ON PARALLEL GENERATION OF COMBINATIONS
IN ASSOCIATIVE PROCESSOR ARCHITECTURES
ZBIGNIEW KOKOSINSKI
*
University of Aizu, Department of Computer Software
Aizu-Wakamatsu, 965-80 Fukushima, Japan
e-mail: kokosin@u-aizu.ac.jp
Abstract
In this paper two new parallel algorithms are presented for generation of (n,k)-combinations. Computations run in associative processor models. Objects are generated in lexicographic order, with
O(1) time per object, in two dierent representations. The rst algorithm uses the conventional
representation of combinations while the second
algorithm generates combinations in the form of
binary vectors and therefore is particularly well
suited for row/column masks generation in associative processors. The algorithms may be also used
for generation of related combinatorial objects like
combinations with repetitions and integer compositions.
Keywords: choice function, combination generation,
associative processing.
1. INTRODUCTION
The rst known algorithm for generating (n,k)combinations published in 1960 is due to Lehmer [27].
In the following years a number of sequential algorithms was developed [7, 9, 26, 33, 38, 40, 41]. Sequential generation methods were reviewed and compared
two times [1, 36]. An increasing interest in parallel
computation systems resulted also in development of
many parallel algorithms for generation of combinatorial objects. The rst parallel algorithm for generating combinations was published in 1984 [44]. In
this adaptive algorithm a new technique for unranking combinations was used, complementary to the rst
such technique given by Lehmer [28]. Then, numerous
parallel solutions to the combination generation problem were published for various models of computations
3
on leave from Politechnika Krakowska,Krak
ow,Poland.
[2, 4, 6, 8, 10, 14, 19, 29, 30, 31, 42, 43]. Simultaneously, the structure of sets of all (n,k)-combinations
and related combinatorial objects like combinations
with repetitions and integer compositions was investigated, and new ranking/unranking techniques were
proposed satisfying various requirements [3, 14, 17, 21,
23, 44]. Most known generation algorithms use the
conventional representation of combinations, which is
not suitable for applications, where fast generation in
the binary representation is required, and therefore objects generated in one representation have to be converted into another. One instance of such application
is massively parallel associative processing [11, 24, 46].
The associative machine model has been shown to
have new applications in many dierent areas of parallel computing including NP-complete problems, processing of data bases, computational geometry, expert
systems, etc. [12, 13, 15]. Many ecient algorithms
developed in these areas explore the power of massive associative processing. However, many associative processors need in their environement two additional hardware components, which are able to perform mask/comparand vector generation eciently: a
fast network generating permutations and a generator of combinations in the binary representation (see
Fig.1). A hardware n-permutation generator, suitable
for pattern vector generation, was presented in [18].
Next, a versatile programmable generator of n-permutations, (n,k)-combinations and (n,m)-partitions
(i.e. partitions of n-element set into at most m nonempty blocks) was proposed [19, 20]. Although that
last generator produces combinations and blocks of
partitions in the binary representation, the time needed
for the generation of a single object becomes too long
for large values of n, because of O(n) circuit (cellular array) propagation time. Therefore, the problem
of nding a better generation technique remains still
open.
In the present paper new combination generators
are proposed. The generation algorithms for both conventional and binary combination representations of
combinations are described which are simultaneously
simple and ecient. Generation of combinations runs
PATTERN INPUT
COLUMN MASK
PERMUTATION
NETWORK
R
O
W
ASSOCIATIVE
M
A
S
PROCESSOR
K
OUTPUT DATA
Figure 1: Applications of masks/pattern generators
in associative processors.
in associative models of computations what coincides with the dedicated processor architecture type.
Consecutive objects are generated in lexicographic order, with constant time per object. Thus, all the
requirements for application in associative processing
are fully satised. The rest of the paper is organized
as follows. The next section introduces combinatorial
objects representations. Section 3 describes models
of computations used throughout this paper. Associative algorithms for generation of combinations in
conventional and binary representation are presented
in section 4. Section 5 contains concluding remarks.
2. REPRESENTATIONS OF
COMBINATORIAL OBJECTS
Let us introduce basic notions used throughout
this paper.
Let < Ai >i2I denote an indexed family of sets
Ai = A, where: A =f0, ... ,ng, I =f1, ... , kg, 1 n,k.
Any mapping f which "chooses" one element from
each set A1 ; :::; Ak is called a choice function of the
family < Ai >i2I [32]. With additional restrictions
we can model by choice functions various classes of
combinatorial objects [14, 16, 18, 19, 20].
