nlHE8fl»i §?> W£8m Bail B3& 8)1$ mmm warn BE SH DH ^B^Kfl UU0] n%£ l»lt§££i H rs£' ■ ».7h ■ I 15 V ■ r-j f ■*K ■ LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 I^6T no. 480-489 cop. 3- C+if 3 UIUCDCS-R-71-U88 December 1971 REFERENCE MANUAL OF FORTRAN PROGRAM ILLOD-(NOR-B) FOR OPTIMAL NOR NETWORKS* by Tomoyasu Nakagawa Hung-Chi Lai IE LIBRARY OFT NOV 9 1972 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPA URBANA, ILLINOIS Report No. U88 REFERENCE MANUAL OF FORTRAN PROGRAM ILLOD-(NOR-B) FOR OPTIMAL NOR NETWORKS* Tomoyasu Nakagawa Hung- Chi Lai Department of Computer Science University of Illinois Urbana, Illinois 6l801 * This work was supported in part by the National Science Foundation under Grant No. NSF GJ-503. Digitized by the Internet Archive in 2013 http://archive.org/details/referencemanualo488naka 11 1. Scope of the program ILLOD-(NOR-B) 2. Set-up of input data 3. Restrictions of the size of the problem k. Examples of input data 5. Program size, flow chart, and notes on possible modifications of the program Appendix. Listing of ILLOD-(NOR-B) 1. THE SCOPE OF THE PROGRAM (i) The FORTRAN program 'ILLOD-(NOR-B) ' derives optimal combinational NOR networks of one output function or a set of output functions. The algorithm is based on the branch- and-bound method. For details, see 'A branch-and-bound algorithm for optimal NOR network' by Nakagawa, T. , and Lai, H. C. , Report No. h?$ f Department of Computer Science, University of Illinois, January 1971. The optimal network means a network with the least cost, where the cost C of the network is defined by the weighted sum of the number of gates and the number of connections, i.e., C=AXR+BXI, R is the number of gates I is the number of connections to the gates (precisely speaking, the connections from external inputs and the interconnections among gates), and A and B are arbitrary non- negative weights. (ii) With this cost function, the program obtains the optimal net- works for the given output function(s). Incompletely specified functions are also acceptable to the program. Suppose a problem consists of m output functions of n variables. Let x ., i=l, ..., n, be external variables, and let f , h=l, ..., m, be the output functions. Before the computation starts, the user must provide the truth table (truth tables for all output functions are condensed into a single table) of the problem. Providing the truth table to the program consists of two steps, namely the specification of external variables, and the specification of output functions. External variables may be specified in either of two ways, (a) an im - plicit specification of external variables , or (b) an explicit specification of external variables . (a) In the case of implicit specification of external variables, the user specifies the number n of external variables along with a parameter which indicates whether or not the uncomplemented variables are available. Reading the value n along with the parameter, the program internally generates the entries of external variables of an ordinary truth table, that is, a truth table which consists of 2 input vectors, as shown in Fig. 1.1. In this truth table, the input vectors are arranged according to the order that an integer j expressed in a binary representation (x_...x ) increases, where x_ is the most significant digit and x is the least significant digit. The uncomplemented variables The complemented variables The output functions r x i X l * * x J * X l ' ' 2 n -l * X l 1 • • • < • * • k» X n x . . n . x J . . n V-l . X n r X l -0 X l ■ ' x J " -2 n -l 1 ' X l < * • * L X n -0 x . . n . x J ' . . n i 2 n -l ' X l J f i A-- . i 1 . . 2 n -l f . i 1 f 2 • • • • ' • f m f°. . m . f J " m '2 n -l . f X m These entries exist only in the case of .logical design problems /where the complemented variables are available as external inputs. Fig. 1.1 The truth table of output functions of n variables. For example, the truth table of a function of three variables is shown in Fig. 1.2. X l x^ x. X l x. X, 00001111 00110011 01010101 11110000^1 These entries exist only in the case of 1100 110 0) logical design prob- lems where comple- 10101010 mented variables are available as input variables . Fig. 1.2 The truth table of a function of three variables. (b) In the case of explicit specification of external variables , the user specifies the entries of external variables of the truth table using additional cards called < external- variable- card > s. The explicit speci- fication of external variables is