MATRIX ALGORITHM OF SOLVING GRAPH CUTTING PROBLEM

Transcription

1 UDC MATRIX ALGORITHM OF SOLVING GRAPH CUTTING PROBLEM V.K. Pogrebnoy TPU Institute «Cybernetic centre» Matrix algorithm o solving graph cutting problem has been suggested. The main algorithm points based on matrix graph presentation were considered. Formalization o the main algorithm procedures deining estimations or selecting relocatable matrix elements and matrix conversion by reciprocal transer o columns and lines was given. Algorithm operation was considered by the example o data transmission graph between the stations o local computer system network. Introduction The problem o graph cutting into minimally connected parts may be reerred to a number o classical problems o graph theory widely used in practical applications. Among the earliest and the most developed applications is the problem o cutting computer circuit given as a graph [1] or in the orm o more adequate model hypergraph [2]. For an ordinary graph [3] the problem o cutting is stated in the ollowing way. Graph G=(S,V) should be cut into parts G =(S,V ), =1,2,...,F, where F is the number o parts into which the graph is cut; S is the vertex set belonging to part; V is the set o arcs incident to vertices S. The set o parts B(G ) is called cutting o graph G, i G B( G )[ G & G = G]; G, Gq B( G )[ G Gq & S Sq = = & V V = V ],, q = 1,2,..., F. q q Here V q is the set o arcs connecting pairs o vertices one o which belongs to the part G, and another one G q. Let us denote V q =g q and call it by a number o arc attachment o parts G and G q. Then a number o arc attachment o cutting graph G is determined by g: F F g = g, q. = 1 q= 1 Traditional criterion o solving the problem o cutting graph G is minimization o a number o arc attachment g at restriction per vertex quantity in parts G. Among the known algorithms o graph cutting there are exact algorithms using techniques o solving the problems o discrete programming and the approximate ones. The approximate algorithms became more widespread. Among them the sequential, iteration and compound algorithms are singled out. In approximate algorithms o sequential type irstly, the graph vertex is selected by a certain criterion then other vertices are attached to it till the irst part obtaining. Then the second part and next parts are ormed rom the rest graph vertices till complete cutting. Iteration algorithms take a certain cutting obtained, or example, by one o sequential algorithms as the original one and then vertices are exchanged rom one part to another in serially connected pairs o parts so that perormance criterion quality is improved. q The analysis o the given algorithm results in the ollowing conclusions: presence o a large variety o algorithms which is generated by a tendency to take into account the speciic character o the investigated objects and graphs used or their description; algorithm alignment on operation directly with a list o vertices, arcs and their attributes; perorming operations o optimization o arc attachment between two parts o graph is o local character and «does not see» what happens to arc attachment between other parts o cutting. The irst two conclusions relect both positive and negative properties o algorithms. So, accounting speciic character results in algorithm uniqueness but allows increasing their eiciency. Using lists is preerable or graphs with a large number o vertices and small number o arcs. As or the third conclusion it relects the disadvantage o such algorithm which is diicult to be surmounted. It is when optimization is ulilled by local enhancements o arc attachments inside an ordinary pair o cutting parts. The cutting algorithm proposed in the given paper is based on graph representation by vertex connection matrix and more complete analysis o estimating eiciency o vertex assignment into subgraphs o cutting. The undamentals o matrix algorithm The cutting problem solution is considered or an ordinary weighted graph G=(S,V,R) represented by the matrix R= r ij, where r ij is the weight o the arc v ij V, i,j=1,2,...,n. Such graph is, or example, a data transer graph [4] between vertices s i S, which correspond in this case to the stations o local area network and weights r ij are the volumes o data transerred between stations s i and s j in network. Matrix R is symmetrical as the weight r ij includes the volume o data which transerred rom station s i to s j and rom s j to s i. Constructing local network o computer system on the basis o several backbones the problem o station distribution on backbones so that the volume o data transerred between stations connected to dierent backbones is minimal occurs. Such problem its to the problem o graph cutting into minimal subgraphs. Vertices included into one subgraph correspond to the stations connected to on backbone. 84

