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combinatoric.F90
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combinatoric.F90
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module combinatoric
!Combinatoric Procedures.
!AUTHOR: Dmitry I. Lyakh (Liakh): [email protected]
!Revision: 2017/05/10
!Copyright (C) 2007-2022 Dmitry I. Lyakh (Liakh)
!LICENSE: BSD 3-Clause
!PROCEDURES:
! - TRNG(i:ctrl,i:ni,i[1]:trn,i[1]:ngt): permutation generator which returns each new permutation.
! - TRSIGN(i:n,i[1]:itr): reorders indices in an ascending order and determines the sign of the corresponding permutation (Bubble). Use MERGE_SORT for fast.
! - i8:FACTORIAL(i:n): factorial of a number.
! - i:NOID(i:m,i:n): this function returns a binomial coefficient (integer).
! - i8:NOID8(i8:m,i8:n): this function returns a binomial coefficient (integer*8).
! - DIVIDE_SEGMENT(i{4|8}:seg_range,i{4|8}:subseg_num,i{4|8}[1]:subseg_sizes,i:ierr): Divides a segment into subsegments maximally uniformly.
! - GPGEN(i:ctrl,i:ni,i[1]:vh,i[1]:trn,i[2]:cil): permutation generator with partial-ordering restrictions, returns each new permutation.
! - TR_CYCLE(i:ni,i[1]:trn,i:nc,i[2]:cyc): decomposes a permutation into permutation cycles and determines the sign of the permutation.
! - l:PERM_TRIVIAL(i:ni,i[1]:trn): checks whether the given permutation is trivial or not.
! - l:PERM_TRIVIAL_INT8(i8:ni,i8[1]:trn): checks whether the given permutation is trivial or not.
! - l:PERM_OK(i:ni,i[1]:trn): checks whether the given permutation is legitimate or not.
! - PERM2TRANS(i:ni,i[1]:trn1,i[1]:trn2,i:ntrp,i[2]:trp): creates a list of elementary transpositions relating one permutation to another.
! - PERMUTATION_CONVERTER(l:seq2pos,i:ni,i[1]:n2o,i[1]:o2n): converts between two permutation representations (N2O and O2N).
! - i:HASH_ARR_INT(i:hash_range,i:ni,i[1]:arr): returns a hash-mask for an integer array.
! - i:HASH_ARR_INT8(i:hash_range,i:ni,i8[1]:arr): returns a hash-mask for an integer*8 array.
! - i:CMP_MULTINDS(i:ml1,i[1]:m1,i:ml2,i[1]:m2): compares two integer multiindices.
! - i:CMP_MULTINDS_INT8(i:ml1,i8[1]:m1,i:ml2,i8[1]:m2): compares two integer*8 multiindices.
! - MULTINDX_CMP(i:ml1,i[1]:m1,i:ml2,i[1]:m2,i:ierr,i[1]:prm1,i[2]:prm2): compares two multi-indices: {-1,0,+1}
! - MULTINDX_MERGE(i:ml1,i[1]:m1,i:ml2,i[1]:m2,i:mlr,i[1]:mr,i:sign_corr): merges two multiindices, providing a potential sign correction.
! - i:CMP_ARRAYS_INT(l:preorder,i:ml1,i[1]:m1,i:ml2,i[1]:m2,i[1]:trn): compares two integer arrays with an optional preodering.
! - i:CMP_ARRAYS_INT8(l:preorder,i:ml1,i8[1]:m1,i:ml2,i8[1]:m2,i[1]:trn): compares two integer(8) arrays with an optional preodering.
! - CLANAL(i:nt,r8:ctol,r8[1]:dta,i:ncl,i[1]:cdta,r8[1]:cmv): cluster analysis of a one-dimensional set of points.
! - RANDOM_PERMUTATION(i:ni,i[1]:trn,l:no_trivial): returns a random permutation (default integer).
! - RANDOM_PERMUTATION_INT8(i8:ni,i8[1]:trn,l:no_trivial): returns a random permutation (integer*8 variant).