used in the case of logical design prob- lems where output functions have many don't-care terms. Suppose that the output functions are defined for a subset of input vectors of the entire truth table of Fig. 1.1. Let x , J=D-m Zo> ••'> J denote these input vectors. The user can 'condense' the truth table of Fig. 1.1 into another table shown in Fig. 1.3. only u input vectors The uncomplemented) variables \ The complemented / variables These entries exist only in the case of logical design problems where the complemented variables are available as external inputs . Fig. 1.3 A 'condensed' truth table having only the input vectors x J , 3=3-,, '>>, 3 , for which the output functions are defined. Using < external- variable- card > s, the user can set up internally the table of Fig. 1.3 in place of Fig. 1.1. (iii) The program is able to find optimal solutions under the following two kinds of constraints: the maximum fan- ins and/or fan-outs of gates, and the maximum level (delay) of the network. In this program, we assume that the fan-out constraint is applied to only the output connections of gates, but not to the connections of each external variable. (iv) Because of the nature of the branch- and-bound method, the first network the program generates may not be optimal. After generating the first network, the program searches for better networks, by backtracking. Each time the program finds a network whose cost is not greater than the previous "best" network, the program prints out the network. Eventually the program enumerates all networks (implicitly). All the networks having the least cost on the output pages are optimal. For difficult problems, however, the program may not terminate the com- putation in a reasonable time. If the user anticipates such excessive com- putation time for a particular problem, he may specify a certain CPU time to be spent on the problem. The program will dump the status data of the com- putation onto a punched card deck after the program spends that prespecified amount of time. This card deck includes all information with which the com- putation of the problem can later be continued at any time. (v) If the user is interested in finding only one optimal network, he may specify with a parameter so that the program searches for only one optimal network, instead of all. Searching for only one optimal solution may cut the computation time to some extent. (vi) The user may specify the initial cost ceiling to a value he thinks is reasonably low, in order to preclude some non-optimal networks which the program may otherwise generate. This reduces the computation time. 2. SET-UP OF INPUT DATA TO THE PROGRAM The input data for any one problem consists of four kinds of cards: (i) < header-card > (ii) < problem-parameter- card > (iii) < external- variable-card > s (iv) < output -function- card > s. The user must prepare the input cards (i), (ii), and (iv) for each problem. The user must prepare the input cards (iii) only when he chooses the ex- plicit specification of external variables for the truth table. The fol- lowing are the formats of these input cards. (i) < Header-card > This is the first card of the input deck for a problem. This card may contain any alphanumeric information in columns 1 ~ 72, which may be used for the identification of the problem but which has no relation to the actual computation. This information will be printed on the front page of the output. Columns 73 ~ 80 must be blank. Columns 73 ~ 80 being blank indicates that the computation starts with the initial solution. (See 'Rerun of the problem' , at the end of this chapter. ) (ii) < Problem-parameter- card > This card specifies the nature of the problem the user wants to solve. There are 11 fields to specify the parameters with characters and numerals. Col. 1: A blank 'b' ' or a character 'S'. -J- A 'b' stands for a blank (no character punched). The 'b' parameter specifies that the problem to be solved is a multiple optimum problem, that is, the program searches for all optimal networks. The ' S' parameter specifies that the problem to be solved is a single optimum problem, that is, the program searches for only one optimal network. Col. 2: A blank 'b', or one of the characters 'X', 'Y', 'U', and 'V . The 'b', 'X', or r Y' parameter indicates an implicit specification of external variables . The ' U' or ' V parameter indicates an explicit speci - fication of external variables . The 'b' or 'X' parameter specifies that only uncomplemented external variables are available