B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), 307-341.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the source and the destination. We show that there exists an n node network on which every routing algorithm of stretch factor s 3 requires at least a total of Omega Gamma n ) bits of routing information, whereas for stretch factor s = 3 the best known upper bound is O(n in total [1]. We show a similar gap for the space complexity of routing schemes on the subclass of networks of diameter 2. Keywords: compact routing in distributed networks, near shortest path routing, routing tables. A preliminary version has been partially published in the proceedings of SIROCCO ....

....They showed that a fraction of 1 Gamma n of all n node labeled graphs supports routing schemes with O(n ) bits in total. For stretch factor s 2, no tight lower bound is known. For stretch factor s = 3, Peleg Upfal s lower bound gives Omega Gamma n 1:1 ) bits, whereas O(n suffices [1]. In the next section, we will show ) are necessary for any stretch factor s 3. 3 Lower Bound of the Memory Requirement We will prove a lower bound of the total number of bits required to describe a routing function of stretch factor less than 3 on n node networks. The model considered for ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307-- 341.

....showed in [8] an optimal lower bound of (n 2 log n) bits, while Buhrman, Hoepman, and Vit anyi showed in [2] that O(n 2 ) bits are sucient for shortest path routing on almost all networks. For s = 3, Peleg Upfal s lower bound gives n 1:1 ) bits, whereas O(n 3=2 log 3=2 n) bits suces [1]. We will show that n 2 ) are necessary for any stretch factor s 3. 4 3 Lower Bound of the Memory Requirement We will lower bound the number of bits required to describe a routing function of stretch factor less than 3 on n node networks. We assume that the names of the nodes are unique ....

Baruch Awerbuch, Amotz Bar-Noy, Nathan Linial, and David Peleg. Improved routing strategies with succint tables. Journal of Algorithms, 11:307-341, 1990.

....by the path. A routing function R is said a shortest path routing function if and only if the paths induced by R are always shortest paths in G. In this paper, we consider shortest path routing function only. Tradeoffs between memory requirement and length of the routing path have been studied in [27, 1, 2]. However, lower bounds given in these papers are not tight for near shortest path routing schemes, and do not include the maximum degree of the network (see [14] for more precisions about lower and upper bounds on memory requirement) 1.3 Memory Requirement Definition 2 (Memory Requirement of a ....

....is important. We will see in Section 3 that if the n vertex labels would have been taken in the set f1; n 2 g, we could not have apply our arguments. However assumption (4) is implicitly used in the literature (e.g. see [27] and [13] even if some upper bounds use longer labels (see [27, 16, 17, 1, 2]) 1.4 Lower Bound on the Memory Requirement This section shortly presents the main results of this paper. Theorem 1 Let , 0 1, be a fixed constant. For every integer d, 3 d n, there exists a graph of order n and maximum degree d that has a memory requirement of Theta(n log d) bits ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

....by the path. A routing function R is said a shortest path routing function if and only if the paths induced by R are always shortest paths in G. In this paper, we consider shortest path routing function only. Tradeoffs between memory requirement and length of the routing path have been studied in [1, 2, 18]. However, lower bounds given in these papers are not tight for near shortest path routing schemes, and do not include the maximum degree of the network as parameter. 1.3 Memory Requirement Definition 2 (Memory Requirement of a Vertex) Given a graph G, a routing function R on G, and a vertex x ....

....x. Note that the limit to f1; ng for the vertexlabeling corresponds to the model of many compact routing schemes, including the interval routing schemes, and is implicitly used in the literature for lower bounds (e.g. see [18] and [8] However, certain upper bounds use longer labels (see [1, 2, 4, 11, 12, 18]) Under this assumption, let us consider the complete graph Kn of order n. In general, in a vertex x, a local routing function Rx on Kn can be stored with Theta(n log n) bits for a random port labeling of x or for a port labeling chosen by an adversary. Indeed, an adversary can choose some ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

....We describe large classes of networks that admit optimal interval routing functions. We also study the case of the usual networks that interconnect the processors of a distributed memory parallel computer. 1 Introduction Compact routing has already been intensively studied (see for instance [2, 7, 8]) There exist many techniques to compress the size of routing tables. The general idea is to group the destination addresses that correspond to the same output port, and to encode the group so that it is easy to check if a destination address belongs to a given group. A very popular solution of ....