2 Control, computer engineering and inormation science The result o solving cutting problem may be presented by the matrix R divided into blocks as it is shown in Fig. 1. Each block is a connection matrix o subgraph G vertices. Elements o matrix R being outside the blocks determine a number o arc attachment o cutting graph G and orm a section area. Matrix shown in Fig. 1 relative to matrix R may be considered as a matrix pattern speciying cutting conditions graph dimension, block composition indicating the dimension o each o them, block binding to numbers o lines and columns o the pattern. Blocks are considered in this case as potential targets o graph vertices at their inclusion into proper subgraph. I such pattern is overlapped on matrix R o graph G the original cutting in which graph vertex numbers coincide with numbers o lines and columns o the pattern is obtained. In Fig. 1 numbers o lines and columns o the pattern are shown on the let and on the top respectively and numbers o vertices are shown on the right and below. In this case index sets J and J q include numbers o lines and columns o the pattern and J * and J q* o matrix R. For initial cutting J =J *, J q =J q*. Fig. 1. Matrix-pattern or cutting problem Quality o cutting or the considered application is determined by a sum o weights o arc attachment o cutting that corresponds to a sum o elements o matrix R situated in the range o pattern section. Let us denote the sum o weights by value r and element set o section region by value V c. It ollows rom this that r = rij. vij Vc It is possible to decrease value r by equivalent conversion o matrix R moving elements rom set V c with higher weights into the region o pattern blocks. Matrix R and matrix R k obtained as a result o k transormation are equivalent i each o them is the vertex matrix o one and the same graph G. On the basis o this the cutting problem consists in conversion o matrix R to the orm in which a sum o weights o section region elements is minimal and a sum o weights o block region elements is maximal, respectively, while moving elements rom section region into block region and saving the equivalence o matrix R. To implement this approach it is necessary to develop a policy o selection and moving elements o matrix R ulilling the conditions o equivalence or solving cutting problem. A proper algorithm is called matrix and includes two main procedures: determination o estimates or selecting relocatable elements; equivalent conversion o matrix R implementing the selected relocation. Implementing this policy two variants o matrix algorithm are singled out. In the irst pattern blocks are considered as ree places or assignment o selected elements. The second variant o algorithm uses pattern overlap on the matrix R. The initial cutting variant obtained in this case is improved by reciprocal element relocation between section and block regions. The second simpler variant o algorithm is suggested urther. Matrix element relocation Relocation is perormed over the elements o matrix R on which matrix pattern is overlapped. The necessity o element relocation is stipulated by a criterion o solving cutting problem according to which a minimal sum o element weights should remain in section region. At pattern overlapping on matrix R we obtain the initial cutting variant. So, operations o element relocation are perormed in conditions when all places in blocks are occupied. Thereore, to perorm the relocation, the target or moved elements should be released. I the released elements are not carried out o the matrix or saving and arranged on the places o moved elements then reciprocal element relocation occurs; such rule o relocation its to the suggested variant o matrix algorithm. Matrix element may move along a line or a column as it is shown in Fig. 1 or element r', situated in the 2 d column o the matrix. I element moves with the change o line and column as it is shown or the element r" situated in (n 1) column then two relocations are ulilled, or example, irstly, along the column and then on a line or on the contrary. To save matrix equivalence the proper line and column should be moved together with element relocation. I relocation is perormed on a line then a column and appropriate line are moved. So relocation o element r' into block G F results in transer o 2 d column into (n 2) column and 2 d line into (n 2) line. Similarly, at relocation o element r' along the column the (n 1) line is transerred into 3 d line and the (n 1) column is transerred into 3 d column. An object o relocation is oten not one but several elements. A number o elements are determined as a rule by dimension o the block where they move. Relocation o 4 elements o the 3 d column into the 1 st column o the block G with the dimension 4 4 is shown in Fig. 1. One o relocated elements alls on block diagonal element in this case it is the 1 element rom the top. I this element has the weight r ij >0 then it is transerred to the place o diagonal element o relocated column. Similarly, i nonzero element alls on diagonal element at 85

