Kóczy, László T.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

A MATHEMATICAL MODEL OF PATH SEARCHING IN GENERAL TYPE SWITCHING NETWORKS* By L. T. K6CZY Institute of Communication Electronics. Technical University. Budapest Received September 13. 1980 Presented by Prof. Dr. S. CSIBI Introduction Let us consider a simple 4-stage folded switching network (Fig. 1.). Path searching has been carried out from the caller switching matrix Al and the called matrix A4 until stage D, also the adjoining free D-D links have been detected: In the figure only the free paths running uninterruptedly from Al to A4 are marked. Now we use the minimum index principle by choosing a "free path": AI-B2-C3-DI-D2 from the caller. Now for the called matrix only one possibility remains: D2-C2-B2-A4. By this we have chosen link B2 - C2 twice, but this fact has not yet been detected. In order to eliminate double selection the algorithm must check all pairs of links between the name stages. After the detection of failure a new "foregoing" path will be chosen: AI-B2-C2-D2-DI The returning path will be now Here the same failure occurs. The next version: DI-C2-B2-A4. AI-B2-C2 -D2-D3 -C2-B2-A4. Failure. * Lecture submitted to the 9th International Te1etraffic Congress 32 L. KOCZY In our very simple example only the fourth version will be sufficient: Al-B2-C2-D2-D3 -C3 -B2-A4. As a matter of course, by using another principle in choosing the paths e.g. the maximum index principle or random choice - much earlier a "good" patr would have been found. It is clear, however, that repeated checking and rejection of paths requires a considerable amount of time. A o 2 3 " n.r?:. 1 Let us consider a second example. Two demands of connection are raised successively, in a 4-s~"6e "one direction" network (Fig. 2.; Al to D2 and A3 to D3. After the eX3mination of link conditions, for the first connection we choose: AI-B2-Cl-D2. In the case of Fig. 2. the second demand must be rejected. In the case, however, when first we choose Al-B2-C2-D2, there remains a free path for the second connection: A3 -B2-Cl-D3. It is again very difficult to check all alternative possibilities fo-r one or several always existing connections, in order to gain a rearrangement strategy performing these examinations successively. In both examples the main problem is common: How to find at least two disjoint paths in the same network, at the same time. (Namely, in folded networks a speach path is always composed of several simple paths, from one end of the network to the other; in the first example of two simple paths.) In the PATH SEARCHING IN SWITCHING NETWORKS 33 first part of this paper a transformation rule for mapping the above mentioned problem into a, though more complicated, end-to-end path searching problem, IS gIven. A B c D ---o 2 ---o o 3 -!> o o o o Fig. 2 Transformation of multiple path searching tasks Immense time consumption of the algorithms sketched in the above section was caused by the impossibility of immediately recognizing double selection of links. For the elimination of the problems we introduce a mapping on the structure of networks, which results in a more complicated network but always in a very simple path searching task. Definitions 1. Switching Network Graph (SN G ) A graph is named a switching network graph if there is a subset of its nodes (start nodes) so, that a. there is no path of length one between any pair of these nodes b. there exists another subset of nodes (termination nodes) so, that there is at least one continuous path between an arbitrary pair when the elements are chosen from both subsets (it is not necessary that these subsets be disjoint or different at all.) 2. Simple Switching Network Graph (SSN G) A graph is named a simple SNG ifthere exist two disjoint subsets of nodes fulfilling condition la and these two fulfilling lb. 3 Periodica Polytechnica EL 25 1. 34 L. KOCZY 3. Switching Path Bundle (SP ) In a SNG we name a set of paths switching path bundle, if all elements of the set are continuous paths from a start node to a termination node, and the elements are disjoint. 