E. Kranakis and D. Krizanc. Lower bounds for compact routing. In 13 Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529-540. Springer-Verlag, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....stretches. The currently best memory stretch tradeoff for arbitrary weighted graphs is achieved by a scheme in which the stretch factor is O( and the memory requirements are O(n O(1) n) bits per vertex [120] Lower bounds for the space efficiency tradeoff of routing schemes were studied in [103,61,79,44,31, 20,58]. More precisely, in [103] it is shown that every routing strategy that guarantees an s stretched routing scheme for every n vertex graph must provide at least a total of 1 1= 2s 4) different routing schemes. Thus for 5, no routing strategy can guarantee for every graph a routing scheme ....

E. Kranakis and D. Krizanc. Lower bounds for compact routing. In Proc. 13 Symp. on Theoretical Aspects of Computer Science, vol. 1046 of LNCS, pages 529--540, Feb. 1996.

....requirements. In [4] Fraigniaud and Gavoille improved the Peleg Upfal s lower bound by proving the existence of 2 Omega Gamma non isomorphic routing functions of stretch factor s 2, including the interesting case of shortest path routing schemes (s = 1) Recently, Krizanc and Kranakis [7] reformulated the proof of [4] in term of Kolmogorov Complexity and showed also a lower bound of Omega Gamma n ) bits for s 2. For stretch factor 1, Gavoille and P erenn es showed in [5] an optimal lower bound of Theta(n log n) bits, while Buhrman, Hoepman, and Vit anyi showed in [2] ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 Annual Symposium on Theoretical Aspects of Computer Science (STACS), C. Puech and R. Reischuk, eds., vol. 1046 of Lecture Notes in Computer Science, Springer-Verlag, Feb. 1996, pp. 529--540.

....build graphs having a compactness of at least n=12 intervals, proving uneventful that interval routing scheme is not compact for all graphs of order n. Study of compactness for smallest classes of graphs, like bounded degree graphs, was rst proposed in [13] Their lower bound has been improved in [17], where Kranakis and Krizanc proved that some bounded degree graphs have a compactness of Omega Gamma n= log n) intervals. In [16] we have proved a tight lower bound of the space requirement for any universal shortest path routing scheme. We showed that there exist graphs where the total space ....

....n log d) bits, where d is the maximum degree of the graphs. In particular if d is constant this implies a lower bound of Omega Gamma n) bits per router for the local routing information. Therefore, we can derive a trivial lower bound of Omega Gamma n= log n) that is not a new result [17]) for the compactness of a bounded degree graph, since a router using interval routing It means that the routing induces only shortest paths. scheme on a graph of compactness k and of bounded degree cannot store more than O(k log n) bits of information. In this paper, we show that this lower ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 Annual Symposium on Theoretical Aspects of Computer Science (STACS), Feb. 1996.

....parameter k is called compactness of a routing table. 3 Motivations and Related Works Many works try to determine routing tables with minimum compactness under several assumptions on the quality of the routing measured in term of length of the routes: shortest path routing [FvLMS98, GG98, GP99, KK96, KR S00] all the routes are shortest paths) bounded stretched routing [EGP98, FGS94, NO97] the ratio between the length of the route and the optimal length) bounded dilation routing [EMZ99, Gav00a, KR S00, NO97, TL97] the length of the longest route for the network) etc. See [Gav00b] ....

Evangelos Kranakis and Danny Krizanc. Lower bounds for compact routing. In 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529--540. Springer-Verlag, February 1996.

....of compact routing since a network of maximum degree and supporting an IRS has its routing table of size O( log n) bits, to be compared with the (n log ) bits of a table returning, for every destination, the output port corresponding to that destination. For more about IRS, we refer to [7, 11, 18, 20, 23, 24], and to the survey [17] For more about compact routing in general, we refer to [13, 14, 15, 19, 21, 26] On the other hand, broadcasting is the information dissemination problem which consists, for an arbitrary node of a network, to send a same message to all the other nodes. As far as the ....

E. Kranakis and D. Krizanc. Lower bounds for compact routing. In 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529-540. Springer-Verlag, 1996.

....tables require O(n) integers. The parameter k is called compactness of a routing table. Many works try to determine routing tables with minimum compactness under several assumptions on the quality of the routing measured in term of length of the routes: shortest path routing [FvLMS98, GG98, GP99, KK96] stretched routing [EGP98, FGS94, NO97] routing with bounded dilation [EMZ99, Gav00a, KR S00, NO97, TL97] etc (cf. Gav00b] Nevertheless, these works have studied only the deterministic case: for each source destination pair, the routing tables encodes a unique possible path only. So, ....

