Complexity Analysis of Routing Algorithms in Computer Networks

Transcription

1 Complexity Analysis of Routing Algorithms in Computer Networks Peter BARTALOS Slovak University of Technology Faculty of Informatics and Information Technologies Ilkovičova 3, 84 6 Bratislava, Slovakia peterbartalos@stonlinesk Abstract There is a huge growth of the Internet users in the last years The network became overloaded One of the necessary processes in computer networks is routing, performed by various algorithms It became important to analyze the complexity of routing strategies This paper includes a complexity analysis of two routing strategies We have showed, that the space complexity of routing with routing tables in network with n nodes is O(nlog ) and the time is O() The space complexity of interval routing with at most k intervals is O(k logn) and the time is O(log(k )), where is the maximum degree of the graph Introduction The routing is one of the latency sources in communications network Routing algorithm has a task to determine the suitable port for delivering message addressed to one network node The aim is to have the fastest routing available in the network The case of every node sending the message to each other by correct delivery can be considered as optimal one It could be achieved by keeping a table of size O (n) by every node, where n is the number of nodes It is not very suitable in larger networks There are lots of methods which memory demand for the data holding required for routing is very low To ensure effective routing with the best possible throughput it is important to identify and analyze complexity of routing strategies In this paper we will use the classic operators of asymptotic notation Supervisor: Mgr Martin Nehéz, Institute of Informatics and Software Engineering, Faculty of Informatics and Information Technologies STU in Bratislava M Bieliková (Ed), IITSRC 5, April 7, 5, pp 69-76

2 7 Peter Bartalos Preliminaries Asymptotic notation (such as O, Ω, o) is used in this paper as it is usual in the computer science, cf [3] As a basic computational model we assume a standard abstract random access machine (RAM), cf [3] We will assume a RAM with logarithmic cost criterion for analysis of space complexities and unit cost for time complexities The network is modeled as a simple unoriented graph G = (V, E), where nodes from V represent processors and edges from E are communication links The maximum degree of G is denoted by For other details see [5, 3] Routing tables Informal description The principle of routing with routing tables is straightforward Every node keeps a table with entries for each other node Using these entries, it could be determined via which outgoing port a message had to be sent m 4 5 Fig Routing tables (message m destined to node 5) Routing with routing tables is today a dominant strategy There are lots of methods to reduce the memory demand for the data holding required for routing The aggregation is one of the most important It is based on that we don t hold information for each node in network but only for subnetworks to which nodes belong This way one can excessively reduce the size of routing tables number of bits number of operations

3 Complexity Analysis of Routing Algorithms in Computer Networks 7 Complexity analysis The Algorithm contains a description of a routing algorithm with routing tables in pseudocode Recall that network is modeled by a graph G = (V, E) with maximum degree Algorithm The algorithm is understood for routing in nodeu V Input: destination node d, message m A data structure table[] is a field which implements the routing table Its size is n and it stores identification labels of ports which are positive integers It is organized in such a manner that in i-th position is stored an ID-number of port via which a message addressed to a node w u is forwarded receive(m, d); if d = = u then OK else p = table[d]; send(m, d) to p; Lemma To store a positive integer j it is required a register with at least log j bits Proof It is possible to vary at most j = n values in n-bit register Taking logarithm of j= n resulted in log j = n It means for storing the j value is an n bit register necessary Indeed, log j bits are necessary, since the number of bits must be an integer Theorem Routing with routing tables requires O( nlog ) bits to store the routing information Proof The size of registers in RAMs depends on the number stored in it 3 We need log bits 4 to keep port numbers We need to store n such values Hence the memory requirements are log = O( n log ) n bits Using routing tables represented by data structure described in previous section the process of finding the corresponding port is very fast The time of executing the algorithm doesn t depend on network parameters It is a constant time performance Theorem Time complexity of Algorithm in RAM model with unit cost criterion is O () 3 in difference with real computers with fixed register sizes 4 from Lemma

