## The Complexity of Interval Routing on Random Graphs (1995)

### Cached

### Download Links

- [archive.cs.uu.nl]
- [comjnl.oxfordjournals.org]
- DBLP

### Other Repositories/Bibliography

Venue: | THE COMPUTER JOURNAL |

Citations: | 32 - 4 self |

### BibTeX

@INPROCEEDINGS{Flammini95thecomplexity,

author = {M. Flammini and J. van Leeuwen and A. Marchetti-Spaccamela},

title = {The Complexity of Interval Routing on Random Graphs},

booktitle = {THE COMPUTER JOURNAL},

year = {1995},

pages = {37--49},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Several methods exist for routing messages in a network without using complete routing tables (compact routing). In k-interval routing schemes (k-IR.S), links carry up to k intervals each. A message is routed over certain link if its destination belongs to one of the intervals of the link. We give some results for the necessary value of k in order to achieve shortest path routing. Even though for very structured networks low values of suce, we show that for 'general graphs' interval routing cannot significantly reduce the space-requirements for shortest path routing. In particular we show that for suitably large n, there are suitable values of p such that for randomly chosen graphs G 6 ,p the following holds, with high probability: if G admits an optimal k-IIS, then k = The result is obtained by means of a novel matrix representation for the shortest paths in a network.

### Citations

713 |
A measure of asymptotic efficiency of tests of a hypothesis based on the sum of observations
- Chernoff
- 1952
(Show Context)
Citation Context ...hat, with sufficiently large probability, the set 5/is sufficiently large. In the sequel we will use the following inequalities: (1-x) , (1-x/2) > e - for 0 )a > 1 - h for h > 1 and 0sdue to Chernoff =-=[11]-=- (the 'Chernoff bounds', see also [5]) on the tails of the binomial distribution: Lemma 12. Let 0sand let m be any positive integer. The probability of at most (1 - )mp successes in m independent tria... |

256 |
Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings
- Angluin, Valiant
- 1979
(Show Context)
Citation Context ...ty, the set 5/is sufficiently large. In the sequel we will use the following inequalities: (1-x) , (1-x/2) > e - for 0 )a > 1 - h for h > 1 and 0sdue to Chernoff [11] (the 'Chernoff bounds', see also =-=[5]-=-) on the tails of the binomial distribution: Lemma 12. Let 0sand let m be any positive integer. The probability of at most (1 - )mp successes in m independent trials of X is less than e -%p/a. The pro... |

133 |
Labelling and Implicit Routing in Networks
- Santoro, Khatib
- 1985
(Show Context)
Citation Context ...s are utilized, and the routing strategy is fully deterministic, see [12, 9]). Many interesting processor interconnection networks have optimum k-interval routing schemes for small k (see for example =-=[13, 9, 14, 15, 16, 17]-=-) and interval routing has been used as an ingredient in various other routing problems (see [18, 19, 20]). Interval routing is not only of theoretical interest but has also found industrial applicati... |

133 |
A trade-off between space and efficiency for routing tables
- Peleg, Upfal
- 1989
(Show Context)
Citation Context ...ction networks have optimum k-interval routing schemes for small k (see for example [13, 9, 14, 15, 16, 17]) and interval routing has been used as an ingredient in various other routing problems (see =-=[18, 19, 20]-=-). Interval routing is not only of theoretical interest but has also found industrial applications, for example in the design of the C104 router chip of the INMOS T9000 Transputer [21] and other pract... |

75 | Routing with polynomial communication-space tradeo
- Awerbuch, Peleg
- 1992
(Show Context)
Citation Context ...,... ,n}. Assuming that the labels I through n are arranged clockwise around a circle, an interval [a, b] represents the node labels from a through b in clockwise order. (Thus for n = 5, the interval =-=[4, 2]-=- represents the node labels 4, 5, 1, and 2.) A k-interval labeling scheme or k-ILS ([40]) consists of two components: (i) a node labeling such that every node gets a unique label, and (ii) at every no... |

