HKU CSIS Tech Report TR-97-10
On the Space and Traffic Problems of Interval Routing Savio S.H. Tse and Francis C.M. Lau Department of Computer Science The University of Hong Kong
fsshtse,
[email protected] May 1997
Abstract Interval routing is a space-efficient method for point-to-point networks. This paper addresses two problems with interval routing schemes (IRS’s) that are optimal—i.e., all the routing paths are shortest paths. The first is a space problem. With up to one interval label per edge, the method has been shown to be non-optimal for arbitrary graphs, where optimality is measured in terms of the longest (routing) path in a graph. In this paper, using a non-planar graph, we prove that even with a relatively large number of labels, interval routing still falls short of being optimal for arbitrary graphs. The bound on the longest path we prove is 3D=2 ; 2, independent of any number of labels up to (log n), where D is the diameter of the graph and n the number of nodes. The second problem with optimal IRS’s is that of traffic imbalance, which may occur among the edges of certain nodes. We illustrate this using the multiglobe graph. If the requirement on shortest paths can be relaxed, we show that it is possible to derive an IRS which uses as little as four labels per edge and which balances the traffic at every node. The length of the longest path due to this IRS matches a known lower bound for the multiglobe graph
p
which is valid for as many as ( n) labels. Correspondence: F.C.M. Lau, Department of Computer Science, The University of Hong Kong,
Hong Kong / Email:
[email protected] / Fax: (+852) 2559 8447. Preliminary versions of parts of this paper have appeared in Proc. CATS’97 (Sydney, Australia) [16].
1
1 Introduction Interval routing was first proposed by Santoro and Khatib [12], and subsequently refined by van Leeuwen and Tan [10]. The idea is to label the nodes by integers (called node numbers) from a cyclicly ordered set, say, f0 1 : : : n ; 1g, where n is
h i, where are node numbers. h i is the set f +1 g if , or f +1 ;1 0 g if . h i is the short form for h i, i.e., the set f g. During routing, a message is
the number of nodes; and the edges by intervals of the form p q
p > q
p p
p
:::q
p < q
p p
p q
p p
:::n
p q
:::q
p
routed along an edge whose interval label contains the destination node number, until the message reaches the destination. An example of interval routing is shown in Figure 1. The figure shows the routing path of a message that travels from node 0
1
2
4
3 a message destined for Node 0
Figure 1: An example of interval routing 2 to node 0. The message first takes the edge to node 3 because 0 is contained in the
interval h3 0i, and then takes the edge to node 4 because 0 is contained in h4 0i, and so on. Clearly, with interval routing, at most O (d) space is needed at a node, where d
is the node’s degree. In general, d is smaller than n, the size of the network, and
we say that the routing information stored at a node as required by interval routing is “compact”. See the survey by Tan and van Leeuwen [13] for an overview of the field of compact routing. One of the main questions in interval routing research is that given G, how to label its nodes and edges so that all the routing paths are shortest paths, where G represents either a specific kind of graphs or arbitrary graphs (general networks). A successful labeling satisfying the condition constitutes an optimum interval rout2
ing scheme (IRS). For a number of specific graphs, optimum IRSs are known to exist [13]. What about arbitrary graphs? Ruˇziˇcka answered this in the negative way by constructing a graph that has no optimum IRS [11]. What then can be done if indeed no optimum IRS exists for a given network? One possibility is to relax the compactness of routing information by allowing more than one interval label to be associated with an edge. An IRS that allows up to M labels per edge is called an M -label IRS, or simply M -IRS. Figure 2 shows a graph which has no optimum 1-label IRS, as proved by Fraigniaud and Gavoille [3], but has optimum IRS if up to two labels per edge are allowed. One such optimum 2-IRS for the graph is shown in the figure.
,6>
>
,2>
(5 4)
(10 14)
< 7 9 > < 20 23
(10 17)
< 24 29
(44 45)
(44 49)
< 40 43
>
33 (v211 ) 10 (w2 )
44 (u3 )
>
>
30 34
>
>
>
>
As shown, the number of labels is between 3 and 4. The traffic load ratio for these nodes are 1:27, 1, 1:06, 1:06, respectively, all very close to 1. These ratios, however, are not much different from those of a shortest-path IRS applied to the same graph, simply because the graph in Figure 6 is relatively small in size and has too few layers. For larger graphs with more layers, the difference would be much greater.
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5 Discussion and Conclusion Theorem 4.1 reveals that it is not necessarily true that by increasing the number of
p
labels, the length of the longest path is reduced. Between four labels and (
)
n
labels, the lower bound on the length of the longest path for the multiglobe graph, 5D=4, remains unchanged. It is interesting to note on the other hand, that increas-
ing the number of labels will certainly bring about a reduction in the average path length, as demonstrated below. Theorem 5.1 Given an arbitrary graph and an M -label IRS applied on the graph, if there is a non-optimal routing path from v1 to v2 , then there exists an (M + 1)-label IRS which can replace the path with an optimal one, while not perturbing any of the other paths. Proof: Suppose the shortest path from v1 to v2 is a1 a2 : : : ak , where a1 =
v1
and
ak
= v2 . For i = 1 : : : k ; 1, if L(ai ai+1 ) = Li does not contain L(ak ), add L(ak ) to
Li
, and remove L(ak ) from the interval (of another edge of ai ) that contains it. Let
this latter interval be L0i . The removal might split L0i into two disjoint intervals. As a result, one interval is added to Li , and L0i might become two intervals. After all the adding and removing along the shortest path, we have an (M + 1)-IRS which provides an optimal routing path from v1 to v2 .
