1 / 29

Hop-Congestion Trade-Offs For ATM Networks

Hop-Congestion Trade-Offs For ATM Networks. Evangelos Kranakis Danny Krizanc Andrzej Pelc Presented by Eugenia Bouts. Abstract & Introduction.

affrica
Download Presentation

Hop-Congestion Trade-Offs For ATM Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Hop-Congestion Trade-Offs For ATM Networks Evangelos Kranakis Danny Krizanc Andrzej Pelc Presented by Eugenia Bouts

  2. Abstract & Introduction • The model we use in this paper is based on the Virtual Path Layout model introduced by Gerstel and Zaks [4, 5]. Messages may be transmitted through arbitrarily long virtual paths. Packets are routed along those paths by maintaining a routing field whose subfields determine intermediary destinations of the packet, i.e. endpoints of virtual paths on its way to the final destination • In such a network it is important to construct path layouts that minimize the hop number (i.e. the number of virtual paths used to travel between any two nodes) as a function of edge congestion (i.e. the number of virtual paths passing through a link).

  3. Notation and definitions • N - arbitrary connected network • VP - virtual path in N is a simple chain in this network (v1 , … , vk) • VC - virtual channel of length k, joining vertices u and v, is a sequence of k VP's • P –layout in the network N, is a collection of virtual paths in N , such that every pair of vertices u ,v of N is joined by a VC composed of VP's from P.

  4. Notation and definitions (cont.) • We assume that traversing any VP is made in a single hop. • HopsN(P, u, v) is the minimum number of hops using VP's in P to go from u to v. • HopsN(P) = MAX{HopsN(P, u, v) | u,v Є N} • We are interested in the hop number for layouts of bounded congestion. • CN(P)- congestion of given layout P as the maximum number of VP's from P passing through any link of N. • HopsN(c)= MIN{HopsN (P) | P : CN(P)=c} (for any c ≥ 1)

  5. A General Lower Bound • Theorem 1: For any network N on n vertices, with maximal degree d, and for any c ≥1 we have the following lower bound: HopsN(c) ≥

  6. A General Lower Bound (cont.) • Proof: Let H= HopsN(c). Choose any vertex of the network as a root, say r. Since the congestion is c and maximum degree of a node is ≤ d, at most dc nodes can be reached from r with one hop. More generally, with at most h hops at most dc + (dc)2 + … + (dc)h vertices can be reached from r. Put h=H and it follows that n ≤ (dc)H+1, which implies that H+1 ≥ □

  7. Chain Graphs • The results in this section are stated for the chain Ln with vertices 0,1,...,n. • Lower Bounds • Theorem2: HopsLn(c) ≥ ½·n1/c for any c ≥ 1

  8. Notions • Notions: • P - a layout for Ln • I = [a,b]Ln( any segment of the chain ) • for J Psuch that • Define the layout PI induced by the layout P on the segment I as the set • Observation: if P has congestion c then PI has congestion <c.

  9. Figure 1- Induced Layout

  10. Lemma • Lemma:For any layout P in Ln with congestion c, there is a vertex u such that • HopsLn(P,0,u) ≥ ½·n1/c • HopsLn(P,u,n) ≥ ½·n1/c • Proof: We prove the statement by induction on c. For c = 1 the result is easy; take as u the midpoint of Ln. Assume the lemma is true for c-1. Let P be an arbitrary layout of Ln with congestion c. Let I=[a,b] be the largest virtual path in layout P. We consider two cases. • |I| ≥ n(c-1)/c • |I| ≤ n(c-1)/c

  11. Lemma (proof) • Case 1: |I| ≤ n(c-1)/c In this case at least n1/c hops are necessary to reach one end-point of the chain Ln and the inequality of the lemma is clearly satisfied. • Case 2: |I| ≥ n(c-1)/c Take u be a mid-point of I. To prove the first inequality assume not and let C=(p1, … ,pk) be a virtual channel in P, of length k< ½·n1/c, joining 1 with u.Let (h0, … ,hk) be the end-points of consecutive VP’s in C.Let hr be the last vertex such hr ≤ a or hr ≥ b. Without loss of generality we may assume that hr ≤ a.Let qr+1 denote the VP joining a with hr+1. thus C’=(qr+1,pr+2, … ,pk) is a virtual channel in the layout PI induced by P on I. By the inductive hypotesis, the length of C’ is at least ½|I|1/(c-1) ≥ ½n1/c. Hence, k≥ ½n1/c, contradiction. □

  12. Lower Bound • Theorem2: HopsLn(c) ≥ ½·n1/c for any c ≥ 1 • Proof: By Lemma there is a vertex u such that • HopsLn(P,0,u) ≥ ½·n1/c • HopsLn(P,u,n) ≥ ½·n1/c • This completes the proof of the theorem. □

  13. Asymptotically Optimal Path Layouts • Theorem3: For any positive integer c, HopsLn(c) ≤c·n1/c • Proof: Let c be a positive integer. For simplicity assume that n1/c is aninteger. The construction can be easily modified in the general case. We construct the layout consisting of nested virtual paths of lengths n(c-1)/c, n(c-2)/c,…,n1/c,1. VP’s of each length form a virtual channel joining both ends of Ln (Figure 2).The layout consisting of all those VP’s has congestion c and hop-number at most cn1/c. □

  14. Figure 2

  15. Theorem 4 • Corollary: For constant c we have • Theorem 4: For any integer k we have HopsLn(kn1/k) ≤ 2k

  16. Theorem 4 - proof • The layout proving this upper bound is givenin Figure3.

  17. Construct VP’s with left end-point 0, of lengths n(k-1)/k,2n(k-1)/k,… ,n1/k,n(k-1)/k. From their right end-points of construct VP’s going right, of lengths n(k-2)/k,2n(k-2)/k,… ,n1/k,n(k-2)/k. Theorem 4 – proof (cont.)