4. Simple switching path (SSP) A SP in a SSNG is a simple SP, if the number of paths is one. Statement A If given a SNG G, there exists an invertible mapping M, so, that G* = M(G) is SSNG and for all SP-s Pin G if the number of elements in SP is less than a given N, p* = M(P) is SSP in G*. We omit the proof, as it has no practical importa~ce in the general case. Now we establish the concrete implementation of the above statement for the two important problems mentioned in the Introduction. 5. Staged one-direction SSN G If the nodes of a SSNG can be partitioned into n> 2 subsets S l' S2' ... , n SjS/1Sj=f/J if if}, U Si=G, the set of graph nodes, and there is no edge 1 1 between any pair of nodes belonging to the same Si' and S 1 is the set of termination nodes; the graph is named a staged one-direction SSN G (DG). Statement B If given a DG G, there exists the mapping M, according to statement A, so that G* is also DG, n*:...: n, and if mi,j (i = 0, ... , n - 1; i <) ~ 11) are the numbers of links between stages Si and SI, mtj= (~j) (N as in definition 3.) Proof Let us construct Si in G* in the following way: If then . (Si+N -1) #Si= N ; PATH SEARCHISG IS SWITCHlSG SETWORKS 35 and IV1(M i .k " and where (eG.c) is an edge between elements of Si and S/i> i), e0.Cl.C2 .... cs is an edge between Si and Sj. Fig. 3 Fig. -f So to all "cuts of between Si and Sj there exists one and only one link beh;;een "* ~j . -Let us examine an example where n = 3. Si = 2 for i = 1, 2. 3 and N 2. The structure is given in 3. G* is to be seen in 4. We pointed out a in G and its equivalent SPP in G* 6. Swged jalded SSN G If a SSNG is partitioned as in def. 5 and there is no edge between any pair nodes belonging to the same Si (i = 1, ... , n 1), but there is at least one edge to any element of Sn going to an (not necessarily different) element of Sri and SI is the set of start and termination nodes; the graph is named a staged folded SSNG ). 3* 36 L. KOCZY Statement C If given a FG G, there exists the mapping M according to statement A, so that S* is DG, n* = n, and if mi •j U = 0, ... , n - 1; 1 <j ~ Il, i =1= j) are the numbers oflinks between stages Si and Sj; the numbers oflinks between stages S7 and Sj U =1= j) are m*.= I.) (m . . ) I.) 2N (N as in definition 3), and S; is equal to the number of N -tu pIes of different pairs in Sn which are linked by SnSn type edges. Proof Let us construct S7 in G* in the following way: If then U=I=n) and and So to all "cut pairs of SP-s" between Si and S j there exists one and only one link between S7 and Sj. An example is shown in Figs. 5 and 6, Il = 3, # Si = 3 for i = 1, 2, 3 and N =1. A folded path and its equivalent is shown on the figures. Conclusion 1 By using the practical version Band C of Statement A we have obtained a way of transforming a very general class of switching networks and path searching problems into a quite simple network and problem. By giving, however, the number of nodes and edges (virtual switching matrices and links) PATH SEARCHING IN SWITCHING NETWORKS 37 we pointed out the difficulties of this method. When considering a simple n-stage network with a constant number of machines in a stage(s),the original. value 115 grows by the transformation to about Fig. 5 H2,3,J Fig. 6 S3 Because of considerations based on the statements of the second part of this paper, more important is the growth ofthe number oUinks. Let us assume, that in the above network, there are t links between two adjoining stages. Now their total number (n - l)t grows to (l1-1)G) (11-1)((t-1)/2 by the transformation. 38 L KOCZY Modelling stages by incidence :matrice,~ In the first part the problem caused by the complicatedness of path or path bundle was discussed, Now we concentrate on the irregularity of networks, but we consider only DG-s and SSP-s, The statements in the first part allow this restriction, ~ -r:ie:. 