.... vertices C and E, the edge (A,D) is never used from A, and E contains its label in the interval on (E,D) and the right one has good properties (shortest paths, one non empty interval per edge on each vertex, and only linear intervals without label of the local vertex in its intervals) 1 2 3 4 5 [2,5] [ 1] 3] 4,1] 3] 2] 5] 1] 3,1] 2] 2] 4,5] 1 2 3 4 5 [4] 1,3] 4,5] 1] 3] 1,2] 4,5] 2,3] 1] 5] 2,3] 4,5] A B C D E A B C D E Fig. 1. Two interval routing functions for the same network. More formally, we define an interval routing function as follows: Definition1. Interval ....

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Baruch Awerbuch, Amotz Bar-Noy, Nathan Linial, and David Peleg. Improved routing strategies with succint tables. Journal of Algorithms, 11:307--341, February 1990.

....and the destination. We show that there exists an n node network on which every routing algorithm of stretch factor s 3 requires at least a total of Omega (n 2 ) bits of routing information, whereas for stretch factor s = 3 the best known upper bound is O(n 3=2 log 2 n) bits in total [1]. We show a similar gap for the space complexity of routing schemes on the subclass of networks of diameter 2. 1 Introduction and Model Routing messages in networks is an important task. For traffic regulation or for minimization of the delivery time of messages, we need routing paths that are ....

B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990).

....the destination. We show that there exists an n node network on which every routing algorithm of stretch factor s 3 requires at least a total of Omega Gamma n 2 ) bits of routing information, whereas for stretch factor s = 3 the best known upper bound is O(n 3=2 log 2 n) bits in total [1]. We show a similar gap for the space complexity of routing schemes on the subclass of networks of diameter 2. Keywords: compact routing in distributed networks, near shortest path routing, routing tables. A preliminary version has been partially published in the proceedings of SIROCCO 97. y ....

....of 1 Gamma n Gamma3 of all n node labeled graphs supports routing schemes with O(n 2 ) bits in total. For stretch factor s 2, no tight lower bound is known. For stretch factor s = 3, Peleg Upfal s lower bound gives Omega Gamma n 1:1 ) bits, whereas O(n 3=2 log 2 n) bits suffices [1]. In the next section, we will show that Omega Gamma n 2 ) are necessary for any stretch factor s 3. 3 Lower Bound of the Memory Requirement We will prove a lower bound of the total number of bits required to describe a routing function of stretch factor less than 3 on n node networks. The ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307-- 341.

.... Of course, the main requirement for these tables is to be as small as possible (for instance a size of Theta(n) for a network of n processors is not realistic as soon as the number of processors is larger that some tens) Compact routing has already been intensively studied (see for instance [1, 7, 8]) In particular, there exist many solutions to compress the size of the routing tables. The general idea is to group in some manner the destination addresses that correspond to the same output port, and to encode the group so that it is easy to check if a destination address belongs to a given ....

.... functions that are defined using intervals: Definition 2 (Interval) An interval of f1; 2; ng denoted [a; b] where a; b 2 f1; 2; ng, is a set of integers i satisfying: ae a 6 i 6 b if a 6 b (linear interval) a 6 i 6 n or 1 6 i 6 b if a b (cyclic interval) 1 2 3 4 5 [2,5] [1] [3] 4,1] 3] 2] 5] 1] 3,1] 2] 2] 4,5] A B C D E 1 2 3 4 5 [4] 1,3] 4,5] 1] 3] 1,2] 4,5] 2,3] 1] 5] 2,3] 4,5] A B C D E Figure 1: Two interval routing functions for the same network. If a = b we note the interval [a] instead of [a; a] we also note ]a; b] the interval [a; b] ....