  18. Theorem 4 – proof (cont.) • Continue in this way with non-overlapping VP's of sizes n(k-j)/k,2n(k-j)/k,… ,n1/k,n(k-j)/k for j=1,…,k. • The layout consisting of all those VP’s has congestion kn1/k and hop-number at most 2k □

  19. Asymptotically Tight Bound • As a corollary we obtain an asymptotically tight bound for congestion log 2 n/log log n. • Corollary: HopsLn (log2n/ loglog n) Θ(log n/loglog n). • Proof:The upper bound is obtained from theorem 4 for k = log n/log log n. Then n1/k= log n and kn1/k= log2n/ loglog n. The lower bound follows immediately from theorem 1. □

  20. Theorem 5 • In case of congestion 2 we have a more precise result: we give a layout for the chain which is optimal up to an additive constant. • Theorem 5: For the chain Ln we have

  21. C3 4 2 4 2 2t 2t … … C1 C2 Theorem 5 - proof • Proof: We first prove the upper bound. Let t= We create 2 virtual channels C1,C2 consisting of non –overlapping paths whose lengths form arithmetic progressions 2,4,6,…,2i,…2t left to right and right to left respectively. Verticles t(t+1) and n-t(t+1) are joined by a third virtual channel C3 consisting of three non-overlapping Vp’s of lengths differing by at most 1.

  22. Theorem 5 – proof (cont.) • We construct the layout P for Ln consisting of all VP’s of length 1 (links of Ln) and all VP’s in chains C1,C2 and C3. • Since all VP’s in the above chains are non-overlapping, layout P has congestion 2. • The distance between end points t(t+1) and n-t(t+1) of channel C3 is • Thus each VP in this channel has length at most

  23. Theorem 5 – proof (cont.) • Consider any pair of vertices u and v in Ln • Case 1: u is inside a VP ( length 2i ) of channel C1 and v is inside a VP ( length 2j ) of channel C2 we have : • Case 2: u is inside a VP ( length 2i ) of channel C1 and v is inside a VP of channel C3 we have : • All other cases when u and v are in VP’s from different channels are similar. • It remains to consider the situation when u and v are in VP’s from the same channel. If this is channel C1 or C2 HopsLn(P,u,v) is less than in case 1.

  24. Theorem 5 – proof (cont.) • Case 3: u and v are inside VP’s of channel C3. • The worst case occures when u and v in different external VP’s of C3.We have • It follows that for any u,v and consequently • This implies the upper bound.

  25. Theorem 5 – proof (cont.) • In order to prove the lower bound, consider any layout P in Ln, with congestion at most 2.Let Q be the set of VP’s in P of length at least 2 and let q be the size of Q. • The overlap of any VP’s from Q can have length at most 1 ,hence a left-right ordering of these VP’s is well defined. • Definitions: • A1,A2,…Ak first k VP’s ordered from left to right • Bk,Bk-1,…B1 last k VP’s in this ordering • If Q is odd let C0 bethe remaining VP in Q • Let ai,bi,c0 be the length of Ai,Bi,C0 respectively • If Q is even c0=0

  26. Theorem 5 – proof (cont.) • We shall prove that HopsLn(P)>X= • Suppose not. • If u is the mid-point of C0 and v is the end–point of this VP we get • If u and v are mid-points of Ak and Bk we get • If u and v are mid-points of Ak-1 and Bk-1 we get • If u and v are mid-points of Ak-i and Bk-i we get

  27. Theorem 5 – proof (cont.) • If u and v are mid-points of A1 and B1 we get • Adding all the above inequalities we get • Since all Vp’s in Q cover Ln, this implies and hence

  28. Theorem 5 – proof (cont.) • Let • Substituting in the latter inequality we get • This is a contradiction because is negative for any t. • It follows that HopsLn(P)>X □

  29. References • [1] B. Awerbuch, A. BarNoy, N. Linial and D. Peleg, “Improved Routing with Succinct Tables” • [2] J. Y. Le Boudec, “The Asynchronous Transfer Mode: A Tutorial” • [3] I. Cidon, O. Gerstel and S. Zaks, ”A Scalable Approach to Routing in ATM Networks” • [4] O. Gerstel and S. Zaks,”The Virtual Path Layout Problem in Fast Networks” • [5] O. Gerstel and S. Zaks, “The Virtual Path Layout Problem in ATM Ring and Mesh Networks” • [6] D. E. McDysan and D. L. Spohn,”ATM: Theory and Applications” • [7] M. de Prycker, “Asynchronous Transfer Mode: Solutions for Broadband ISDN • [8] N. Santoro and R. Khatib,”Labeling and Implicit Routing in Networks”

More Related