71 | Compact distributed data structures for adaptive routing
- AWERBUCH, BAR-NOY, et al.
- 1989
(Show Context)
Citation Context ...of =slogdv = C (2 log 3) bits (this follows from the inequality for the geometric and arithmetic means using d +... + d, = 2). It is an intriguing question whether this bound can be reduced to e.g. C =-=(2)-=-. lraigniaud and Gavoille [21] (see also [32]) recently proved that any general shortest-path routing scheme requires/(2) bits total at the nodes. Practical networking environments have introduced the... |

71 |
Designing networks with compact routing tables
- Frederickson, Janardan
- 1988
(Show Context)
Citation Context ...l edges are utilized, and the routing strategy is fully deterministic, see [39, 40]). Many interesting processor interconnection networks have optimum k-interval routing schemes for small k (see e.g. =-=[36, 40, 23, 25, 26, 19, 13]-=-) and interval routing has been used as an ingredient in various other routing problems (see [2, 3, 32]). Interval routing is not only of theoretical interest but has also found industrial application... |

69 |
Improved routing strategies with succinct tables
- Awerbuch, Noy, et al.
- 1990
(Show Context)
Citation Context ...nection networks have optimum k-interval routing schemes for small k (see e.g. [36, 40, 23, 25, 26, 19, 13]) and interval routing has been used as an ingredient in various other routing problems (see =-=[2, 3, 32]-=-). Interval routing is not only of theoretical interest but has also found industrial applications, e.g. in the design of the C104 router chip of the INMOS T9000 Transputer [31] and other practical ap... |

69 | Interval routing
- Leeuwen, Tan
- 1987
(Show Context)
Citation Context ... endpoint of an edge may be assigned different sets of up to k edge labels. When none of the edge labels contains the interval In, 1], i.e. no interval 'wraps around', then the k-ILS is called linear =-=[6, 7]-=-. Given a k-ILS, a routing strategy is devised as follows. Let a message m be destined for some node v, suppose it has reached some node u on its way, and suppose usv. Then an edge e = (u, ul) inciden... |

58 | The compactness of interval routing
- Gavoille, Peleg
- 1999
(Show Context)
Citation Context ...l edges are utilized, and the routing strategy is fully deterministic, see [39, 40]). Many interesting processor interconnection networks have optimum k-interval routing schemes for small k (see e.g. =-=[36, 40, 23, 25, 26, 19, 13]-=-) and interval routing has been used as an ingredient in various other routing problems (see [2, 3, 32]). Interval routing is not only of theoretical interest but has also found industrial application... |

42 |
Efficient message routing in planar networks
- Frederickson, Janardan
- 1989
(Show Context)
Citation Context ...l edges are utilized, and the routing strategy is fully deterministic, see [39, 40]). Many interesting processor interconnection networks have optimum k-interval routing schemes for small k (see e.g. =-=[36, 40, 23, 25, 26, 19, 13]-=-) and interval routing has been used as an ingredient in various other routing problems (see [2, 3, 32]). Interval routing is not only of theoretical interest but has also found industrial application... |

37 | On multi-label linear interval routing schemes, in: Graph-Theoretic Concepts in Computer
- Kranakis, Krizanc, et al.
- 1994
(Show Context)
Citation Context ...ach optimal k-IRS scheme of G one has k = f2(n/log n). Clearly this bound holds also for linear interval routing, and thus it improves the recent/2(n /3) lowerbound due to Kranakis, Krizanc, and Ravi =-=[30]-=-. The technique for proving the bound has been used recently by Gavoille and Guvremont [27] to construct graphs G with n nodes that actually require k = O(n). (ii) Next we consider the problem of boun... |