for the network. The 'Y' parameter specifies that both uncomplemented and complemented variables are available for the net- work. If the user specifies the 'b', 'X', or 'Y' parameter, the program sets up the truth table by generating a set of 2 input vectors fx!:, ..., x ), for j=0, ..., 2 -1, in the case of a 'b' or 'X' parameter, or (x^, ..., x , x"i , ..., x J ), for 3=0, ..., 2 -1, in the case of a 'Y' parameter. The ' U' parameter specifies that only uncomplemented external variables are available for the network. The ' V parameter specifies that both un- complemented and complemented variables are available for the network. If the user specifies either of ' U' and 'V parameters, then he must prepare n < external- variable- card > s that follow the < problem-parameter-card > . The program sets up the truth table by reading these < external-variable- card > s. Cols. 3 ~ h: An integer N which is right- justified. N specifies the number n of external variables of the output functions. Be sure to write n (not 2n) for N, in the case of both uncomplemented and complemented variables available. (A 'Y* or 'V parameter in column 2.) Furthermore, the value of N must agree with the number of < external- variable-card > s, in the case of an explicit specification of external variables. (A 'IT' or 'V parameter in column 2.) Cols. 5 ~ 8: An integer M which is right- justified. M specifies the number m of the output functions to be realized simul- taneously. The value of M must agree with the number of < output- function- card > s. Cols. 9 ~ 12: An integer A which is right-justified. A is the value of the non-negative weight for the number of gates in the cost function. (See Table 1 'Typical combinations of values A and B for different optimization problems ' . ) Cols. 13 ~ 16: An integer B which is right- justified. B is the value of the non-negative weight for the number of inputs in the cost function. (See Table 1. ) Cols. 17 ~ 20: An integer C which is right- justified. C_ sets the initial value of the cost ceiling. If the user specifies a positive C Q , the initial value of the cost ceiling C is set to C„. If the user does not specify this field (i.e., leaving blank), then the program sets the initial value of the cost ceiling C automatically to 12 X A + 99 X B."^ + This cost ceiling, for example, precludes networks of more than 12 gates in the case of minimizing the number of gates primarily, or precludes networks of more than 99 connections in the case of minimizing the number of connections primarily. Optimization Problem Values of A and B minimizing the number of gates only. minimizing the number of gates primarily, then minimizing the number of connections secondarily. minimizing the number of connections only. minimizing the number of connections primarily, then minimizing the number of gates secondarily. minimizing the sum of the number of gates and the number of connections. A = 1 and B = A = 100 and B = 1 A = and B = 1 A = 1 and B = 100 A = B = 1 TABLE 1 Typical combinations of values A and B for different optimization problems. 10 Setting a lower cost ceiling at the beginning of the computation will eliminate some non-optimal networks. But setting the cost ceiling too low may cause the problem to become infeasible. On the other hand, setting the cost ceiling sufficiently high at the beginning of the computation guarantees the feasibility of the problem, yet the computation time may become longer. Cols. 21 ~ 2k: An integer FANIN which is right- justified. FANIN is the maximum number of fan- ins to each gate. If the user does not specify this field, then the program sets the number of fan- ins to a sufficiently large number (100). Cols. 25 ~ 28: An integer FANOUT which is right- justified. FANOUT is the maximum number of fan-outs from each gate. If the user does not specify this field, then the program sets the number of fan-outs to a sufficiently large number (100). Notice that the fan-out constraint is applied to the output connections originating from each gate, but not to the connections of each external variable. Cols. 29 ~ 32: An integer LMAX which is right- justified. IMAX is the maximum level of gates. If the user does not specify this field, the program sets the maximum number of levels to a sufficiently large number (100). Cols. 33 ~ 36: An integer TMAX which is right- justified. TMAX is the time in seconds assigned to this problem. If the user does not specify this field, then TMAX = 36OO (1 hour) is assumed by the program. When the actual CPU time