! - RANDOM_COMPOSITION(l:ordered,i:irange,i:ni,i[1]:trn): returns a random sequence of ni natural numbers from the range [1..irange] without repeats.
! - MERGE_SORT_INT(i:ni,i[1]:trn): fast sorting algorithm for an integer array (default integer).
! - MERGE_SORT_KEY_INT(i:ni,i[1]:key,i[1]:trn): fast sorting algorithm for an integer array, based on integer keys (default integer).
! - MERGE_SORT_INT8_S(i8:ni,i8[1]:trn): fast sorting algorithm for an integer*8 array (returns the permuation sign separately).
! - MERGE_SORT_INT8(i8:ni,i8[1]:trn): fast sorting algorithm for an integer*8 array.
! - MERGE_SORT_KEY_INT8(i8:ni,i8[1]:key,i8[1]:trn): fast sorting algorithm for an integer*8 array, based on integer keys (integer*8).
! - MERGE_SORT_REAL8(i:ni,r8[1]:trn): fast sorting algorithm for a real*8 array.
! - MERGE_SORT_KEY_REAL8(i:ni,r8[1]:key,i[1]:trn): fast sorting algorithm for an integer array, based on real*8 keys.
! - MERGE_SORT_CMPLX8(i:ni,c8[1]:trn): fast sorting algorithm for a complex*8 array.
! - MERGE_SORT_KEY_CMPLX8(i:ni,c8[1]:key,i[1]:trn): fast sorting algorithm for an integer array, based on comlex*8 keys.
! - MERGE_SORT_KEY_GEN(i:ni,*[1]:key,i[1]:trn,cmp_i): fast sorting algorithm for an array of generic keys (requires comparator).
!NOTES:
! - As a rule, a permutation is passed as an integer array (0:n), where the 0th element is the sign of the permutation {1..n}.
! - TRSIGN uses the bubble algorithm => will be very slow for long permutations.
! - For some subroutines, the permutation {1..n} must contain all integers from the segment [1..n].
implicit none
private
!GLOBAL PARAMETERS:
real(8), parameter, public:: DP_ZERO_THRESH=1d-13 !certified guaranteed precision of a double-precision real number (one order lower than the epsilon)
!ABSTRACT INTERFACES:
abstract interface
!Generic comparison:
function cmp_i(obj1,obj2) result(cmp)
integer:: cmp !out: result of comparison: {-1,0,+1}, see dil_basic.F90
class(*), intent(in), target:: obj1 !in: object 1
class(*), intent(in), target:: obj2 !in: object 2
end function cmp_i
end interface
!GENERIC INTERFACES:
interface divide_segment
module procedure divide_segment_i4
module procedure divide_segment_i8
end interface divide_segment
interface merge_sort
module procedure merge_sort_int
module procedure merge_sort_int8_s
module procedure merge_sort_int8
module procedure merge_sort_real8
module procedure merge_sort_cmplx8
end interface merge_sort
interface merge_sort_key
module procedure merge_sort_key_int
module procedure merge_sort_key_int8
module procedure merge_sort_key_real8
module procedure merge_sort_key_cmplx8
end interface merge_sort_key
!PROCEDURE VISIBILITY:
public trng
public trsign
public factorial
public noid
public noid8
public divide_segment
private divide_segment_i4
private divide_segment_i8
public gpgen
public tr_cycle
public perm_trivial
public perm_trivial_int8
public perm_ok
public perm2trans
public permutation_converter
public hash_arr_int
public hash_arr_int8
public cmp_multinds
public cmp_multinds_int8
public multindx_cmp
public multindx_merge
public cmp_arrays_int
public cmp_arrays_int8
public clanal
public random_permutation
public random_permutation_int8
public random_composition
public merge_sort
public merge_sort_key
public merge_sort_int
public merge_sort_key_int
public merge_sort_int8_s
public merge_sort_int8
public merge_sort_key_int8
public merge_sort_real8
public merge_sort_key_real8
public merge_sort_cmplx8
public merge_sort_key_cmplx8
public merge_sort_key_gen
!--------------------------------
contains
!IMPLEMENTATION:
!---------------------------------------
subroutine trng(ctrl,ni,trn,ngt)
!Permutation generator: Returns each subsequent permutation.