36 | Memory requirement for routing in distributed networks
- Gavoille, Pérennès
- 1996
(Show Context)
Citation Context ...the geometric and arithmetic means using d1+ ···+dn = 2e). It has been an intriguing question whether this bound can be reduced to e.g. O(n2 ). In a series of papers Fraigniaud, Gavoilles and Pérennès=-=[1,2,3]-=-haveshown Received October 3, 1996; revised January 21, 1998 THE COMPUTER JOURNAL, Vol. 41, No. 1, 1998 that graphs can be constructed that require θ(nlog n) bits of routing information in as many as ... |

33 |
Random graphs
- Bollobs
- 1985
(Show Context)
Citation Context ...ity of interval routing in random graphs. We adopt the 6n,p model of random graphs, which consists of all graphs with node set V -- { 1,... , n) and edges chosen independently with probability p (see =-=[10]-=-). We will assume throughout that Inl ff-------n ( p ( n -'5-e for some e ) 0. The (lower) bound on p guarantees that a graph in 6n,p is indeed connected, with high probability ([10], Chapter IV). We ... |

28 | Memory requirement for universal routing schemes
- Fraigniaud, Gavoille
- 1995
(Show Context)
Citation Context ...the geometric and arithmetic means using d1+ ···+dn = 2e). It has been an intriguing question whether this bound can be reduced to e.g. O(n2 ). In a series of papers Fraigniaud, Gavoilles and Pérennès=-=[1,2,3]-=-haveshown Received October 3, 1996; revised January 21, 1998 THE COMPUTER JOURNAL, Vol. 41, No. 1, 1998 that graphs can be constructed that require θ(nlog n) bits of routing information in as many as ... |

24 |
Space-efficient message routing in c-decomposable networks
- Frederickson, Janardan
- 1990
(Show Context)
Citation Context |

23 | Upper and lower bounds ,for routing schemes in dynamic networks
- AFEK, GAFNI, et al.
- 1989
(Show Context)
Citation Context ...k-1 (k-1)!p ks1=1 k=l c--1 nC-2PC-1 E 2nl-2Pl-I -- -- l=l Summing over all possible nodes vj SI, we obtain Prob(there exists a jsi s.t. A(i, j, u)) _-2n -. Using lemma 9 and the fact that c(nPn)c- = o=-=(1)-=- we obtain, for suitably large n: Prob(d(vi,u) = c- 1 and for each jsi, -,A(i,j,u)) _> Prob( d( vi , u ) = c - 1) - Prob( there exists a jsi s.t. A( i, j, u ) ) _> -- (rip) c-1 2(np) - _ 4c(np)2C_n_s=... |

22 | Worst case bounds for shortest path interval routing
- Gavoille, Guévremont
- 1998
(Show Context)
Citation Context ...near interval routing, and thus it improves the recent/2(n /3) lowerbound due to Kranakis, Krizanc, and Ravi [30]. The technique for proving the bound has been used recently by Gavoille and Guvremont =-=[27]-=- to construct graphs G with n nodes that actually require k = O(n). (ii) Next we consider the problem of bounding the values of k for which random graphs on n nodes have an optimal k-IRS. We show that... |

22 |
Boolean routing
- Flammini, Gambosi, et al.
- 1993
(Show Context)
Citation Context ...ther practical applications [22, 23, 24, 25, 26]. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like ‘prefix routing’ [27], ‘Boolean routing’, see =-=[28, 29, 30]-=- or ‘multi-dimensional’ interval routing [31]). Most of the results cited above show that interval routing can efficiently be used for specific topologies and, in general, for networks having particul... |

21 | Searching Among Intervals and Compact Routing Tables, Algorithmica - Frederickson - 1996 |

21 | Compact routing methods: A survey
- Leeuwen, Tan
- 1994
(Show Context)
Citation Context ... be coded more succinctly at the nodes, usually with a different regime for the routing decisions as well. Tables typically use in the order of d log n bits per node, hence O(e log n) bits total. See =-=[7, 8]-=- for an overview. We will consider the most popular of these techniques called ‘interval routing’ and will demonstrate that for general networks it is not likely to work as well. Interval routing is b... |