3 line relocation then it is transerred to diagonal element o the relocated line. Such rule o interaction o nonzero element and diagonal element is also used at transer o lines and columns with released elements. Element selection or relocation On the basis o tendency to move elements with maximal total weights rom section region into block region let us introduce preerence judgment o elements or their reciprocal transer. For this purpose each column o matrix is divided into groups o elements according to their belonging to the lines o one block so that group o elements corresponds to lines J, belonging to block G. or example, or the matrix in Fig. 1, 1 st group o all columns consists o irst three elements and the last one o elements {n 2, n 1, n}. At such partitioning one o the groups o each column belongs to a block and all the rest belong to section region. Let us introduce the ollowing notations. Numbers o lines and columns o matrix-pattern are denoted by indices i and j: j is the number o column or line moved into block; i is the number o column or line excluded o the block. Block numbers o matrix-pattern are denoted by indices and q: q is the number o block into which column j is relocated; is the number o block rom which column j is excluded; J q is the set o line numbers o block q; J is the set o line numbers o block. Preerence judgments are calculated relative to the groups o elements situated in section region. Sum o weights o elements in group J q o column j which is transerred into block q is denoted by value α ~ qj, α = r, α = α ξ. (1) qj ij qji qj qji i Jq Here ξ qji is the weight r ij o element (i,j) o group J q which at transer o column j into block q on the place o column i, alls on its diagonal element; α qji is the sum o weights o elements in column j ater its relocation into block q on the place o column i. Similarly or column i which moves into block on the place o column j, the values ~ α i and α ij are calculated: α = r, α = α ξ. (2) i ij ij i ij i J Decrease o element weight sums which occurs in blocks q and excluding appropriate columns i and j rom them is determined by values β qi and β j : β = r, β = r. (3) qi ij j ij i Jq i J The diagram o reciprocal transer o columns j and i and determination o proper estimates in agreed notations is given in Fig. 2. Diagonal elements o columns i and j in blocks q and are hatched. Elements o columns i and j which all on diagonal elements in blocks being transerred are marked out as well. Beore column transer the elements marked out in them move in the column to the place o diagonal element that is shown by arrows in Fig. 2. The places o marked out elements are taken as diagonal elements ater column transer. Fig. 2. Diagram o column reciprocal transer Estimates introduced in (1) (3) allow obtaining sum preerence judgments μ ij or pairs o columns i and j relative to their reciprocal transer, μij = αqji βj + αij βqi. (4) Estimates μ ij >0 show how much element sum weight increases in blocks ater reciprocal transer o columns j and i or, respectively, how much element sum weight decreases in section region. Estimates μ ij are calculated only or elements o pattern section region as element move in blocks does not change the sum o their weights. I matrix R is symmetrical in this case then estimates μ ij are calculated or one part o section region top or bottom. Matrix algorithm and example o its application Ater presentation o graph by adjacency matrix R algorithm operation includes ulillment o the ollowing stages. 1. Pattern ormation and its overlap on matrix R. Pattern is ormed on the basis o inormation about a number o cutting blocks, dimension o each o them and their place in the pattern. Places o block situation in the pattern are determined in a random sequence regardless o block dimension. As a result o pattern overlapping on the matrix R the initial variant o cutting in which J =J * ; J q =J q* ;,q=1,2,...,f is obtained. 2. Calculation o estimates μ ij. Estimates are calculated by the expressions (1) (4). I matrix R is symmetrical then calculation is carried out or one part o section region symmetrical relative to matrix diagonal top (over diagonal) and bottom (under diagonal). Among estimates μ ij >0 a maximal one is selected and appropriate columns i and j are preerred or their reciprocal transer. In this case the conversion to ulilling stage 3 is carried out. Without estimates μ ij >0 algorithm stops operating and ixes the obtained variant o cutting. 3. Reciprocal transer o columns i and j. Columns and proper lines are transerred by the rules given above and the diagram given in Fig. 2. Reciprocal transer o lines and columns is attended by updating o index sets J * and J q*. Ater that the conversion to ulilling the second stage o algorithm is carried out. 86