spent for this problem exceeds TMAX, the program stops the computation, and dumps the current results of the computation onto a card deck through PUNCH statements. (See 'Rerun of the problem', in the end of this section. ) 11 (iii) < External- variable-card > s. In combination with the 'U' or • V parameter in column 2 of the < problem-parameter- card >, the n < external-variable-card > s specify the entries of external variables of the truth table of Fig. 1.3. Each card . a, i.e., { x p contains the binary representation of external variable x., i.e., (x, , ^2 ^ u XLp , ... , x. ), starting from column 1 of the card. The maximum number of digits in a binary representation is limited to 32. (This means the maximum number of input vectors is 32. ) If the actual number of digits is less than 32, then put a termination symbol '/' (a slash) on the right of the right-most digit of the binary representation on the first < external- variable-card >. The remaining columns after the termination symbol '/' in the first card, as well as the same columns in the following cards, may contain any alphanumeric information which may be used for identification. This information will not be printed on the output pages. In the case of the ' V parameter, the program generates the binary representations corresponding to complemented variables by taking negation of the entries of the < external- variable- card > s. Therefore the user must not provide < external- variable- card > s representing the complemented variables, x.. (iv) < Output-function-card > s. The m < output -function- card > s specify the set of m output functions to be realized simultaneously. Each card contains the binary representation of one output function f, , starting from column 1 of the card. Use a symbol '*' to denote don't-care terms, if any. The maximum number of digits in a binary representation is limited to 32. The actual number of digits must be 2 in the case of an implicit 12 specification of external variables, or must be the same as defined by the < external- variable- card > s in the case of an explicit specification of external variables. The remaining columns, up to column 72 (inclusive), may contain any alphanumeric information -which may be used for identification. This information will not be printed on the output pages. The following is the input card sequence for the execution of the pro- gram using typical JCL statements for the IBM 360/75. The Program r The input cards \ //ID . . . // EXEC FORT // FORT.SYSIN DP * FORTRAN Program ILLOD-(NOR-B) ~n " // EXEC LKED // EXEC GO, REGION = 3^8 K //GO.SYSIN DD * < header-card > I < problem-parameter-card > & s (m s \< header-card > }if any sThe first problem yihe second problem I The last J problem / Fig. 2.1 Input card sequence for the execution of the program ILLOD-(NOR-B). 13 Rerun of a problem (i) Organization of the card deck punched by the program. When the actual CPU time spent on a problem exceeds TMAX, the program dumps the current status data of computation onto a card deck through PUNCH statements. This deck consists of the following cards. The first three cards are dummy, containing triple punches in columns 1 through 80. These cards are punched merely for the purpose of separating the card deck from the previous problem, so they should be discarded. The fourth card (this becomes the first card of the actual deck for the rerun, since the first three cards are discarded ) is a duplicate of the < header-card > of this problem except the field of columns 73 ~ 80; the field of columns 73~80 is filled with a 5 character word 'RERUN' followed by a blank 'bbb'. The fifth card (this becomes the second card of the actual deck) is a duplicate of < problem-parameter- card > of this problem except two fields, C„ and TMAX; the field of C is being replaced by the cost of the best feasible solution found thus far, and the field of TMAX is cleared with blanks . The succeeding cards contain all other information necessary for con- tinuing the computation starting from the point where the program halted the computation due to the time limit TMAX. These cards are given se- quence numbers in columns 76 ~ 80, starting from the sequence count 1, for assuring the proper card sequence. (ii) Set-up of the input cards from the card deck punched by the pro- gram. The card deck explained in (i) above is ready for running. (Notice that the first three cards are discarded.) The user may specify the time Ik limit again for this run, by putting a time limit (in seconds) in the field of TMAX of the < problem-parameter- card > which is included in this card deck. The integer TMAX must be specified as the sum of the time already- spent in the previous runs and the time to be spent in this run, for this problem. (iii) Running The program accepts the above card deck as an ordinary input deck. The user can put this deck anywhere in the entire input card deck shown in Fig. 2.1. 