19 | Prefix routing schemes in dynamic networks
- Bakker, Leeuwen, et al.
- 1993
(Show Context)
Citation Context ...S T9000 Transputer [31] and other practical applications [28, 43, 44, 45, 46]. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like 'prefix routing' =-=[8]-=-, 'Boolean routing', see [12, 15] or 'multi-dimensional' interval routing). Most of results cited above show that interval routing can be efficiently used for specific topologies and, in general, for ... |

19 |
Interval routing schemes, Research Report 94-04, LIP, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07
- Fraigniaud, Gavoille
- 1994
(Show Context)
Citation Context ...ed that ‘almost all’ graphs can be routed with routing tables of O(n 2 ) bits, using a formalization of the problem in terms of Kolmogorov complexity theory. For an overview of these developments see =-=[6]-=-. We are interested in the effectiveness of concrete routing methodologies instead. Practical networking environments have introduced the need for devising routing methods with provably smaller tables... |

18 |
Routing with compact routing tables
- Leeuwen, Tan
- 1983
(Show Context)
Citation Context ...dge labels assigned to the edges at that node are a partition of 1,...,n (thus the ‘intervals’ at every node are disjoint, all edges are utilized, and the routing strategy is fully deterministic, see =-=[12, 9]-=-). Many interesting processor interconnection networks have optimum k-interval routing schemes for small k (see for example [13, 9, 14, 15, 16, 17]) and interval routing has been used as an ingredient... |

13 | Local memory requirement of universal routing schemes
- Fraigniaud, Gavoille
- 1996
(Show Context)
Citation Context ...the geometric and arithmetic means using d1+ ···+dn = 2e). It has been an intriguing question whether this bound can be reduced to e.g. O(n2 ). In a series of papers Fraigniaud, Gavoilles and Pérennès=-=[1,2,3]-=-haveshown Received October 3, 1996; revised January 21, 1998 THE COMPUTER JOURNAL, Vol. 41, No. 1, 1998 that graphs can be constructed that require θ(nlog n) bits of routing information in as many as ... |

13 |
Transputers and Routers: Components for concurrent machines. Inmos
- May, Thompson
- 1991
(Show Context)
Citation Context ...ms (see [18, 19, 20]). Interval routing is not only of theoretical interest but has also found industrial applications, for example in the design of the C104 router chip of the INMOS T9000 Transputer =-=[21]-=- and other practical applications [22, 23, 24, 25, 26]. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like ‘prefix routing’ [27], ‘Boolean routing’... |

12 |
Interval labeling schemes for chordal rings
- Flammini, Gambosi, et al.
- 1994
(Show Context)
Citation Context |

10 |
Performance Benefits from Locally Adaptive Interval Routing
- Hofestidt, Klein, et al.
(Show Context)
Citation Context ... routing is not only of theoretical interest but has also found industrial applications, e.g. in the design of the C104 router chip of the INMOS T9000 Transputer [31] and other practical applications =-=[28, 43, 44, 45, 46]-=-. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like 'prefix routing' [8], 'Boolean routing', see [12, 15] or 'multi-dimensional' interval routing)... |

8 | A characterization of networks supporting linear interval routing - Fraigniaud, Gavoille - 1994 |

7 |
On devising boolean routing schemes
- Flammini, Gambosi
- 1997
(Show Context)
Citation Context ...other practical applications [28, 43, 44, 45, 46]. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like 'prefix routing' [8], 'Boolean routing', see =-=[12, 15]-=- or 'multi-dimensional' interval routing). Most of results cited above show that interval routing can be efficiently used for specific topologies and, in general, for networks having particular symmet... |