4 7 Peter Bartalos Proof Algorithm consists of 4 operations It is necessary to carry out exactly these 4 = O() operations for optional input, ie constant amount, independent on input 3 Interval routing 3 Informal description Interval routing is a space efficient routing strategy used in computer networks This method gaining a great interest because of accessible implementation chips available on the market The basis of interval routing idea is to label all of the nodes with integer from one set (for example {,, 3,, n}) and labeling of all arcs with interval (in range of number of nodes) These intervals means that a message addressed to a node labeled with u, via a port labeled with interval which includes a node u is forwarded 4 [3, 5] [, 3] [] [, ] 3 [4] [5] [, 5] [5] [4] m [, 3] 5 Fig Interval routing (message m destined to node 5) Definition : Interval labeling scheme (ILS) of graph G = (V, E) is a scheme, where: a node labeling is an assignment of unique labels to nodes of V for each node v V, an edge labeling is an assignment of disjoint intervals to arcs e I(v), where I(v) denotes the set of arcs outgoing from v

5 Complexity Analysis of Routing Algorithms in Computer Networks 73 An interval routing scheme, denoted also IRS, in a graph is a valid ILS The validity means that the routing strategy guarantees that the message will always be delivered to its destination using that ILS Proposition [] The validity of every ILS can be checked in O(n ) time, where n is the number of nodes of the graph Intervals used in labeling are understood as cyclic: [a, b] = {a, a +,, V,,, b} for a > b In case of IRS using only linear interval I = [a, b], where a b, it is called a linear (LIRS) If the labels of all outgoing arcs are grouped into at most k intervals then the scheme is denoted as k-irs The memory requirements of interval routing schemes depends also on k The Compactness of a graph G is the smallest integer k such that G supports k-irs Proposition [] For general graphs, the problem of deciding whether G supports -IRS is NP-complete 3 Complexity analysis Let Ψ (u) be the number of intervals assigned to a node u Let Ψ denote max( Ψ (u) ) over all nodes u V (G) Lemma It holds: Ψ k, hence Ψ = O ( k ) Proof Let us denote φ i (u) the number of intervals which belongs to port i in node u We know that we denote every port with at most k intervals in k-irs, so φ i (u) k ϕ i ( u) represents Ψ (u) Since Ψ( u) = ϕ i ( u) k = k deg( u) k for ports ports ports each node u V, hence Ψ k, what implies 5 Ψ = O ( k ) The pseudo code of a routing algorithm is as follows Algorithm : The algorithm is understood for routing in node u V Input: destination node D, number of intervals in node Ψ Output: port number Other variables: index of interval I Algorithm uses these functions: int getleftborder(int i) returns the lower bound of interval stored in i-th position in sequence P = (a,,a ψ ) of lower bounds of intervals 5 from definition of O-notation

6 74 Peter Bartalos int divide(int R) returns R / for R > and for R equals I = Ψ / ; R = I / ; While() if(getleftborder( I )) <= D ) if(getleftborder( I + )) > D ) return( I ); else I = I + R ; R = divide( R ); else I = I - R ; R = divide( R ); Theorem 3 Provided having a graph G and let u V(G) Assume that it is possible to create k-irs for G We need than O ( k deg( u)log n) bits to keep routing information in node u The space complexity for a whole network is O( k log n), since we can substitute deg(u) by Proof For a convenience, let us assume only a linear k-irs The proof for a general (ie nonlinear) k-irs can be done by a similar manner In all nodes we will denote the i-th interval in node [a i, b i ] for i {,, Ψ } and the port to which this interval belongs we will denote p i Let us assume that intervals [a i, b i ] are increasingly sorted and linear Recall that P = (a,,a ψ ) is a sequence of lower bounds of intervals and P = (p,,p ψ ) is the sequence of ports such that p i is a ID-number of port via which a message addressed to a node included in interval [a i, b i ] is forwarded Than we only need to store these two sequences for routing It is not necessary to keep upper bounds b i of intervals We can evaluate them: b i = a i+ - for i < Ψ and b i = n for i = Ψ Storing the lower bound requires O(log n) bits and the number of port O (log ) bits In both sequences we have exactly Ψ members Totally we need Ψ( O (log n) + O(log )) bits As < n, therefore also log < log n It yields Ψ( O(log n) + O(log )) = Ψ O(log n) = O( Ψ log n) = O( k log n) n Proposition 3 [] A more accurate coding allows us to use only O( k log ) bits k per node Assume the encoding of subsection 3, finding the interval which belongs to the destination node requires looking in at most Ψ intervals Using sequential searching the time complexity is O (Ψ) However P is sorted, we can reduced the time We obtain the following results Theorem 4 Time complexity of Algorithm in RAM model with unit cost criterion is O (log( k ))