If a suplementary conditions: I =f1, ... , ng and
ai 2 f0; 1g are satised then any choice function =
< ai >i2I , that belongs to the indexed family
< Ai >i2I , is called binary choice function of this
family. Sets of all binary choice functions, with the
number of ai = 1 equal to k, are binary representations of the set of all k-subsets (combinations) of the
set A (in these cases we deal in fact with indexed sets
Bi = f0,1g Ai ).
If a suplementary condition: ai < aj , for i < j,
and i, j 2 I , are satised then any choice function
=< ai >i2I , that belongs to the indexed family
< Ai >i2I , is called increasing choice function of this
family. Sets of all increasing choice functions are representations of the set of all k-subsets (combinations)
of the set A. In the conventional representation of
combinations we deal in fact with indexed sets Ci =
fi, ... ,n-k+ig Ai , i 2 I .
If a suplementary condition: ai aj , for i < j,
and i, j 2 I , is satised then any choice function
=< ai >i2I , that belongs to the indexed family
< Ai >i2I , is called nondecreasing choice function of
this family. Sets of all nondecreasing choice functions
are representations of the set of all k-subsets with repetitions (combinations with repetitions) of the set A. In
the conventional representation of combinations with
repetitions we deal in fact with indexed sets Di = f1,
... ,n-k+1g Ai .
P
If a suplementary condition: ki=1 ai = n, is satised then any choice function =< ai >i2I that
belongs to the indexed family < Ai >i2I is called
k-composition of the integer n. Set of all such choice
functions represents the set of all k-compositions of
the integer n. For given 0n and 1k, the number of all
choice functions equals n+k0k01 1 .
Let us introduce now lexicographic order on the
set of all choice functions of the family < Ai >i2I .
For given choice functions =< d1 ; :::; dk > and
=< g1 ; :::; gk >, we say that is less then according to the increasing lexicographic order, if and only
if there exists i 2 f1, ... ,kg, satisfying di < gi , and
dj = gj , for every j < i.
For given choice functions =< d1 ; :::; dk > and
=< g1 ; :::; gk >, we say that is less then according to the decreasing lexicographic order, if and only
if there exists i 2 f1, ... ,kg satisfying di > gi and
dj = gj , for every j < i.
From the above denitions result immediately the
following properties of choice functions and , respectively:
Property 1
0 1
Any given c.f. t ; 1 < t nk , may be obtained
from the c.f. t01 , preceding it in lexicographic order, by incrementing the rightmost element t01 [g ]
< (n 0 k + 1), and seting all (at most k-1) elements
t01 [h], h > g , to the same value t01 [g ] + 1.
A
1
A[1]
...
...
κ
A[k]
k
Algorithms, which implement both transformations,
are presented in section 4. They can be used for parallel generation of , , and sequences in lexicographic order.
3. MODELS OF COMPUTATIONS
Parallel algorithms described later in this paper
run in associative processor models.
The basic model (see Fig.2) consists of single memory cell S and associative memory block A of size k,
with memory cells lineary ordered, and containing respective order numbers as their address keys. Cell S
and cells in block A are considered to be elementary
processors. As most parallel algorithms, the two combination generation algorithms presented in this paper require an interprocessor communication pattern.
In particular, we need a single source processor (cell
S) to sent identical data to a subset of destination
processors (cells) in the block A. This sort of communication operation is called one-to-subset broadcast
(single-node broadcast) and together with single-node
accumulation operation is used in several important
parallel algorithms including matrix-vector multiplication, vector inner product, Gaussian elimination,
and shortest path [25].
All processors of the block A and processor S are
connected to a bus which is used for data transmission.
In order to perform one-to-subset broadcast operation
(i.e. to determine destination processors) the memory
cells have to execute associative range matching in
processor block A. Only one one-to-subset broadcast
is performed at a time.
The basic model is used for the generation of control sequences in both algorithms COMBGEN and
BINCOMBGEN. A slightly simplied version of this
model is used exclusively in the algorithm BINCOMBGEN. This second model consist of one associative
O
U
T
P
U
T
converter
Property 2
Similar transformation of corresponding c.f. t01
into t , may be obtained by two independent substring reversal operations : < t01 [g + t01 [g ] 0 1],
t01 [g + t01 [g ]] >r and < t01 [g + t01 [g ] + 1] ,...,
t01 [n] >r (as a result substrings 01 and 13 03 are received, respectively).
λ
data
bus
address
range A
S
S
Figure 2: Basic model of associative computations.
memory block B of size n, with memory cells lineary
ordered, and containing respective order numbers as
their address keys. The only operation performed in
this model is one-to-subset broadcast but the source
is restricted to values from the set f0,1g. Thus, the
second model can be considered as a degenerate version of the basic model.