! CTRL - control argument. The first call must have CTRL<>0 and initialized TRN!
! Then, on a regular output, CTRL will be 0 unless permutations are over (CTRL=-1).
! NI - number of indices to permute;
! TRN - current permutation (TRN(0) is the sign of the permutation);
! NGT - work structure. Do not change it outside this subroutine!
!NOTES:
! - Permutation index values are not restricted (any array of integer numbers).
implicit none
integer, intent(in):: ni
integer, intent(inout):: ctrl,trn(0:*),ngt(0:*)
integer:: j,k
if(ctrl.ne.0) then !first call: NGT initialization. The original permutation is returned.
ngt(0)=0; do j=1,ni; ngt(j)=j-1; enddo
ctrl=0
else !subsequent call: a new permutation is returned.
k=1 !maybe k=2 will accelerate
do while(k.le.ni)
if(ngt(k).ne.k-1) call transp(k,ngt(k)+1)
if(ngt(k).ne.0) then
call transp(k,ngt(k))
ngt(k)=ngt(k)-1
return
else
ngt(k)=k-1
k=k+1
endif
enddo
ctrl=-1
endif
return
contains
subroutine transp(m,n)
implicit none
integer:: m,n,l
l=trn(m); trn(m)=trn(n); trn(n)=l; trn(0)=-trn(0)
return
end subroutine transp
end subroutine trng
!------------------------------------------------------------------
subroutine trsign(n,itr) !ITR - permutation of the length N
!This subroutine orders integers in ITR in an ascending order.
! ITR - integers. ITR(0) will be the sign of the ordered permutation.
! N - number of integers to be ordered.
!NOTES:
! - Permutation index values are not restricted (any array of integers).
! - Bubble algorithm (only for small arrays).
implicit none
integer, intent(in):: n
integer, intent(inout):: itr(0:*)
integer:: k,l,isgn
k=1
isgn=+1
do while(k.lt.n)
if(itr(k).gt.itr(k+1)) then
l=itr(k); itr(k)=itr(k+1); itr(k+1)=l; isgn=-isgn
if(k.gt.1) then; k=k-1; else; k=2; endif
else
k=k+1
endif
enddo
itr(0)=isgn
return
end subroutine trsign
!----------------------------------------------------
recursive function factorial(n) result(fctrl) !returns N! for N>=0, and -1 otherwise
implicit none
integer(8):: fctrl
integer, intent(in):: n
integer(8):: k
if(n.ge.0) then
fctrl=1_8
do k=2_8,int(n,8)
fctrl=fctrl*k
if(fctrl.lt.0_8) then; write(*,*)'ERROR(combinatoric:factorial): integer(8) overflow!'; stop; endif !trap
enddo
else
fctrl=-1_8
endif
return
end function factorial
!------------------------------------------------------------------------------------------------------
integer function noid(m,n) !returns the number of unique distributions of N objects on M places
implicit none
integer, intent(in):: m,n
integer:: k,l
if(n.gt.m.or.n.lt.0.or.m.lt.0) then
noid=0
return
elseif(n.eq.m.or.n.eq.0) then
noid=1
return
endif
noid=1; l=m
do k=1,n; noid=(noid*l)/k; l=l-1; enddo
if(noid.le.0) then; write(*,*)'ERROR(combinatoric:noid): integer overflow:',m,n,noid; stop; endif !trap
return
end function noid
!----------------------------------------------------------------------------------------------------------
integer(8) function noid8(m,n) !returns the number of unique distributions of N objects on M places
implicit none
integer(8), intent(in):: m,n
integer(8):: k,l
if(n.gt.m.or.n.lt.0.or.m.lt.0) then
noid8=0_8
return
elseif(n.eq.m.or.n.eq.0) then
noid8=1_8
return
endif
noid8=1_8; l=m
do k=1_8,n; noid8=(noid8*l)/k; l=l-1_8; enddo
if(noid8.le.0_8) then; write(*,*)'ERROR(combinatoric:noid8): integer*8 overflow: ',m,n,noid8; stop; endif !trap
return
end function noid8
!---------------------------------------------------------------------------
#ifndef NO_PHI
!DIR$ ATTRIBUTES OFFLOAD:mic:: divide_segment_i4
#endif
subroutine divide_segment_i4(seg_range,subseg_num,subseg_sizes,ierr) !SERIAL
!A segment of range <seg_range> will be divided into <subseg_num> subsegments maximally uniformly.