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Baruch Awerbuch, Amotz Bar-Noy, Nathan Linial, and David Peleg. Improved routing strategies with succint tables. Journal of Algorithms, 11:307--341, February 1990.

....referenced papers, the bounds on the memory requirement of the routing schemes are always given in term of n 1=k where k 1 is an integer function raising with the stretch factor. Table 1 presents results in an other way so that one can compare them as a function of the stretch factor. ffl In [1], the stretch factor is at most 2 k Gamma 1, and the routing scheme allows non uniform cost on the arcs. Another routing scheme is proposed in [1] It locally requires at most O(k(d n 1=k ) log n) memory bits on any vertex of degree d for every stretch factor at most 2 Delta 3 k Gamma ....

....raising with the stretch factor. Table 1 presents results in an other way so that one can compare them as a function of the stretch factor. ffl In [1] the stretch factor is at most 2 k Gamma 1, and the routing scheme allows non uniform cost on the arcs. Another routing scheme is proposed in [1]. It locally requires at most O(k(d n 1=k ) log n) memory bits on any vertex of degree d for every stretch factor at most 2 Delta 3 k Gamma 1. Therefore, this routing scheme is efficient in the case of bounded degree networks. ffl In [2] the stretch factor is at most 16k 2 , and the ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

.... Complexity 99 of the routing tables on one hand, and the length of the routing paths on the other hand (see for instance [14] Some of them use quite sophisticated techniques that allow to drastically reduce the memory requirement for routing, up to a small increase of the length of the routes [3], nay no increase at all [10] for particular types of networks. However, compact routing protocols may suffer from a major drawback if reducing the space complexity of encoding the routes is up to the price of a large increase in the time complexity for computing these routes. This increase can be ....

AWERBUCH, B., BAR-NOY, A., LINIAL, N., AND PELEG, D. Improved routing strategies with succint tables. Journal of Algorithms 11 (Feb. 1990), 307--341.

....the destination. Moreover, some consider adaptive routing while others restrict their study to non adaptive routing. Similarly, some papers focus on one to one routing while others deal with one to many routing. Actually, one can find so many ways of defining routing functions in the literature [6, 15, 35, 48] that it is often difficult to figure out whether the known results about deadlock avoidance apply to these routing functions. In this paper, we describe a general framework for the deadlock avoidance problem in wormhole routed networks. This theory is grounded on a very general description of ....

....torus in the circuit switched mode applies to wormhole routing too, but it makes use of a flag modifying the usual XY routing for the purpose of the algorithm. Compact routing. Many researches have been performed in order to reduce the size of the memory requirement for routing (see for instance [6, 35]) Many solutions are based on headers of length dlog 2 ne for an n node network, that is headers which do not contain the destination addresses only. Such routing functions can be formalized as R : H Theta V 7 E where H is any set of headers (H is different from V in general) Multicast ....

B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

....locally on each router. The main requirement for these tables is to be as small as possible (for instance a size of Theta(n) for a network of n processors is not realistic as soon as the number of processors becomes large) Compact routing has already been intensively studied (see for instance [2, 11, 12]) There exist many solutions to compress the size of the routing tables. The usual approach consist in group the destination addresses which correspond to the same output port, and to encode the group so that it will be is easy to check whether or not a destination address belongs to a given ....

....for each vertex x, associate intervals to each edge e 2 out(x) The number of intervals associated to e 2 out(x) on x is denoted by k(x; e) A message located on x, and of destination y is routed by y through e 2 out(x) if and only if y belongs to one of the intervals associated to e on x. 4 3 4 [2][5] 2,5] 5 1 [5] 2,3] 4,5] 1,2] C D A [4,5] 2,3] E B [3] 1,3] 1] 4] 4,5] 1] 2 3 [2] 1] 3] B [4,5] 1 2 5 [1] 3] 4,1] C [3,1] D A E [2] Figure 1: Two interval routing functions for the same network. For instance, on Figure 1, we have indicated two interval routing ....