4. ALGORITHMS
Construction of both presented algorithms is based
on the observation that despite of applying dierent
approaches to the generation task various generation
algorithms for given class of objects reveal a common
control structure. For instance, the common control
structure of permutation generation algorithms was
discovered by Sedgewick [39] and this structure was
used for the construction of a permutation generator
[18].
In this paper we assume that the common control structure for (n,k)-combinations in both representations, (n-k+1,k)-combinations with repetitions and
(n-k,k+1)-compositions of integers is the structure of
(n-k+1,k)-combinations with repetitions itself. The
properties of the sequence of combinations with repetitions as nondecreasing choice functions are a key
factor of our parallelization method. Therefore the
sequence of choice functions has been chosen as a
basic control sequence for the generation. Actually,
other related objects can be obtained from c.f. by
certain conversion operations.
Both presented algorithms use consequently uniform one-to-subset broadcast operations described in
section 3. In order to produce control sequences the
algorithms operate on the set of associative memory
locations A and single memory cell S. The range of the
subset of destination cells in set A is determined in
parallel by associative compare range operation which
requires O(1) time. In the algorithm COMBGEN the
procedure OUTPUT performs a conversion, which is
required to produce from the control sequence in table
A proper output sequences.
A pseudocode of the parallel algorithm COMBGEN for generating combinations in the ordinary representation is as follows:
Algorithm COMBGEN
Input : n - size of the set, k - size of the subsets.
Output: Table K with the consecutive choice functions .
Method: In table S future values of A subsequences
are computed and stored in advance. Computations
begin with S=1. Then, the rst function in the
table A is obtained (steps 1-2), and next value of S
is determined (step 3). In step 4 the rst output is
produced. Next, consecutive values A and S are produced and output sequences are computed (step 5).
Computations run until the last c.f. is generated,
i.e. IND=0.
/1-3 initialization phase/
1. MAX:=n-k+1; IND:=1; S:=1;
2. ONE2SUBSET(S,A,IND,k);
3. S:= A(IND)+1;
4. do in parallel
4.1. OUTPUT;
4.2. IND:=k;
5. while IND>0 do
5.1. ONE2SUBSET(S,A,IND,k);
5.2. do in parallel
5.2.1. OUTPUT;
5.2.2. if A[IND]<MAX then
5.2.2.1. S:= A(IND)+1;
5.2.2.2. IND:=k;
else
5.2.2.3. IND:=IND-1;
ONE2SUBSET(ONE,SET,LEFT,RIGHT)
/one-to-subset broadcast/
1. for I:=LEFT to RIGHT do in parallel
SET[I]:=ONE;
OUTPUT /conversion and output/
1. for I:=1 to k do in parallel K[I]:=A[I]+I-1;
2. output K;
Exemplary sequences generated by the algorithm,
for n=6, k=3, are depicted in Table I. In columns 3
and 4 (Tables S and A) transformations of the control
sequence are shown. The bold font in these columns
points out the source and the destination memory cells
in all one-to-subset broadcasts between S and A.
TABLE I
Sequences generated by algorithms COMBGEN
and BINCOMBGEN (n=6, k=3)
No.
IND
S
A=L
K
B
1
1
1
111
123
111000
2
3
2
112
124
110100
3
3
3
113
125
110010
4
3
4
114
126
110001
5
2
2
122
134
101100
6
3
3
123
135
101010
7
3
4
124
136
101001
8
2
3
133
145
100110
9
3
4
134
146
100101
10
2
4
144
156
100011
11
1
2
222
234
011100
12
3
3
223
235
011010
13
3
4
224
236
011001
14
2
3
233
245
010110
15
3
4
234
246
010101
16
2
4
244
256
010011
17
1
3
333
345
001110
18
3
4
334
346
001101
19
2
4
344
356
001011
20
1
4
444
456
000111
0
Exemplary output sequences obtained from the
conversion procedure and representing combinations
are shown in column 5 (Table K).
Constant delay between objects is provided by execution of conditional step 5.2.2 in constant time. If
constant delay is not essential further speedup may
be achieved through a hardware implemention if, for
IND=k, all one-to-one broadcasts are replaced by increment operations.
Algorithm BINCOMBGEN
Input : n - size of the set, k - size of the subsets.
Output: Table B with the consecutive choice functions .
Method: Computations in tables S and A run identically as in algorithm COMBGEN. Table B is initialized at the same time as tables A and S (steps 1-3).
In step 4 the rst output is produced. Then, in step
5, the alternate modication of tables A and S take
place, while table B is modied in one or two phases
(phase 1: steps 5.2 and 5.3; phase 2: steps 5.4.2.1 and
5.4.2.2). Computations run until the last -sequence
is generated, i.e. IND=0.