....schemes and its generalizations. However, most of theses works try to give a compact representation of all the shortest paths. Although these schemes extend the deterministic case, they su er by the fact that many general lower bounds for deterministic routing established in [FvLMS98, GP99, GP96, KK96, KKU95] apply as well for the adaptive case. Indeed, these lower bounds are based on the uniqueness of the shortest paths between speci c subset of nodes in some worst case graphs. Thus, on these graphs all shortest paths routing would consist to route along one shortest path as in deterministic ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), C. Puech and R. Reischuk, eds., vol. 1046 of Lecture Notes in Computer Science, SpringerVerlag, February 1996, pp. 529-540.

....of compact routing since a network of maximum degree and supporting an IRS has its routing table of size O( log n) bits, to be compared with the (n log ) bits of a table returning, for every destination, the output port corresponding to that destination. For more about IRS, we refer to [6, 10, 16, 18, 20, 21], and to the survey [15] For more about compact routing in general, we refer to [11, 12, 13, 17, 19, 24] On the other hand, broadcasting is the information dissemination problem which consists, for an arbitrary node of a network, to send a same message to all the other nodes. The ....

E. Kranakis and D. Krizanc. Lower bounds for compact routing. In 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529-540. Springer-Verlag, 1996.

....to route with stretch factor at most s on any n node network. In [5] Fraigniaud and Gavoille improved this lower bound by proving the existence of 2 n 2 ) non isomorphic routing functions of stretch factor s 2, including the case of shortest path routing (s = 1) Krizanc and Kranakis [9] reformulated the proof of [5] in terms of Kolmogorov Complexity and showed also a lower bound of n 2 ) bits for s 2. For stretch factor 1, Gavoille and P erenn es 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 ) ....

Evangelos Kranakis and Danny Krizanc. Lower bounds for compact routing. In 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529-540. Springer-Verlag, 1996.

....Unfortunately, Gavoille and Gu evremont in [9] build graphs having a compacity of at least n=12 intervals, proving uneventful that interval routing scheme is not compact in general. Study of compacity for smallest classes of graphs was first proposed in [8] Their lower bound has been improved in [12], where Kranakis and Krizanc proved that some bounded degree graphs have a compacity of Omega Gamma n= log n) intervals. In [10] we have proved a tight lower bound for any universal shortest path routing scheme. We showed that there exist graphs where the total space required to store the routing ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Feb. 1996.

....one can not do better than routing tables for Theta(n) nodes. In other words, there is no hope to find a universal routing scheme more compact than routing tables for n node 2 networks of maximum degree d. This general result improves previous works on the compact routing problem studied in [27, 13, 23]. A straightforward consequence of this result is that there exists a n node network of maximum degree 3 that requires Omega Gamma n= log n) intervals on almost all the links to code a shortest path routing function using the interval routing scheme. This already improves by a factor log n the ....

....of intervals that one can find on each arc, minimized over all shortest routing functions on G. Finding the compactness of G consists in finding the appropriate vertex labeling on G. The problem of finding the worst case of compactness for graphs of order n has been addressed in [24, 9, 20] and in [4, 19, 23] for graphs of bounded degree. Here we give a general lower bound for every degree d, and we give a tight lower bound for the compactness of graphs of maximum degree bounded by a constant. Theorem 1 showed that the best shortest path routing scheme on graphs of order n and maximum degree bounded ....

[Article contains additional citation context not shown here]

E. Kranakis and D. Krizanc, Lower bounds for compact routing, Tech. Rep. TR-95-18, Carleton University, July 1995. To appear in STACS '96.

....the total routing information is Omega Gamma n 2 log d) bits. In other words, there is no hope to find a universal routing scheme more compact than routing tables for n node networks of maximum degree d. This general result improves previous works on the compact routing problem studied in [8, 16, 18]. 1.2 A General Model of Routing Function A point to point communication network is described by a finite connected symmetric digraph G = V; E) In the following, we denote by n the number of vertices of such a graph. The vertices represent the nodes or the routers of the network: we assume that ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, Tech. Rep. TR-95-18, Carleton University, July 1995. To appear in STACS '96.

....requirements. In [4] Fraigniaud and Gavoille improved the Peleg Upfal s lower bound by proving the existence of 2 Omega (n 2 ) non isomorphic routing functions of stretch factor s 2, including the interesting case of shortest path routing schemes (s = 1) Recently, Krizanc and Kranakis [6] reformulated the proof of [4] in term of Kolmogorov Complexity and showed also a lower bound of Omega (n 2 ) bits for s 2. For stretch factor 1, Gavoille and P erenn es showed in [5] an optimal lower bound of Theta(n 2 log n) bits, while Buhrman, Hoepman, and Vit anyi showed in [2] that ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in STACS '96.