7 4 -Fig,1) Fig. Y r 11 '1 1 1 1 l1 1 1 1 1 1 [' 0 , 01 o 1 1  1 OD 1 o 1 0 1_ l11GOlj 1 1 1 0 0 1 1 1 0 G 000 1 1 000 1 1 What are the main features of an irregular netv\iork ': 1. The dimensions of the switching matrices) even in the same stage. 2. Not all crosspoints exist really (e.g. because of failures this can be time-variant). 3. Some links connect not adjoining stages. Such links are called passing In our algorithm we model SM-s by binary incidence matrices. simple 11 x m SM is modelled by a 11 x m matrix full with 1 - s (figure 7). If in a SM there are empty places of cross points, there we put 0-5 in matrix (Fig. 8). Now the question arises, how' to manage different SM-s belonging to the same stage together. It means no difficulty, if we merge two SM -s into one in the way, that the resulting SM has as many inputs and outputs as the two original ones together, and there are no crosspoints on the places where an input of the first SM meets an output of the other (figure a). So, with the help of the abo ve method (Fig. 9), they can be modelled together by one matrix. PATH SEARCHING IN SWITCHING NETWORKS 39 By using this idea an arbitrary number of SM-s can be merged into one matrix. What about passing links? We assume on each passing link 1 x 1 virtual SM-s in each stage, where the link passes through. By this method an arbitrary SSNG with n stages can be modelled by n matrices (and the link structure), each of them belonging to one stage. ~ t '.- '--' t r [1~ [11 Fig. 10 Path searching in SSN G-s Let us consider now one stage with m inputs and n outputs. The input and output link bundles can be modelled by state vectors, where 1 means a free, 0 a busy link. It is easy to see, that if a link state vector ;; _ 1 and the stage matrix F i are gIven, I:' = I: 10F • I • I 1 results in a vector, which features the possibilities to reach the links between the ith and (i + 1 )th stages - taking into consideration only the state of the previous link bundle and the structure of SM-s in stage i. The symboi "0" stands for the conjunctive composition of [aij]o[b jk] = [ciJ , Cik = v (aij;\ bjk ). j A SSNG consisting of S stages is given, the stages featured by matrices are F l' F1 , ... , Fs. The state of the network is given by the link state vectors fl' ... , is- l' A SSP searching task can be defined by vectors fo and is, the indicator vectors of possible starting and termination points in the network (1 indicates "good" points). Then a complete free path map can be gained by the following algorithm: P6=f6 pT = (pT-l oFJ;\ fT i=l, ... ,s 40 L. KOCZr where " 1\ " stands now for vectorial conjunction: As we stated above, Pi _ 1 of i is the vector of reachable links in stage i, it must be, however, masked by the state of these links J;. There is still another problem. stage i-I stage I Fig. 11 [0 1 0 0] p = 1000 I 000 1 0010 The sequence of the same links considered as OUtputs of the ith stage and as inputs of the (i + 1 )th stage is not the same. So their suitable permutation must be produced by permutation matrices (Fig. 11). As the link permutation matrices are structure determined similarly to the stage matrices, it is an obvious aim to merge them pair by pair. For permutation matrices conjunctive composition and matrix multiplication are identical, so we can write: Here we assumed the associativity of conjunctive composition, which property can easily be proven: ([aij]o[bjk])o[Ckl ] = [v (a ij 1\ bjk)]o[Ckl ] = j = [v (( v )uij 1\ bjd (1\ Ckl )] = [v (v (aij 1\ bjk 1\ Ckl ))] = k j k j = [v (aij 1\ b jk 1\ Ckl )] = l v t v ) b jk 1\ Ckl ( 1\ aij)] = j,k j k Statement of this part of the paper is Statement D. If given a SSNG G with structural matrices I i belonging to the subsets Si (i = 1, ... , s), where Ii are gained from PpF if i> 1, and 11 = F l' and Pi are the permutation matrices to ordering the sequence of F i - 1 columns so, as to be in accordance with that of the lines in F i, finally Fi are the incidence matrices PATH SEARCHING IN SWITCHI.