15 3. RESTRICTIONS ON THE SIZE OF THE PROBLEM The restrictions on the size of the problem are as follows. 1. The number t of input vectors in the truth table is 32 or less. 2. The number n of external variables. Because of t < 32, n is 5 or less in the case of completely specified functions. In the case of incompletely specified functions, however, any n < 20 is acceptable if only uncom- plemented variables are available, or n < 10 if both uncom- plemented and complemented variables are available, provided that the truth table is defined by the < external- variable- card > s. 3. The number R of gates: The number R of gates in the network must not exceed UO-n at any time during the computation, in the case of uncomplemented variables only available (a 'b', 'X', or 'U' parameter). In the case of both uncomplemented and complemented variables available, the limit is lowered to ^0-2n ( a ' Y', or 'V parameter). Should the current network exceed these limits, the program prints a message and terminates the computation of the current problem. k. The size of the possible cover list. An array of 3000 entries is reserved for the possible cover list (PC-list). The array can be divided into at most 500 segments, where each segment will contain the possible covers of one un- covered component P. Should the current number of P's exceed 500, or should the current number of possible covers in the PC- list exceed 3000, the program prints a message and immediately 16 terminates the computation of the current problem. The user may loosen these restrictions by enlarging the size of appropriate arrays reserved by the program. The complete instructions for enlarging the size of arrays for this purpose are given in the be- ginning of the program listing. IT k. EXAMPLES OF INPUT DATA The following eight examples will illustrate the way of setting up the input data cards to the program. Problem 1 : A function of three variables: f = (x_x v x x ) x . The problem is to find one network which has the minumum number of gates, assuming that only the uncomplemented variables are available. The < problem-parameter-card > must be: v Col. 1 s, single optimum solution Col. 2 b, uncomplemented variables only available Cols. 3- k 3, value of N Cols. 5- 8 1, value of M Cols. 9-12 1, value of A Cols. 13-16 o, value of B. Therefore the input data deck for this problem is P \ I - » ) C Fig. k.l The input data deck of Problem 1. The first card is the , and the second card is the < problem-parameter- card >. The third card is the < output -function- card > in which the first 8 digits are the binary representation of the given function. (AB + A'B' )C appearing in the remaining columns of the < output-function-card > is used for an identification of the given function. (This comment is applied in the succeeding figures also. ) 18 Problem 2 : A function of four variables: f=x l X 2 (*3 X k v X 3 *lp v X l X 2 X 3 X U v X l X 2 ^ X 3 v *1|)' The problem is to find one network which has the minimum number of inputs to all gates, among those which have the minimum number of gates, assuming that only the uncomplemented variables are available. The < problem- parameter- card > must be: < v Col. 1 s, single optimum solution Col. 2 *>, uncomplemented variables available Col. 3- k fc, value of N Col. 5- 8 1, value of M Col. 9-12 100, value of A Col. 13-16 1, value of B. Therefore the input data deck for this problem is PKUrtLt*! 2. HJkCTIUn A • H ( C ' U+C() • 7+AB- 1 C • l) T + AH TC + r T» KE XAOfcClH AL = 0nK! S 4 1 1 (J 1 0000 11010001011 hEX=06BB Fig. ^. 