2
This paper opens a new avenue for interval routing research: the derivation of interval routing schemes that can balance traffic. One could try to identify graphs or properties of graphs that are most prone to traffic imbalances under interval routing. In Section 4, we defined traffic balance as an attribute of a node, which is in terms of routes that originate from the node. In real operation, other routes may pass through this node and hence add to the traffic load of this node. A more accurate measure would consider all routes, whether originating from this node or not. Consider Figure 3. The lower bound results that are based on non-planar graphs appear to be a constant function. It might be worthwhile to try to replace this by a decreasing function. The segments beyond beyond
M
n = ( D log(n=D) ) for
D
=
(
1
O n3
18
M
= (n=D ) for
D
1
= (n 3 ), and
) are still missing a non-trivial bound.
The tightness of these existing lower bounds for M -IRS (M
>
1) is not so clear as
there are no upper bounds to match. So far, the longest-path analysis and the shortest-path analysis both focus on the worst case. Practically, longest-path analyses is not as significant as average-path analyses. The traffic burden of one longest path is not really that great if most of the other paths are close to being longest. In other words, the lower bound on averagepath length is more important than the lower bound on longest-path length. The average-path analysis is expected to be more complicated; the technique of counter proof can no longer be used, and random graphs, or statistical analysis techniques will have to be used instead.
References [1] M. Flammini, G. Gambosi, and S. Salomone, Interval routing schemes, Proc. 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS’95), 279–290, 1995. [2] M. Flammini, G. Gambosi, U. Nanni, and R. Tan, Multi-Dimensional Interval Routing Schemes, Proc. the 9th International Workshop on Distributed Algorithms (WDAG’95), 1995. [3] P. Fraigniaud and C. Gavoille, Interval Routing Schemes, Research Report No. 94-04, Laboratoire de L’Informatique du Parallˇelism, Ecole Normale Supˇerieure de Lyon, France, 1994. To appear in Algorithmica. [4] P. Fraigniaud and C. Gavoille, Universal Routing Schemes, Journal of Distributed Computing, 10:65–78, 1997. [5] P. Fraigniaud and C. Gavoille, Local Memory Requirement of Universal Routing Schemes, Research Report No. 96-01, Laboratoire de L’Informatique du Parallˇelism, Ecole Normale Supˇerieure de Lyon, France, 1996. [6] C. Gavoille and E. Guˇevremont, Worst Case Bounds for Shortest Path Interval Routing. Research Report No. 95-02, Laboratoire de L’Informatique du Parallˇelism, Ecole Normale Supˇerieure de Lyon, France, 1995. 19
[7] C. Gavoille, On Dilation of Interval Routing, Research Report RR-1145-96, LaBRI, University of Bordeaux, France, November 1996. [8] E. Kranakis, D. Krizanc, and J. Urrutia, Compact Routing and Shortest Path Information, Proc. 2nd Colloquium on Structural Information & Communication Complexity (SIROCCO’95), 101–112, 1995. ˇ [9] R. Kr´aˇloviˇc, P. Ruˇziˇcka, D. Stefankoviˇ c, The Complexity of Shortest Path and Dilation Bounded Interval Routing, Technical Report, Department of Computer Science, Comenius University, August 1996. [10] J. van Leeuwen and R.B. Tan, Interval routing, The Computer Journal, 30:298– 307, 1987. [11] P. Ruˇziˇcka, A Note on the Efficiency of an Interval Routing Algorithm, The Computer Journal, 34:475–476, 1991. [12] N. Santoro and R. Khatib, Labelling and Implicit Routing in Networks, The Computer Journal, 28:5–8, 1985. [13] R.B. Tan and J. van Leeuwen, Compact Routing Methods: A Survey, Technical Report RUU-CS-95-05, Dept. of Computer Science, Utrecht University, 1995. (Also in Proc. Colloquium on Structural Information and Communication Complexity (SICC’94), 99–109, 1994.) [14] S.S.H. Tse and F.C.M. Lau, Some Lower-Bound Results on Interval Routing in Planar Graphs, Technical Report TR-97-05, Department of Computer Science, University of Hong Kong, April 1997. (Preliminary version: Lower Bounds for Multi-label Interval Routing, Proc. 2nd Colloquium on Structural Information & Communication Complexity (SIROCCO’95), 123–134, 1995.) [15] S.S.H. Tse and F.C.M. Lau, On the Space Requirement of Interval Routing, Technical Report TR-96-11, Department of Computer Science, The University of Hong Kong, 1996. [16] S.S.H. Tse and F.C.M. Lau, Two Lower Bounds for Mult-Label Interval Routing, Proc. of Computing: The Australasian Theory Symposium (CATS’97), Syd20
ney, Australia, February 1997, in Australian Computer Science Communication, 19(2):36–43. [17] S.S.H. Tse and F.C.M. Lau, More on the Efficiency of Interval Routing, Technical Report TR-97-04, Department of Computer Science, The University of Hong Kong, March 1997. [18] S.S.H. Tse and F.C.M. Lau, An Optimal Lower Bound for Interval Routing in General Networks, Technical Report, Department of Computer Science, The University of Hong Kong, June 1997.
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