4 Control, computer engineering and inormation science Let us show algorithm operation by the example o weighted ordinary graph containing 10 vertices. The graph should be divided into 3 subgraphs one o which contains 4 vertices and two other ones contain 3 vertices. Matrix R o the given graph and matrix pattern constructed or the given cutting conditions matched with it are given in Fig. 3, a. The initial variant o cutting obtained in this case is characterized by element weight sums in blocks (numerator) and in section region (denominator) equal 15/50. To improve the ratio 15/50 the estimates μ ij are calculated; the estimate with maximal value is selected among them. In this case it is estimates μ 5,3 =μ 9,3 =9. Let us select any o them, or example μ 5,3 that assumes relocation o element (2,5) with weight r 2,5 =2 into block J 1 and element (4,3) with weight r 4,3 =7 into block J 2. To relocate these elements the reciprocal transer o columns j=5 and i=3 and appropriate lines is carried out. The result o transer is given in Fig. 3, б. The ratio o weight sum is equal to 24/41. μ 9,1 =14 is maximal estimate μ ij or the given variant o cutting. Ater reciprocal transer o column j=9 and i=1 the cutting variant given in Fig. 3, в, with weight sum ratio 38/27 turns out. μ 8,6 =3 is maximal estimate μ ij in this case. As a result o reciprocal transer o columns j=8 and i=6 we obtain cutting given in Fig. 3, г, or which there are no estimates μ ij >0. Thereore, algorithm completes its operation with variant o cutting J 1* ={0,9,2,5}, J 2* ={4,3,8}, J 3* ={7,6,1} and ratio o weight sum 41/24. I the graph used as an example is interpreted as a graph o data transer [4] the vertices o which are stations o local network s i, i=0,1,2,...,9, and arc weights r ij correspond to volumes o data transerred between stations s i and s j then as a result o solving cutting problem we obtain three subsets o stations {s 0,s 9,s 2,s 5 }, {s 4,s 3,s 8 }, {s 7,s 6,s 1 }, each o that is connected to one o three network backbones. At such variant o network construction the majority o data, in this example it is 41 units, is Fig. 3. Example o solving the problem o graph cutting 87

5 transerred between stations inside sets loading appropriate backbone. In this case data may be transerred simultaneously (parallel) in all backbones. Other data at the rate o 24 units are transerred between stations connected to dierent backbones. At such transers two or three backbones are loaded simultaneously that results in increasing network load and, respectively, time or data transer. Conclusion Advantages o suggested algorithm in comparison with the known ones operating with arc list are stipulated by graph presentation in matrix orm at which section region is speciied in explicit orm. It allows easily observing («seeing») the changes occurring in section region at matrix conversion and calculating preerence judgments or ulilling regular transormation on the basis o analysis o current state o the whole section region. The simplest scheme o element relocation rom section region into block region based on reciprocal transer o columns and appropriate lines is implemented in algorithm. Other more complicated schemes connected, or example, with the ormation o relocation close circuits are the subject o urther investigations and they are not considered in the given article. Matrix algorithm is applicable or graphs with asymmetrical matrix including an unweighted one. It is obvious that use o this algorithm turns out to be more preerable or graphs with high degree o vertices connectivity as the volume o calculations carried out by algorithm does not depend on coeicient o matrix illing with nonzero elements. Along with achieving the main goal ormalization o matrix algorithm o solving cutting problem the matrix technique has another no less important value. It consists in the act that matrix orm o cutting problem presentation creates possibilities or deeper understanding o its nature, improving analysis techniques o section region at optimization, studying algorithm dependence on various strategies o determining estimates or making decision. Matrix method may be taken as a basis o algorithm development or various types o graphs and cutting conditions. In particular, bipartite graphs are o interest; their matrix presentations, in comparison with the examined ones, assume more eicient operations o matrix conversion when solving cutting problem. REFERENCES 1. Shtein M.E., Shtein B.E. Methods o computer-aided design o digital equipment. Moscow: Sovetskoe Radio, p. 2. Kornienko A.V., Pogrebnoy V.K. Model and algorithm or cutting digital computing devices into unctional blocks // Upravlyayushie Systemy i Mashiny P Kristoides N. Graph theory. Algorithm approach. Moscow: Mir, p. 4. Pogrebnoy A.V. Determining data transer volumes in computer system network or the speciied model o program load // Bulletin o the Tomsk Polytechnic University V P Received on