2 The input data deck of Problem 2. Problem 3: A function of three variables: f =x _x x s/ x_ (x vx ). J. £. j j- d. _5 The problem is to find all networks which have the minimum number of gates, among those which have the minimum number of inputs to all gates, assuming that only the uncomplemented variables are available and that the maximum levels (delay) of the network is four. The < problem-parameter- card > must be: 19 ( Col. 1 b, multiple optimum solutions Col. 2 b, uncomplemented variables are available Cols. 3- k 3, value of N Cols. 5- 8 1, value of M Cols. 9-12 1, value of A Cols. 13-16 100, value of B Cols. 29-32 k, value of IMAX. Therefore the input data deck for this problem is PRfMLtf"" 6. FUivCTI'JN A:-JC+'A ' ( 6 ' +C ' ) 11 I ID!) U 11100001 'ABCT/TiTHT^+C 1 Fig. If. 3 The input data deck of Problem 3. Problem k : A one-bit adder. The output functions are f =x $ x 0- x and f = x..x v x P x o v x q x i * The problem is to find all networks which have the minimum number of inputs to all gates, among those which have the minimum number of gates, assuming that both uncomplemented and complemented variables are available. We want to exclude networks of more than ten gates. The < problem-parameter- card > must be: 20 Col. 1 b, multiple optimum solutions Col. 2 Y, both uncomplemented and complemented variables available Cols. 3- k 3, value of N Cols. 5- 8 2, value of M Cols. 9-12 100, value of A Cols. 13-16 1, value of B ^ Cols. 17-20 1099* value of C~, excluding any networks of more than ten gates. Therefore the input data deck for this problem is PKOHLEtf 4-. Ijivb— B I T AOUfcR Y 3 2 100 1 1 ^ c 110 1001 SUP" 00010111 CAKKY Fig. h.k The input data deck of Problem k- Problem jp : A one-bit adder. The output functions are f. = x $ x $ x and f p = x x v x x v x x . The problem is to find all networks which have the minimum number of inputs to all gates, among those which have the minimum number of gates, assuming that only uncomplemented variables are available, and that both the maximum fan- in and the maximum fan-out are three. We want to exclude any networks of more than eleven gates. The < problem-parameter- card > must be: 21 Col. 1 To, Col. 2 b, Cols. 3- h 3, Cols. 5- 8 2, Cols. 9-12 100, Cols. 13-16 1, Cols.. 17-20 1199, Cols. 21-24 3, Cols. 25-28 3, multiple optimum solutions uncomplemented variables are available value of N value of M value of A value of B value of C Q , excluding any network of more than eleven gates maximum fan- ins maximum fan- outs. Therefore the input data deck for this problem is '3 Z 1CK)_ 1 1 1 q c 11 01 L01001 000101 1 1 SUi* C A K K Y Fig. 1+.5 The input data deck of Problem 5« Problem 6: A function of four variables: f = z ( % 6, 8, 12, Ik, 15 ) + d(3, 9) The problem is to find one network which has the minimum number of gates, assuming that only uncomplemented variables are available. Let us input this problem to the program with an implicit specification of external variables. The < problem-parameter-card > must be: 22 r Col. 1 S, single optimum solution Col. 2 b, uncomplemented variables are available Cols. 3- k k, value of N Cols. 5- 8 1, value of M Cols. 9-12 1, value of A Cols. 13-16 0, value of B. Therefore, the input data deck for this problem is PRUBLEM 6. FUNCTlUN WITH TWO UNSPECIFIED TRE"iS s 4 i i_ o 000-01101-001011 Fig. k.6 The input data deck of Problem 6. Problem 7' The same problem as Problem 6. Let us input this problem to the program in an alternative "way, i.e., with an explicit specification of external variables. The < problem- parameter-card > must be: r Col. 1 S, single optimum solution Col. 2 U, uncomplemented variables are available, the input vectors are defined by < external- input- card > s Cols. 3- k k, value of N Cols. 5- 8 1, value of M Cols. 9-12 1, value of A Cols. 13-16 0, value of B. Therefore, the input data deck for this problem is 23 PKIHLfcM 7. FUNCTION WITH TWl.J INSPbC I b I bO TfcKNS 0(3,9 Si I 4 1 1 0000000 1111111/ V AK I A ri L b # 1 ( : 1 1 1 1 0001111 VAklAH L t a 2 00 100 110 1 1.00 1 1 Oil .'010 100 10 10 1 0000 1 1010010 1 1 V A K I i\ r) L b # 6 V A K 1 A B L b « <* I-UIMC"! Idlvl # 1 Fig. h.7 The input data deck of Problem 7. Problem 8 : A network of five functions of eight variables, The functions are defined by the following truth table. x l 1 X 2 1 X 3 1 x lt 1 X 5 1 x 6 1 X 7 1 x 8 1 f l 1 1 1 1 f 2 1 1 1 f 3 1 1 1 f ^ 1 1 1 1 1 1 1 f 5 1 1 1 1 The problem is to find one network which have the minimum number of inputs to all gates, among those which have the minimum number of gates, assuming that only uncomplemented variables are available, the maximum fan- ins and 2k and fan-outs are three, and the maximum level (delay) of the network is three. For a trial run, let us allot 120 seconds of running time to the problem. The < problem-parameter-card > must be: single optimum solutions uncomplemented variables are available, external input vectors are defined by < external- variable- card > s. value of N value of M value of A value of B value of FANIN value of LMAX value of TMAX. Col. 1 s, Col. 2 u, Cols. 3- h 8, Cols. 5- 8 5, Cols. 9-12 100, Cols. 13-16 1, Cols. 21-24 3, Cols. 29-32 3, Cols. 33-36 120, V. Therefore, the input data deck for this problem is PROBLEM M. FIVE FUNCTIONS OF EIGHT VARIABLES SU 8 5 100 1 3 3 120 1000000 0/ VARIABLE # 1 01000000 VARIABLE #2 00100000 V A K _IA BJL E j¥ 3 00010000 VARIABLE #A 0001000 VARIABLE # b 000100 VARIABLE #6 000000 10 VARIABLE lil 00000001 VARIABLE'^8 1101001 FUNCTION #1 00101100 FUNCTION nz 00010011 FUNCTION #3 11110111 FUNCTION fr4 000011 11 FUNCTION &b Fig. k.Q The input data deck of Problem 8. 