sense Nonblocking (WSNB):If the path selection must follow a routing algorithm to maintain the nonblocking connecting capacity. • Rearrangeable Nonblocking (RNB)

8. Focus of this paper • To determine the minimum number of wavelengths required for a WDM network to be wide-sense nonblocking for arbitrary multicast assignments. • In other words, to determine the condition on which any multicast assignment can be embedded in a WDM network on-line under the routing algorithm.

9. Assumption • Each link in the network is bidirectional. • No wavelength converter facility is available in the network. Thus, a connection must use the same wavelength throughout its path. • No light splitters are equipped at each routing nodes.

10. Outline • Introduction • Linear Arrays • Rings (unidirectional, bidirectional) • Meshes and Tori • Hypercubes • Conclusions

11. Linear array There are only two possible directions for anyconnection in a linear array and the routing algorithm is unique.

12. Theorem 1 • The necessary and sufficient condition for a WDM linear array with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengthsww =N －1.

13. Proof of Theorem 1 • Sufficiency (ww N －1): • The connections on the same link in different directions may use the same wavelength. • There are at most N －1 connections in the same direction in a multicast assignment.

14. Proof of Theorem 1 • Necessity (ww N －1):

15. Wavelength assignment algorithm in a linear array TR: the wavelengths used for rightward connections. TL: the wavelengths used for leftward connections. TW: available wavelengths • Step 1: TL = TR =Φ, |TW| = N －1. • Step 2: When a new rightward (leftward) connection is requested, assign it a wavelength in TW－TR (TW－TL), and add this wavelength to TR (TL). • Step 3: When an existing rightward (leftward) connection is released, delete it from TR (TL).

16. Outline • Introduction • Linear Arrays • Rings (unidirectional, bidirectional) • Meshes and Tori • Hypercubes • Conclusions

17. Unidirectional Rings • The routing algorithm is unique. • Assume the directional of a ring is counter-clockwise.

18. Theorem 2 • The necessary and sufficient condition for a unidirectional WDM ring with N nodes to be wide-sense nonblocking for any multicast assignment is the number of wavelengths ww= N.

19. Conflict graph • Given a collection of connections • G=(V,E) : an undirected graph, whereV={v: v is a connection in the network}E={ab: a and b share a physical fiber link} (a, bcan’t use the same wavelength.) • The chromatic numberc(G) of G is the minimum number of wavelengths required for the corresponding connections.

20. Conflict graph (contd.) • Find c(G) is a NP-complete problem. • c(G) can be efficiently determinedfor conflict graphs for multicast communication in most of the networks.

21. Proof of Theorem 2 • Sufficiency (wwN) : Since there are a total of N connections in any multicast assignment. • Necessity (wwN) : Consider the multicast assignment:PN = {i (i 1) mod N : 0  i  N-1}The conflict graph of PN is KN.Therefore wwc(KN)=N.

22. P6 Conflict graph=K6

23. Tu: currently used wavelengths Tn: available wavelengths

24. Bidirectional Rings • There are two possible paths for a connection between any two nodes: clockwise or counter-clockwise. • The shortest path routingalgorithm is adopted.

26. Theorem 3 • The necessary and sufficient condition for a bidirectional WDM ring with N nodes to be wide-sense nonblocking for any multicast assignment under shortest path routing is the number of wavelengths ww=N/2.

27. Proof of Theorem 3 • Sufficiency (wwN/2) : • In a multicast assignment, there are at most N connections. • We can divide the N connections in a multicast assignment into N/2pairs with the connections in each pair using the same wavelength.

29. Proof of Theorem 3 • Necessity (wwN/2) :

30. Proof of Theorem 3 (0,2) (4,1) (1,3) (2,4) (3,0) • Necessity (wwN/2, N is odd) :PN = {i (i+(N1)/2) mod N : 0  i  N1}

32. Outline • Introduction • Linear Arrays • Rings (unidirectional, bidirectional) • Meshes and Tori • Hypercubes • Conclusions

33. Meshes and Tori • Definition 1:Under the row-major shortest path routing, for a connection request ((x0, y0), (x1, y1)) in a mesh or a torus,the path is deterministically from node (x0, y0) to node (x1, y1) in row x0 alongthe shortest path first, then to node (x1, y1) in column y1 along the shortest path.