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Baruch Awerbuch, Amotz Bar-Noy, Nathan Linial, and David Peleg. Improved routing strategies with succint tables. Journal of Algorithms, 11:307--341, February 1990.

....To alleviate the space requirements of routing tables, compact routing schemes were introduced: in [SK85] for arbitrary networks and in [FJ88, FJ89, FJ90] for planar and c decomposable networks. Trade offs between the space requirements for every node and the length of the routes were proposed in [ABNLP90, AP92, PU88]. A popular compact routing method, interval routing, is to group together the destination nodes corresponding to the same output port of a given node in intervals. Just as for table routing, this method requires that a header of only O(log n) bits be added to the forwarded message. This routing ....

Baruch Awerbuch, Amotz Bar-Noy, Nathan Linial, and David Peleg. Improved routing strategies with succint tables. Journal of Algorithms, 11:307--341, February 1990.

.... 2] Omega Gamma n 2 ) Theorem 1] O(n 2 log n) 2 s 3 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O(n log n) O(n 2 log n) 3 s 16 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O(n log n) O(n 1 1=blog 2 (s 1)c log 2 n) [1] 16 s 87 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O(n 1=b p s=4c log 2 n) 2] O(n 1 1=blog 2 (s 1)c log 2 n) 1] s 87 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O( p sn 1=b p s=4c log 2 n) 2] O(s 3 n ....

.... 2 log n) 3 s 16 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O(n log n) O(n 1 1=blog 2 (s 1)c log 2 n) 1] 16 s 87 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O(n 1=b p s=4c log 2 n) 2] O(n 1 1=blog 2 (s 1)c log 2 n) [1] s 87 Omega Gamma n 1= 2s 4) y [31] Omega Gamma n 1 1= 2s 4) 31] O( p sn 1=b p s=4c log 2 n) 2] O(s 3 n 1 1=b(s Gamma3) 12c log n) 31] s = O(log n) Omega Gamma19 y Omega Gamma n) 31] O(e p log n log 3=2 n) 2] O(n log 4 n) 31] s = O( p n) ....

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

....router has to store locally Theta(n log d) bits of memory, where d is the degree of the router (i.e. the number of output ports) and n is the number of nodes in the network. Therefore, this scheme is impractical when dealing with large networks. It was shown in a series of papers (see, e.g. [1, 2, 3, 25]) that there is a tradeooe between the memory requirements of a routing scheme and the worst case stretch factor it guarantees. In [25] it was shown that any universal routing strategy achieving stretch factor s 1 must use a total of Omega Gamma n 1 1 2s 4 ) bits of routing information in the ....

....is shown that there exist some graphs on which every routing scheme of stretch factor s 3 requires at least Omega Gamma n 2 ) bits. 5 Comparison with General Routing Schemes This section compares the PIR1 strategy with other general strategies. Let HCPk be the routing strategy developed in [2] called Hierarchical Covering Pivot. Theorem 5.1 (Awerbuch et al. 1] For every n node weighted graph G, and for every integer k 1, and there exists R 2 HCPk on G such that Memory G (R) O(k n 1 1=k log 2 n) and StretchG (R) 2 k Gamma 1. Moreover R is constructible in polynomial ....

Awerbuch, B., Bar-Noy, A., Linial, N., and Peleg, D. Improved routing strategies with succint tables. Journal of Algorithms 11 (Feb. 1990), 307341.

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), 307-341.

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B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved routing strategies with succint tables, Journal of Algorithms, 11 (1990), pp. 307--341.

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AWERBUCH, B., BAR-NOY, A., LINIAL, N., AND PELEG, D. Improved routing strategies with succint tables. Journal of Algorithms 11 (Feb. 1990), 307--341.

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