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Address calculation algorithms for ordered sets of combinations
Information Processing North-Holland
ADDRESS
15 December
Letters 17 (1983) 251-253
CALCULATION
Seiichi NISHIHARA
ALGORITHMS
and Katsuo
FOR ORDERED
SETS
1983
OF COMBINATIONS
IKEDA
Instituteof InJormutlon Sciences and Electronics. CJnioerslt.v of Tsukuha, Sakura -mura, Niihari-gun,
Ibarukr 305.
Japun
Communicated by Shuzo Yajima Received 15 June 1983 Rewsed 12 July 1983
Kepvords:
Combination,
address
calculation,
ordered
set
1. Introduction
Our problem is to find algorithms for calculating addresses of given combinations arranged in lexicographic order. Consider the set of all combinations of M integers, { 1,. . . ,M}, taken N at a time. The size of the set is (E). Let (u,,. . . ,uN) and (v),..., vN) be two different N-combinations whose components are arranged in ascending order, i.e., u,< uI+,, v, < vI+, for 1
Firstly we consider the ordered sets of combinations of M integers taken N at a time. Next the algorithm is extended to the case that permits repetitions. 2.1. Addressing
of combinations
without repetitions
Assume an N-combination U = (u,, . . , u N ) is given, where ui < u,,, for 1 < i < N. Let ui) denote the set of all combinations S;(u,,..., V=(v,,.... vN), satisfying (v 13. ..1 v,>+(u,,
. ..1 u,),
where (v,, . . , v,) comprises the first i components of V. In other words, S,(u,, . . ,u,) is given as S,(u ,,‘.”
u,)={v~v+
I,... ,u,_,,u,+l,..., u, + 1 +(N
-i))},
is where (u,,. . .,ui_,, u,+l,...,u,+l+(N-i)) the first combination, i.e., the one having the lowest address, of those combinations whose first i components are equal to (u,, . . . , u,-, , u, + 1). Notice that the size of Si(u,, . . . , u,) is equal to the number of (N - i + l)-combinations taken out of M-u,elements{u,+1,u,+2 ,..., M},i.e.,
2. The algorithms In this section, algorithms for calculating the address of a given combination are presented. 0020-0190/83/$3.00
0 1983, Elsevier Science Publishers
where 1x1 denotes the size of set X. The set of all combinations greater
B.V. (North-Holland)
(i.e.,
+) 251
Volume
17. Number
Ordered
tables
5
INFORMATION
of combinations
without
repetitions.
where
PRCK‘ESSING
M
LEI-ERS
Ordered
table of combinations
=X.N=4
M=6.N=3
addr
add1
with repetitions
-
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Notice a i b.
that
Example
the binomial
1. Let
M = 8.
coefficient
N = 4.
-
(c)
for
6
is zero
which
if
the
ordered table is shown as in Table 1. The address of the given combination U = (1. 3, 6, 7) is calcuthan
U is given
S,(u,)uS,(u,, where
as uz)u
lated
.
u s,(u
I,....
Uh)’
(2)
S, n S, = 0 for i f j.
Thus;
theJaddress (u,, . , u N), denoted the table size minus in set (2); therefore,
of a given combination U = by addr,., (U). is equal to the number of combinations by using
by using
addr,,,((l,3,6,7)) =
(1)
= 23. 252
(3) as follows: =
8-1 4-l+l
x-3 i -,4-2+1 i
Volume
17, Number
2.2. Addressing
5
INFORMATION
of combinations
PROCESSING
with repetitions
Next the algorithm given in Section 2.1 is extended to the case of combinations with repetitions permitted. Table 2 shows an example of an ordered table where M = 6, N = 3. In general, the first combination located at the address 1 is (1,. . . , I), while the last one is (M,. , M). The size of the ordered table is equal to the number of N-combinations of M elements with repetitions permitted, i.e, ( M+K-’ ) [l]. Let U=(u ,,..., uN) be a given combination, where ui < u, + , for 1 < i -C N. Let Si(u,. . . , u,) denote the set of all combinations with repetitions permitted, V = (v,, . . . ,vN), satisfying (v ,“.‘,
v,>> (u ,,...’