34. Meshes • Definition 2:For a connection in a mesh under row-major shortest path routing, if the connection goes right at the first step from the source, we refer to it as a rightward connection. If the connection goes left at the first step, we refer to it as a leftward connection. Otherwise, we refer to it as a straight connection.

35. Theorem 4 • The necessary and sufficient condition for a WDM mesh with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths ww = p(q 1)

36. Proof of Theorem 4 • Sufficiency (ww p(q 1)) : • Wavelengths: w0, w1, …, wp(q 1) 1 • Let Ri ={wi(q 1), wi(q1)+1, …, w(i+1)(q 1)1},0  i  p－1 . (|Ri|= q 1) • Let the connections destined to row i use the wavelengths within range Ri. • The connections destined to the same column will use different wavelengths.

37. Proof of Theorem 4

38. Proof of Theorem 4 • Sufficiency (ww p(q 1)) (continued…) • Among all the connections to the same row, if the sources of two connections are in different rows, they can use the same wavelength. (by row-major routing) • Consider those connections originated from the same row and destined to the same row. Since each row can be considered as a linear array with q nodes. By Thm 1, q-1 wavelengths are sufficient for WSNB. Hence ww p(q 1).

39. Necessity (wwp(q1)) : Node (0,0) is the source of rightward p(q 1)connections. They must share the link (0,0)  (0,1).So p(q 1) wavelengths are required.

40. TRi: currently used wavelengths for rightward connections to row i.TLi : currently used wavelengths for leftward connections to row i.

41. Torus • Definition:A torus network is a mesh with wrap-around connections in both the x and y directions. This allowed the most distant processors to communicate in 2 hops.

43. Theorem 5 • The necessary and sufficient condition for a WDM torus with p rows and q columns to be wide-sense nonblocking for any multicast assignment under row-major shortest path routing is the number of wavelengths ww = pq/2.

44. Proof of Theorem 5 • Sufficiency (ww pq/2) : • (Similar to meshes) Divide the wavelengths to sets R0~Rp-1, and let the connections destined to row i use the wavelengths within set Ri. • We need only to consider those connections originated from the same row and destined to the same row. Since each row can be considered as a bidirectional ring with q nodes. By Thm 3, q/2 wavelengths are sufficient for WSNB. Hence ww pq/2.

45. Necessity (wwpq/2) when q is even: Consider the connection (0,0)  (i, j), where 1  i  p-1, 0  j  q-1. There arep (q/ 2)connections passed through the link(0,0)(0,1) or (0,0)(0, q-1). q=6

46. q=5 • Necessity (wwpq/2) when q is odd: Consider the connection(0, j)  (i, j+(q-1)/2modq), where 0  i  p-1, 0  j  q-1. The connection(0, j)  (i, j+(q-1)/2modq)must pass through the node (0, j+(q-1)/2modq). So there are p connections passthrough the link(0,j)  (0,j+(q-1)/2modq). By Thm 3, a ring with q nodes need (q-1)/2+1 wavelengths. Hence pq/2 wavelengths isnecessary.

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48. Outline • Introduction • Linear Arrays • Rings (unidirectional, bidirectional) • Meshes and Tori • Hypercubes • Conclusions

49. Hypercubes • Definition: (e-cube routing)In an n-cube with N=2n nodes, let each node b be binary-coded as b = bnbn-1…b2b1, where the ist bit corresponds to the ith dimension.In e-cube routing, a route from node s = snsn-1…s2s1 to node d = dndn-1…d2d1 is uniquely determined as follows:

50. Hypercubes • Definition: (e-cube routing) (contd.)s = snsn-1…s2s1 snsn-1…s2d1 snsn-1…d2d1 … sndn-1…d2d1dndn-1…d2d1 =dNote that if si=di, no routing is needed along dimension i .