7 Complexity Analysis of Routing Algorithms in Computer Networks 75 Proof It was showed in [5] that the complexity of binary searching is O(log n) for n element list Assuming the Algorithm the resulting complexity is O (log Ψ) = O(log( k )) 4 Conclusions The main results of this paper are summarized in Table Tab Summary Routing tables Interval routing Space complexity O ( nlog ) O( k log n) Time complexity O () O (log( k )) Our results can be used mainly to estimation of latency of message passing in widearea networks References Aho, A, Hopcroft, J, Ullman, J: Design and Analysis of Computer Algorithms Addison-Wesley 974 Bakker,E M, Leeuwen, J, Tan, R B: Linear Interval routing Department of Computer Science, Utrecht University, 99 3 Bartalos, P: Complexity analysis of routing algorithms in computer networks Bachelor thesis (in Slovak) Faculty of Informatics and Information Technologies, STU December 4 4 Bodlaender, H L, Leeuwen, J, Tan, R B, Thilikos, D M: On Interval Routing Schemes and Treewidth Department of Computer Science, Utrecht University Cormen, T, Leiserson, C, Rivest, R: Intorduction to lgorithms MIT Press, 99 6 Flammini, M, Leeuwen, J, Spaccamela, A M: The Complexity of Interval Routing on Random Graphs Comput J 4 ()(998) Fraigniaud, P, Gavoille, C: Interval routing schemes Algorithmica (998) Gavoille, C, Nehéz, M: Interval routing in reliability networks Theoretical Compurer Science Vol 333, No 3, (5) Gavoille, C, Nehéz, M: On Interval Routing in Random Meshes In: Proc Eurocomb '3 (J Fiala ed), ITI Charles University Prague (3), 47-5 Gavoille, C, Peleg, D: The compactness of interval routing for almost all graphs SIAM J Comput 3, No 3 (),, 76-7

8 76 Peter Bartalos Gavoille, C, Peleg, D: The compactness of interval routing Université Bordeaux 999 Gavoille, C: A survey on interval routing SIAM J Discrete Math, No 4 (999), Gruska, J: Foundations of Computing Int Thomson Computer Press Lau, F C M, Tse, S S H: A Lower Bound for Interval Routing in General Networks Networks 9, No (997), Lau, F C M, Tse, S S H: On Two-label Interval Routing Department of Computer Science The University of Hong Kong Lau, F C M, Tse, S S H: Two Lower Bounds for Multi-Label Interval Routing In: Proc of SIROCCO '95, Carleton Scientific Press, Lau, F C M, Tse, S S H: On the Space and Traffic Problems of Interval Routing Department of Computer Science The University of Hong Kong Lau, F C M, Tse, S S H: More on the Efficiency of Interval Routing Department of Computer Science The University of Hong Kong Nehéz, M: On Subgraphs with Compactness in Random Meshes In: Proc 3rd International Conference APLIMAT 4, Dept of Mathematics, Faculty of Mechanical Engineering, Slovak University of Technology, Bratislava (4), Leeuwen, J, Tan, R B: Interval routing Comput J 3 (987), 98-37