7 | Vitnyi: Optimal Routing Tables
- Buhrman, Hoepman, et al.
- 1996
(Show Context)
Citation Context ...ormation in as many as θ(n) nodes in any general shortest-path routing scheme, thus proving that the overall worst-case bound of O(n 2 log n) bits cannot be improved. On the other hand Buhrman et al. =-=[4]-=-(seealso[5])have argued that ‘almost all’ graphs can be routed with routing tables of O(n 2 ) bits, using a formalization of the problem in terms of Kolmogorov complexity theory. For an overview of th... |

7 | Network fault-tolerance with interval routing devices
- Vounckx, Deconinck, et al.
- 1993
(Show Context)
Citation Context ...g is not only of theoretical interest but has also found industrial applications, for example in the design of the C104 router chip of the INMOS T9000 Transputer [21] and other practical applications =-=[22, 23, 24, 25, 26]-=-. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like ‘prefix routing’ [27], ‘Boolean routing’, see [28, 29, 30] or ‘multi-dimensional’ interval rou... |

4 |
Combinatorial Problems in Information Networks and Distributed Datastructuring
- Bakker
- 1991
(Show Context)
Citation Context ... endpoint of an edge may be assigned different sets of up to k edge labels. When none of the edge labels contains the interval In, 1], i.e. no interval 'wraps around', then the k-ILS is called linear =-=[6, 7]-=-. Given a k-ILS, a routing strategy is devised as follows. Let a message m be destined for some node v, suppose it has reached some node u on its way, and suppose usv. Then an edge e = (u, ul) inciden... |

4 |
Deadlock-Free Fault-Tolerant Wormhole Routing in Mesh based Massively Parallel Networks
- Vounckx, Deconinck, et al.
- 1994
(Show Context)
Citation Context ...g is not only of theoretical interest but has also found industrial applications, for example in the design of the C104 router chip of the INMOS T9000 Transputer [21] and other practical applications =-=[22, 23, 24, 25, 26]-=-. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like ‘prefix routing’ [27], ‘Boolean routing’, see [28, 29, 30] or ‘multi-dimensional’ interval rou... |

4 |
Minimal Deadlock-Free Compact Routing in Wormhole Switching based Injured Meshes", internal report KULeuven - ESAT
- Vounckx, Deconinck, et al.
- 1994
(Show Context)
Citation Context ...g is not only of theoretical interest but has also found industrial applications, for example in the design of the C104 router chip of the INMOS T9000 Transputer [21] and other practical applications =-=[22, 23, 24, 25, 26]-=-. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like ‘prefix routing’ [27], ‘Boolean routing’, see [28, 29, 30] or ‘multi-dimensional’ interval rou... |

2 |
Spaccamela. Lower bounds on interval routing
- Flammini, Leeuwen, et al.
- 1994
(Show Context)
Citation Context ...evelop the main results (ii) concerning the minimum number of intervals required by IRS schemes for random graphs. Some conclusions follow in section 5. The paper is a revised and extended version of =-=[17]-=-. 2 A Lowerbound Technique for k-Interval Routing Schemes Let G = (V, E) be a graph with I V I= n. We want to be able to determine lowerbounds on the required value of k for routing in G according to ... |

2 |
Memory requirement for universal routing schemes, Research Rep. 95-05, Lab. de l'Informatique du ParMllisme, Ecole NormMe Suprieure de
- Fraigniaud, Gavoille
- 1995
(Show Context)
Citation Context ... (this follows from the inequality for the geometric and arithmetic means using d +... + d, = 2). It is an intriguing question whether this bound can be reduced to e.g. C (2). lraigniaud and Gavoille =-=[21]-=- (see also [32]) recently proved that any general shortest-path routing scheme requires/(2) bits total at the nodes. Practical networking environments have introduced the need for devising routing met... |

2 | Space-efficient and fault-tolerant message routing in outerplanar networks - Frederickson, Janardan - 1988 |