.... In [4] Fraigniaud and Gavoille improved the Peleg Upfal s lower bound by proving the existence of 2 Omega Gamma n 2 ) non isomorphic routing functions of stretch factor s 2, including the interesting case of shortest path routing schemes (s = 1) Recently, Krizanc and Kranakis [7] reformulated the proof of [4] in term of Kolmogorov Complexity and showed also a lower bound of Omega Gamma n 2 ) bits for s 2. For stretch factor 1, Gavoille and P erenn es showed in [5] an optimal lower bound of Theta(n 2 log n) bits, while Buhrman, Hoepman, and Vit anyi showed in [2] ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), C. Puech and R. Reischuk, eds., vol. 1046 of Lecture Notes in Computer Science, Springer-Verlag, Feb. 1996, pp. 529--540.

.... ) 1 vertices, that is Theta(4 k = p k) Its compactness is thus at least Omega Gammaast n) Its diameter is 6, and its maximum degree Gamma 2k Gamma1 k Gamma1 Delta Gamma 1 = Theta(n) Many works deal with the asymptotic behavior of the maximum compactness of a graph of order n [6, 13, 14, 18, 19]. Up to our knowledge, the best results are a tight bound of Theta(n) for cubic graphs, and a lower bound of Omega Gamma p n) for cubic planar graphs [14] We are ready now to study the properties satisfied by the usual graphs considered as candidates for interconnecting processors of parallel ....

Evangelos Kranakis and Danny Krizanc. Lower bounds for compact routing. In 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), February 1996.

....build graphs having a compactness of at least n=12 intervals, proving uneventful that interval routing scheme is not compact for all graphs of order n. Study of compactness for smallest classes of graphs, like bounded degree graphs, was rst proposed in [13] Their lower bound has been improved in [17], where Kranakis and Krizanc proved that some bounded degree graphs have a compactness of Omega Gamma n= log n) intervals. In [16] we have proved a tight lower bound of the space requirement for any universal shortest path routing scheme. We showed that there exist graphs where the total space ....

....n 2 log d) bits, where d is the maximum degree of the graphs. In particular if d is constant this implies a lower bound of Omega Gamma n) bits per router for the local routing information. Therefore, we can derive a trivial lower bound of Omega Gamma n= log n) that is not a new result [17]) for the compactness of a bounded degree graph, since a router using interval routing 1 It means that the routing induces only shortest paths. scheme on a graph of compactness k and of bounded degree cannot store more than O(k log n) bits of information. In this paper, we show that this ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Feb. 1996.

....link are not consecutive integers. Unfortunately, it was proven in [8, 11, 27] that the number of intervals per link necessary to route in any network is not bounded when using shortest path routing. Actually, it was recently proven in [19] that this number is 1 Theta(n) it was even noticed in [26] that in fact Theta(n) routers require such an amount of intervals) Since the C104 routing chip allows a few intervals per link only, this means that there exist networks such that no shortest path routing protocol can be implemented on them using C104 chips. In this sense, this chip is not ....

....of Theta(n log n) bits for the local memory requirement. Finally, Section 6 summarizes the state of the art of memory requirement of universal routing schemes. Note that simpler arguments to derive the Omega Gamma n 2 ) bound on the global memory requirement have been recently presented in [26]. These arguments are based on constructions taken from [12] but make use of the Kolmogorov Complexity theory rather than counting arguments. 2 Statement of the Problem We consider the standard model of point to point communication networks described as finite connected symmetric digraphs G = ....

E. Kranakis and D. Krizanc, Lower bounds for compact routing, in 13 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), C. Puech and R. Reischuk, eds., vol. 1046 of Lecture Notes in Computer Science, Springer-Verlag, Feb. 1996, pp. 529--540.

No context found.

E. Kranakis and D. Krizanc. Lower bounds for compact routing. In 13 Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1046 of Lecture Notes in Computer Science, pages 529-540. Springer-Verlag, 1996.

No context found.

E. Kranakis and D. Krizanc, Lower bounds for compact routing, Tech. Rep. TR-95-18, Carleton University, July 1995. To appear in STACS '96.

No context found.

KRANAKIS, E., AND KRIZANC, D. Lower bounds for compact routing. In 13 Annual STACS (Feb. 1996), C. Puech and R. Reischuk, Eds., vol. 1046 of Lecture Notes in Computer Science, Springer-Verlag, pp. 529--540.

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