25 The computation times and the number of backtracks for the above eight problems are as follows. Problem Computation Time Number of BTKS Problem 1 0.15 Seconds k BTKS Problem 2 ^.57 217 Problem 3 1.08 68 Problem h 37-06 2659 Problem 5 28.^9 2102 Problem 6 1.30 18 Problem 7 1.33 18 Problem 8 ^23.87 21+520 26 5. PROGRAM SIZE AND THE FLOW CHART OF THE PROGRAM The program 'ILLOD-(NOR-B) • is coded for the IBM 360/75 in FORTRAN IV language. The program consists of a main program MAIN, nineteen subroutines ADDO, CHKRDY, CNTCHK, COSTBD, DELO, GITOGJ, GTCHK, LMPLMT, LNITLZ, NFIMP, OUTPUT, PPUNCH, PRECHK, PREDGJ, PROBLM, RESTOR, SELECT, SOLSET, and SUBNET, and two timing subroutines STLMEZ and KTIMEZ which are provided with FORTRAN compiler. The entire program occupies 278K bytes of core storage in which 117K bytes are for the 'program' and l6lK bytes are for the 'data'. Fig. 5.1 shows the organization of the main program MAIN. The major functions of these subroutines are as follows. The Major Functions of Subroutines ADDO: Enters one component of a gate output as an uncovered component into a list that maintains all uncovered components of this gate. This subroutine is called by the main program, and the subroutines IMPLMT, NFIMP, PROBLM, and RESTOR. CHKRDY: Given a feasible network S, checks redundant gates and con- nections, in part by itself and in part by calling two other subroutines, CNTCHK and GTCHX. If the subroutine CHKRDY finds any redundancies, it removes them, generating a better network. Fin ally it lowers the cost ceiling C to the cost of the new network. CNTCHK: Checks a certain type of redundancy in the network. This subroutine is called by the subroutine CHKRDY. 27 C INITLZ 3 PROBLM — .sk. > rerun)- r K = 1 I BTK = C PRECHK (cov) 1 c SELECT C COSTBD 2 > ■ The name ins ide (f 3 1 is a subroutine to be called. ^0 ( t RESTOR ^ ) s * v_M. Fig. 3 The flow chart of the main program ILLOD-(NOR-B). Notes 1 COV means that the current intermediate solution S K is a feasible solution. VTO means that the current intermediate solution S R violates one of the following conditions. (i) The cost of S„ exceeds the cost ceiling K (ii) Gates in S T . violate the fan- in/ fan- out constraint (iii) Gates in S T . violate the level constraint. RERUN means that the problem is a continuation of the previous run. TLIM means that the CPU time exceeds the prespecified time limit TMAX. 28 COSTBD: Accepts possible covers which meet the two conditions: (l) they will not apparently raise the cost of the current intermediate solution over the cost ceiling, and (2) they will not apparently violate the level constraint. Then COSTBD orders the accepted possible covers according to the implementation priority of possible covers (IPPC). In the case where there are no acceptable possible covers, the program control transfers to RETURN 1, which in turn, transfers to the < Backtrack > of the main program. DELO: Deletes one component of a gate output from the list that main- tains uncovered components of this gate. This subroutine is called by the subroutines IMPLMT, NFIMP, and PESTOE. GITOGJ: Classifies gates which are chosen for a gate under consider- ation by the subroutine PREDGJ into two groups, i.e., gates whose j-th components are for all j such that j-th component of gate GJ is 1, and gates whose j-th components is * for at least one j such that j-th com- ponent of gate GJ is 1. GTCHK: Checks a certain type of redundancy in the network. This subroutine is called by the subroutine CHKRDY. IMPLMT: Implements a possible cover, generating the augmented inter- mediate network. Should the resulting network violate the level constraint, then the program control transfers to RETURN 1, which, in turn, transfers to the < Backtrack > point of the main program. INITLZ: Clears all working storages. 