2 |
Fault-Tolerant Compact Routing based on Reduced Structural Information in Wormhole-Switching based Networks
- Vounckx, Deconinck, et al.
- 1994
(Show Context)
Citation Context |

1 |
On interval routing schemes and treewidth, extended abstract
- Bodlaender, Tan, et al.
- 1995
(Show Context)
Citation Context ...re indeed the more common situation if we allow dynamic edge costs and require that a graph can be optimally routed by a k-IRS for every cost assignment to the edges. Recent work of Bodlaender et al. =-=[9]-=- shows that graphs G which have a node labeling such that for some fixed k, an optimal k-IRS can be designed for G for every choice of positive edge costs, have a treewidth bounded in terms of k. It i... |

1 |
Boolean routing, in: A. Schiper (Ed.), Distributed Algorithms
- Flaremini, Gambosi, et al.
- 1993
(Show Context)
Citation Context ...other practical applications [28, 43, 44, 45, 46]. Several variants are being explored as well, to obtain even more flexible compact routing schemes (like 'prefix routing' [8], 'Boolean routing', see =-=[12, 15]-=- or 'multi-dimensional' interval routing). Most of results cited above show that interval routing can be efficiently used for specific topologies and, in general, for networks having particular symmet... |

1 |
On the complexity of devising interval routing schemes
- Flaremini, Gambosi, et al.
- 1994
(Show Context)
Citation Context ... In section 2 we develop the technique for proving lower- and upperbounds on the number of edge labels (intervals) needed in an optimal k-IRS. It extends ideas used in Flaremini, Gambosi and Salomone =-=[12, 14, 16]-=- for proving complexity results concerning the hardness of devising interval routing schemes with a minimum number of intervals and some approximation algorithms for this problem. We apply the techniq... |

1 |
Interval routing schemes, in: E. W1Qlayr and C
- Flaremini, Gambosi, et al.
- 1995
(Show Context)
Citation Context ...n destinations to an edge can be represented by at most n intervals on the edge). But it appears to be much harder to decide whether a network admits an optimal k-IRS for some smaller value of k (see =-=[16]-=-). Some results exist that show that 1-IRS [34, 35, 37] and 2-IRS schemes [38] cannot even come close to optimality for some classes of networks. Indeed, when no specific assumption about the topology... |

1 | Interval routing schemes, Research Rep 94-04, Lab. de l'Informatique du ParMlhsme, Ecole Normale Suprieure de - FrMgniaud, Gavoille - 1994 |

1 |
Treewidth; Computations and Approzimations
- Kloks
- 1994
(Show Context)
Citation Context ...have a node labeling such that for some fixed k, an optimal k-IRS can be designed for G for every choice of positive edge costs, have a treewidth bounded in terms of k. It is known from work of Kloks =-=[29]-=- that random graphs that have sufficiently many edges necessarily have a treewidth of O(n) and thus require k = O(n) for routing under this, stronger, dynamic edge costs model. The paper is organized ... |

1 |
Communication, synchronization, fault tolerance
- Hoepman
- 1996
(Show Context)
Citation Context ... as many as θ(n) nodes in any general shortest-path routing scheme, thus proving that the overall worst-case bound of O(n 2 log n) bits cannot be improved. On the other hand Buhrman et al. [4](seealso=-=[5]-=-)have argued that ‘almost all’ graphs can be routed with routing tables of O(n 2 ) bits, using a formalization of the problem in terms of Kolmogorov complexity theory. For an overview of these develop... |

1 |
A survey of compact routing techniques
- Vounckx, Deconinck, et al.
- 1993
(Show Context)
Citation Context ... be coded more succinctly at the nodes, usually with a different regime for the routing decisions as well. Tables typically use in the order of d log n bits per node, hence O(e log n) bits total. See =-=[7, 8]-=- for an overview. We will consider the most popular of these techniques called ‘interval routing’ and will demonstrate that for general networks it is not likely to work as well. Interval routing is b... |