!The length of each subsegment will be returned in the array <subseg_sizes(1:subseg_num)>.
!Any two subsegments will not differ in length by more than 1, longer subsegments preceding the shorter ones.
implicit none
integer, intent(in):: seg_range,subseg_num
integer, intent(out):: subseg_sizes(1:subseg_num)
integer, intent(inout):: ierr
integer:: i,j,k,l,m,n
ierr=0
if(seg_range.gt.0.and.subseg_num.gt.0) then
n=seg_range/subseg_num; m=mod(seg_range,subseg_num)
do i=1,m; subseg_sizes(i)=n+1; enddo
do i=m+1,subseg_num; subseg_sizes(i)=n; enddo
else
ierr=-1
endif
return
end subroutine divide_segment_i4
!---------------------------------------------------------------------------
#ifndef NO_PHI
!DIR$ ATTRIBUTES OFFLOAD:mic:: divide_segment_i8
#endif
subroutine divide_segment_i8(seg_range,subseg_num,subseg_sizes,ierr) !SERIAL
!A segment of range <seg_range> will be divided into <subseg_num> subsegments maximally uniformly.
!The length of each subsegment will be returned in the array <subseg_sizes(1:subseg_num)>.
!Any two subsegments will not differ in length by more than 1, longer subsegments preceding the shorter ones.
implicit none
integer(8), intent(in):: seg_range,subseg_num
integer(8), intent(out):: subseg_sizes(1:subseg_num)
integer, intent(inout):: ierr
integer(8):: i,j,k,l,m,n
ierr=0
if(seg_range.gt.0_8.and.subseg_num.gt.0_8) then
n=seg_range/subseg_num; m=mod(seg_range,subseg_num)
do i=1_8,m; subseg_sizes(i)=n+1_8; enddo
do i=m+1_8,subseg_num; subseg_sizes(i)=n; enddo
else
ierr=-1
endif
return
end subroutine divide_segment_i8
!-------------------------------------------
subroutine gpgen(ctrl,ni,vh,trn,cil)
!This subroutine generates all unique permutations of NI items,
!in which the items belonging to the same host are always ordered.
!INPUT:
! - ctrl - control argument: at the begining must be <> 0; 0 - next permutation; -1 - permutations are over.
! - ni - number of items;
! - vh(1:ni) - index hosts;
!INPUT(dummy)/OUTPUT:
! - trn(0:ni) - current permutation (trn(0) is the sign);
! - cil(0:1,0:ni) - connected list (for internal use only, do not set or change it outside!).
!OUTPUT:
! - trn(0:ni) - current permutation (trn(0) is the sign).
!NOTES:
! - The first permutation is created here (fully ordered one).
! Its sign is +1. Each next permutation is generated from the previous one.