29 NFIMP: Implements a type NWG possible cover. This subroutine is called by the subroutine PRECHK. OUTPUT: Has five entry points, PAGE, LINE, TRUTH, CKT, and LEVCAL, besides OUTPUT itself. OUTPUT assigns mnemonic names to external variables and gates for the purpose of print-out. PAGE ejects one page on the printer. LINE skips a specified number of lines on the printer. TRUTH prints the outputs of gates in a form of truth table. CKT prints the network. LEVCAL calculates the levels of gates. PPUNCH: Dumps the current result of the computation onto a card deck through PUNCH statements. PRECHK: Checks whether or not the current intermediate solution is a feasible network. If it is a feasible network, then the program control transfers to RETURN 2, which, in turn, transfers to the < Solution > point of the main program. If it is not yet a feasible network, and if it has type NWG components, then the subroutine PRECHK covers some (or all) of them by introducing new gates so that the resulting intermediate solution has no more type NWG components. The subroutine PRECHK calls the subroutine NFIMP for covering each type NWG- component with one new gate. PREDGJ: Lists all external variables and all gates which could be possible covers of uncovered components of a gate, and which do not violate the fan- in and/or fan-out constraint. PROBLM: Reads input data for one problem, coverting the data into a 30 form of truth table. (In the case when we prepare the input data for a rerun, see the subroutine SOLSET. ) RESTOR: Reconstructs the intermediate solution S_. SELECT: Selects one uncovered component according to the selection criterion of uncovered components (SCUC). Then it lists the possible covers of this uncovered component. SOLSET: Is called by the subroutine PROBLM, in the case when we prepare the input data for a rerun. From the input. data, this subroutine reconstructs the status data of computation at the time when the execution was abandoned because of exceeding the time limit. SUBNET: Has two entry points, PVALUE and RSTRCT, besides SUBNET it- self. This subroutine is called by the subroutine CHKRDY. SUBNET generates a detailed information on the topology of the given network. PVALUE calculates the outputs of gates of the given network. RSTRCT checks whether or not the network derived by the subroutine CHKRDY violates any prespecified constraints, i.e., the fan- in/ fan- out constraint and/or the level constraint. Notes On Possible Modifications Of The Program ILLOD-(NOR-B) The following three modifications of the algorithm could be made in order to improve the program, if users desire so: (i) a modification of the selection criterion of uncovered components (SCUC), (ii) a modification of the implementation priority of possible covers (IPPC), (iii) a modi- fication of the redundancy check of feasible networks. Changing the 31 program to accomodate the modifications (i), (ii), or (iii) can be accom- plished by rewriting only these subroutines as described below. Change Of The SCUC The subroutine PRECHK selects type NWG components and covers them by introducing new gates, when the current intermediate solution contains type NWG components. Therefore, if the user wants to change this part of the SCUC (i.e., the scheme of covering type NWG components), then he must re- write the subroutine PRECHK. The subroutine SELECT selects an uncovered component P, when the current intermediate solution has only the uncovered components of types other than NWG. Therefore, if the user wants to change this part of the SCT5S- (i.e., the scheme of selecting one uncovered component when the given intermediate solution has only the uncovered components of types other than type NWG), then he must rewrite the subroutine SELECT. Change Of The IPPC The subroutine COSTBD gives the ordering of possible covers of the selected uncovered component . Therefore, if the user wants to change the implementation priority of possible covers, he must rewrite the subroutine COSTBD. Change Of The Redundancy Check The subroutine CHKRDY checks redundant gates and redundant connections in the given network, in part by itself and in part by calling two sub- routines CNTCHK and GTCHK; the detailed explanation of this scheme is given 32 in section 3 of 'A b ranch- and-bound algorithm for optimal NOR network' by Nakagawa, T., and Lai, H. C, Report No. ^38, Department of Computer Science, University of Illinois, January 1971- If the user wants to change the scheme of redundancy check, then he must rewrite the subroutine CHKRDY, along with two subroutines CNTCHK and GTCHK. I 33 Program Listing g < X X p- ~ UJ (- £ a. c b a: 3 •a a H. 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