LETTERS
15 December
1983
In both (3) and (3’), the computation of the binomial coefficient (t) is efficiently performed by using Algorithm 19 in [2]. However, notice that (i!,) should be zero when a < b. Example
2. Let M = 6 and N = 3, for which the ordered table with repetitions permitted is shown as Table 2. The address of the given combination U = (1,4,6) is calculated by (3’) as follows:
addi& (( 1,4,6)) =
ul>,
where (v,, . . . ,v,) comprises the first i components of V. In other words, S:(u,, . . ,u,) is given as Si(u,,.
..,u,)={V~v~(u,
,...,
u,_,,u,+l,...,
u; + I)}, where (u ,,..., u,_,, u, + l,..., u, + 1) is the first combination, i.e, the one having the lowest address, of those combinations whose first i components are equal to (u,, . . , u i _ ,, u, + 1). Notice that the size of S:(u,, . . . , u,) is equal to the number of (N - i + I)-combinations with repetitions permitted taken out of M - ui elements {uI + 1, u, + 2,. . , M}. i.e.,
The set of all combinations U is given as S;(u,)us;(ul.
uz)u
...
greater (i.e, s) than
uS’,(u,,...,u.),
(2’)
whereS{nSi=Qfori#j. Thus, the address of a given combination with repetitions permitted, U = (u,, . . . , u,), is equal to the table size minus the number of combinations in set (2’); therefore, by using (l’), addr’,~, (U) = [ M +i
-
; I=1
- ’)
[“,“:,“Nl-ij.
(3’)
3. Conclusion Algorithms that calculate the addresses of given N-combinations of M integers arranged in lexicographic order are discussed. Two formulas for combinations with and without repetitions are presented. Our techniques may be applicable for searching data composed of more than one attribute-value pair.
Acknowledgment
The authors would like to thank Professor J.W. Higgins of the University of Tsukuba and the referee for their valuable suggestions.
References
[l] D.E. Knuth, The Art of Fundamental Algorithms 2nd ed., 1973). [2] Algorithm 19 (binomial Algorithms from Comm.
Computer Programming, Vol. 1: (Addison-Wesley, Reading, MA, coefficients), 19-P l-0, ACM (1960).
Collected
253
ADDRESS
15 December
Letters 17 (1983) 251-253
CALCULATION
Seiichi NISHIHARA
ALGORITHMS
and Katsuo
FOR ORDERED
SETS
1983
OF COMBINATIONS
IKEDA
Instituteof InJormutlon Sciences and Electronics. CJnioerslt.v of Tsukuha, Sakura -mura, Niihari-gun,
Ibarukr 305.
Japun
Communicated by Shuzo Yajima Received 15 June 1983 Rewsed 12 July 1983
Kepvords:
Combination,
address
calculation,
ordered
set
1. Introduction
Our problem is to find algorithms for calculating addresses of given combinations arranged in lexicographic order. Consider the set of all combinations of M integers, { 1,. . . ,M}, taken N at a time. The size of the set is (E). Let (u,,. . . ,uN) and (v),..., vN) be two different N-combinations whose components are arranged in ascending order, i.e., u,< uI+,, v, < vI+, for 1
Firstly we consider the ordered sets of combinations of M integers taken N at a time. Next the algorithm is extended to the case that permits repetitions. 2.1. Addressing
of combinations
without repetitions
Assume an N-combination U = (u,, . . , u N ) is given, where ui < u,,, for 1 < i < N. Let ui) denote the set of all combinations S;(u,,..., V=(v,,.... vN), satisfying (v 13. ..1 v,>+(u,,
. ..1 u,),
where (v,, . . , v,) comprises the first i components of V. In other words, S,(u,, . . ,u,) is given as S,(u ,,‘.”
u,)={v~v+
I,... ,u,_,,u,+l,..., u, + 1 +(N
-i))},
is where (u,,. . .,ui_,, u,+l,...,u,+l+(N-i)) the first combination, i.e., the one having the lowest address, of those combinations whose first i components are equal to (u,, . . . , u,-, , u, + 1). Notice that the size of Si(u,, . . . , u,) is equal to the number of (N - i + l)-combinations taken out of M-u,elements{u,+1,u,+2 ,..., M},i.e.,
2. The algorithms In this section, algorithms for calculating the address of a given combination are presented. 0020-0190/83/$3.00
0 1983, Elsevier Science Publishers
where 1x1 denotes the size of set X. The set of all combinations greater
B.V. (North-Holland)
(i.e.,
+) 251
Volume
17. Number
Ordered
tables
5
INFORMATION
of combinations
without
repetitions.
where
PRCK‘ESSING
M
LEI-ERS
Ordered
table of combinations
=X.N=4
M=6.N=3
addr
add1
with repetitions
-
15 Ikcemhrr
19X3
pcrmltted.
uhere
_ addr’
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2
3
4
36
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-
Notice a i b.
that
Example
the binomial
1. Let
M = 8.
coefficient
N = 4.