\'G NETWORKS 41 indicating with elements 1 the meeting of a line and a column if the links (edges) corresponding to them are allowed to participate in the same SSP, furthermore given are the mask vectors 1; (i> 1), with elements 1 corresponding to links allowed to participate in an arbitrary SSP; all solutions for the path searching problem given by vector Po, where a path is searched to Sn, are determined by the algorithm: .' Po given, pT = (pT- loIJ 1\ iT; i= 1, ... , S where Pi indicates the links taking part in a path by elements 1. The proof is obvious from the above considerations. An example is shown in Fig. 12. It is obvious, that the analogue of statement D: namely, the solution of the problem when the indicator vector p~ for the last stage is given and paths are searched from arbitrary elements of stage 1 is also true; then p~ given, i=1, .. . ,S Statement E At the conditions of statement D, subsets of S 1 and Ss are given by the indicator vectors Po, Ps; then all solutions for the path searching task are given by: going on with the example in Fig. 12, let pT be [010]. Then first we get p~ ... p~ and finally Wo •.. W3 as visible in Fig. 13. Conclusion 2 By statement E we have an algorithm for parallelly finding all paths in a SSNG. The algorithm operates with binary matrices and so with logical operations. Even so, the algorithm requires a high number of operations, i.e. 42 L. KOCZr much time. When the original problem is more complicated and we must use the transformation. of statement A, the dimensions of stage matrices grow even more. This is a serious difficulty of practical application. It is suggested to use a special hardware besides the control processor ensuring fast matrix operations. Such hardware equipment does exist, as e.g. [1100001 001100 000011j r:, [ ~ ~ g g g g] 001100 001100 o 0 0 0 1 1 000011 12 [100] 1 0 Cl 010 010 001 001 Fj [1 0000 0j 001000 000100 000010 010000 000001 [g g ~ ~ g gj 000010 010000 000001 P3 rbb~~ggl b~g 001100 010 o 0 0 0 1 1 001 110000 100 000011 001 [110000] 001100 000011 /, [g] [i i! [i [t l [b]Hil o 1 0 1 1 1 I Po g b g g b g / / 0J P1\~®@1I '---_-'-----J ~~ ~ (p/o li)T o Po P3 o o o P2 -Fig. 12 PATH SEARCHING IS SWITCHIl,G NETWORKS _~ 0 -~i 0 P2~-Po o o o o Fig. 13 o P3 o 43 cellular automaton fields. In any case, however, we have given a complete model of path searching, adequate for a very general class of switching networks. Summary In the stored program control of switching networks. one of the most interesting algorithmic problems is the full examination of path conditions at a given time point and state of the system. Although theoretically. there exists no problem, it is impossible from the practical point of view to examine sequentially all paths in consideration. as the time consumption of this method is enormous. In most existing systems this problem was solved by developing switching networks of specific structure suitable for fast path searching. It is. however, sometimes impossible to satisfy such conditions. because of other reasons, as e.g. traffic considerations. Moreover. these fast algorithms are heuristic and suitable only for a given type of network. and a given problem group. There is. e.g. the question: is there a possibility to rearrange existing speach connlXtions so. that a new connection could be established in a virtually blocked network? This paper intends to give a mathematical model that allows a switching network of arbitrary structure and Il-fold path searching parallelly. We are aware of the fact. that even a compact algorithm able to solve such a universal task needs much time and memory capacity, so using conventional control processors real time functioning is impossible. There exist. howevel'. such special hardware facilities that connected to one of several conventional processors. form an able control s}stem. • dr. U.szl6 T. K6CZY H-1521 Budapest 

A MATHEMATICAL MODEL OF PATH SEARCHING IN GENERAL TYPE SWITCHING NETWORKS

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A MATHEMATICAL MODEL OF PATH SEARCHING IN GENERAL TYPE SWITCHING NETWORKS

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Periodica Polytechnica (Budapest University of Technology and Economics)