/1-3initialization phase/
1. MAX:=n-k+1; IND:=1; S:=1;
2. do in parallel
2.1. ONE2SUBSET(S,A,IND,k);
2.2. ONE2SUBSET(0,B,1,n);
3. do in parallel
3.1. S:= A(IND)+1;
3.2. ONE2SUBSET(1,B,1,k);
4. do in parallel
4.1. output B;
4.2. IND:=k;
5. while IND>0 do
5.1. do in parallel
5.1.1. ONE2SUBSET(S,A,IND,k);
5.1.2. v:=IND+S;
5.2. ONE2SUBSET(0,B,v-2,v-2);
5.3. ONE2SUBSET(1,B,v-1,v-1);
5.4. if A(IND)<MAX then
5.4.1. S:= A(IND)+1;
5.4.2. if IND<k then
5.4.2.1. do in parallel
5.4.2.1.1. ONE2SUBSET(0,B,v,n);
5.4.2.1.2. IND:=k;
5.4.2.2.ONE2SUBSET(1,B,v,IND+S-2);
else
5.4.3. IND:=IND-1;
5.5. output B;
ONE2SUBSET(ONE,SET,LEFT,RIGHT)
/one-to-subset broadcast/
1. for I:=LEFT to RIGHT do in parallel
SET[I]:=ONE;
Theorem 1
Algorithm COMBGEN generates, in the conventional representation, all (n,k)-combinations in the lexicographic order with constant time per combination
in an associative model with k+1 processors, each of
constant size. Thus, the algorithm COMBGEN is optimal.
The algorithm BINCOMBGEN preserves the basic
control structure of the algorithm COMBGEN. In the
algorithm -sequences corresponding to -sequences
are derived by performing analogous one-to-subset
broadcast operations on additional set of associative
memory locations, i.e. the table B of size n. In the
column 6 of Table I transformations of the output
-sequence (Table B) are depicted; italic and bold
fonts indicate destinations of all one-to-subset broadcasts performed in the table B. It is worth to notice
that lexicographic orders of c.f. and do not match
(B is generated in reverse lexicographic order). In
order to generate -sequences in lexicographic order,
negation of the binary sequence in B is sucient.
Algorithm BINCOMBGEN produces -sequences
in O(1) time per object. However, it turns out that
the most frequent degenerate broadcasts performed
sequentialy in steps 5.2 and 5.3 (shown in Table 1
in italics) can be easily replaced by two simultaneous
logical operations (negations): B[v-2]:= : B[v-2] and
B[v-1]:= : B[v-1], what leads to an improvement of
time parameters in a practical hardware implementation. The details of such implementation are outside
the scope of this paper. Constant delay between objects can be obtain in a similar way as in algorithm
COMBGEN.
Theorem 2
Algorithm BINCOMBGEN generates, in the binary representation, all (n,k)-combinations in the reverse lexicographic order with constant time per combination in an associative model with k+1 processors,
each of constant size. Thus, the algorithm BINCOMBGEN is optimal.
5. CONCLUDING REMARKS
Two algorithms for (n,k)-combination generation
in two dierent representations has been described in
this paper. Although originally designed in order to
support fast associative processing, they may be usefull also in solving classes of combinatorial problems
in other models of computations. In both algorithms
generation time for producing a single object is independent of parameters n and k. Unlike other genera-
tion methods the algorithm BINCOMBGEN produces
combinations in a binary representation and no timeconsuming conversion from the conventional representation is required.
The algorithms provide the parallelization of computations on the level of single combinatorial object,
satisfying all postulates listed in [5]. However, they
can be used in adaptive combination generation too,
enabling further parallelization on the set of objects
level. In this case standard unranking techniques for
combinations may be applied with a little eort for
programming a number of generators working in parallel [21, 23]. Splitting the generation task in adaptive
generation algorithm is much easier to accomplish in
associative model then in linear array model, since in
the latest model it is necessary to know states of six
registers involved in computations and message passing in each of k systolic processors in order to program
one combination generator [4, 5].
Integer compositions can be generated in O(1) time
per object using algorithm COMBGEN providing that
known transformation from (n,k)-combinations [34, 35,
37] is included in OUTPUT procedure. The solution
presented for the combinations in the binary representation can be seen as a special case of parallel generation of n-compositions of the integer k, with a restiction on component size (the component size is less or
equal 1). Solving a more general problem of integer
composition generation with restrictions on component sizes remains an open problem.
Acknowledgement
The author thanks professor Adam Kapralski, who
encouraged him to take up this research. The research
was conducted as a part of \Generation of combinatorial congurations in parallel" project and supported
by University of Aizu grant R-1-3/96.
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