-
(c)
for
6
is zero
which
if
the
ordered table is shown as in Table 1. The address of the given combination U = (1. 3, 6, 7) is calcuthan
U is given
S,(u,)uS,(u,, where
as uz)u
lated
.
u s,(u
I,....
Uh)’
(2)
S, n S, = 0 for i f j.
Thus;
theJaddress (u,, . , u N), denoted the table size minus in set (2); therefore,
of a given combination U = by addr,., (U). is equal to the number of combinations by using
by using
addr,,,((l,3,6,7)) =
(1)
= 23. 252
(3) as follows: =
8-1 4-l+l
x-3 i -,4-2+1 i
Volume
17, Number
2.2. Addressing
5
INFORMATION
of combinations
PROCESSING
with repetitions
Next the algorithm given in Section 2.1 is extended to the case of combinations with repetitions permitted. Table 2 shows an example of an ordered table where M = 6, N = 3. In general, the first combination located at the address 1 is (1,. . . , I), while the last one is (M,. , M). The size of the ordered table is equal to the number of N-combinations of M elements with repetitions permitted, i.e, ( M+K-’ ) [l]. Let U=(u ,,..., uN) be a given combination, where ui < u, + , for 1 < i -C N. Let Si(u,. . . , u,) denote the set of all combinations with repetitions permitted, V = (v,, . . . ,vN), satisfying (v ,“.‘,
v,>> (u ,,...’
LETTERS
15 December
1983
In both (3) and (3’), the computation of the binomial coefficient (t) is efficiently performed by using Algorithm 19 in [2]. However, notice that (i!,) should be zero when a < b. Example
2. Let M = 6 and N = 3, for which the ordered table with repetitions permitted is shown as Table 2. The address of the given combination U = (1,4,6) is calculated by (3’) as follows:
addi& (( 1,4,6)) =
ul>,
where (v,, . . . ,v,) comprises the first i components of V. In other words, S:(u,, . . ,u,) is given as Si(u,,.
..,u,)={V~v~(u,
,...,
u,_,,u,+l,...,
u; + I)}, where (u ,,..., u,_,, u, + l,..., u, + 1) is the first combination, i.e, the one having the lowest address, of those combinations whose first i components are equal to (u,, . . , u i _ ,, u, + 1). Notice that the size of S:(u,, . . . , u,) is equal to the number of (N - i + I)-combinations with repetitions permitted taken out of M - ui elements {uI + 1, u, + 2,. . , M}. i.e.,
The set of all combinations U is given as S;(u,)us;(ul.
uz)u
...
greater (i.e, s) than
uS’,(u,,...,u.),
(2’)
whereS{nSi=Qfori#j. Thus, the address of a given combination with repetitions permitted, U = (u,, . . . , u,), is equal to the table size minus the number of combinations in set (2’); therefore, by using (l’), addr’,~, (U) = [ M +i
-
; I=1
- ’)
[“,“:,“Nl-ij.
(3’)
3. Conclusion Algorithms that calculate the addresses of given N-combinations of M integers arranged in lexicographic order are discussed. Two formulas for combinations with and without repetitions are presented. Our techniques may be applicable for searching data composed of more than one attribute-value pair.
Acknowledgment
The authors would like to thank Professor J.W. Higgins of the University of Tsukuba and the referee for their valuable suggestions.
References
[l] D.E. Knuth, The Art of Fundamental Algorithms 2nd ed., 1973). [2] Algorithm 19 (binomial Algorithms from Comm.
Computer Programming, Vol. 1: (Addison-Wesley, Reading, MA, coefficients), 19-P l-0, ACM (1960).
Collected
253