implicit none
!------------------------------------------
integer, parameter:: MAX_ITEM=16384 !max allowed number of items (because of the permutation sign determination, see below)
!------------------------------------------
integer, intent(in):: ni,vh(1:ni)
integer, intent(inout):: ctrl,trn(0:ni),cil(0:1,0:ni)
integer:: i,j,k,l,m,n,k1,k2,k3,k4,k5,k6,ks,kf,ierr
if(ni.gt.0) then
if(ni.gt.MAX_ITEM) then
write(*,*)'ERROR(combinatoric:gpgen): legnth of the permutation exceeds the maximal value: ',MAX_ITEM,ni
stop
endif
if(ctrl.ne.0) then
call first_call()
else
!free the last box:
m=vh(ni); n=ni
do while(vh(n).eq.m)
call free_item(trn(n)); n=n-1
if(n.eq.0) exit
enddo
!get the next composition:
if(n.gt.0) then
ks=-1
do while(n.gt.0)
! write(*,'(''DEBUG1: '',128(i1,1x))') trn(1:n) !debug
! write(*,'(4x,128(i2,1x))') (k6,k6=0,ni); write(*,'(4x,128(i2,1x))') cil(0,0:ni) !debug
! write(*,'(4x,128(i2,1x))') cil(1,0:ni) !debug
if(ks.lt.0) then
i=trn(n); call free_item(i); j=cil(1,i)
if(j.gt.0) then
m=cil(1,j); call engage_item(j); trn(n)=j; ks=+1
endif
else
if(vh(n).eq.vh(n-1)) then
j=m
if(j.gt.0) then
m=cil(1,j); call engage_item(j); trn(n)=j
else
ks=-1
endif
else
j=cil(0,0); m=cil(1,j); call engage_item(j); trn(n)=j
endif
endif
! write(*,'(''DEBUG2: '',128(i1,1x))') trn(1:n) !debug
! write(*,'(4x,128(i2,1x))') (k6,k6=0,ni); write(*,'(4x,128(i2,1x))') cil(0,0:ni) !debug
! write(*,'(4x,128(i2,1x))') cil(1,0:ni) !debug
n=n+ks
if(n.gt.ni) then !success
call determine_trn_sign
return
endif
enddo !n
ctrl=-1 !end
else
ctrl=-1
endif
endif
else
ctrl=-1
endif
return
contains
subroutine first_call()
implicit none
integer:: j1
trn(0)=+1; do j1=1,ni; trn(j1)=j1; enddo
cil(0:1,1:ni)=-1; cil(0:1,0)=(/ni+1,0/)
ctrl=0
return
end subroutine first_call
subroutine free_item(it)
implicit none
integer, intent(in):: it
integer:: j1,j2
if(it.lt.cil(0,0)) then
if(cil(0,0).le.ni) then
cil(0:1,it)=(/0,cil(0,0)/); cil(0,cil(0,0))=it; cil(0,0)=it
else
cil(0:1,it)=(/0,0/); cil(0:1,0)=it
endif
elseif(it.gt.cil(1,0)) then
if(cil(1,0).ge.1) then
cil(0:1,it)=(/cil(1,0),0/); cil(1,cil(1,0))=it; cil(1,0)=it
else
cil(0:1,it)=(/0,0/); cil(0:1,0)=it
endif
else
if(it-cil(0,0).le.cil(1,0)-it) then !moving up: insert the new item between the minimal and maximal elements
j1=cil(0,0); do while(cil(1,j1).lt.it); j1=cil(1,j1); enddo
j2=cil(1,j1); cil(0:1,it)=(/j1,j2/); cil(1,j1)=it; cil(0,j2)=it
else !moving down: insert the new item between the minimal and maximal elements
j1=cil(1,0); do while(cil(0,j1).gt.it); j1=cil(0,j1); enddo
j2=cil(0,j1); cil(0:1,it)=(/j2,j1/); cil(1,j2)=it; cil(0,j1)=it
endif
endif
return
end subroutine free_item
subroutine engage_item(it)
implicit none
integer, intent(in):: it
integer:: j1,jd,ju
jd=1; ju=1
if(it.eq.cil(0,0)) then
j1=cil(1,it)
if(j1.gt.0) then; cil(0,j1)=0; cil(0,0)=j1; else; cil(0,0)=ni+1; endif
jd=0
endif
if(it.eq.cil(1,0)) then
j1=cil(0,it)
if(j1.gt.0) then; cil(1,j1)=0; cil(1,0)=j1; else; cil(1,0)=0; endif
ju=0
endif
if(jd*ju.eq.1) then; cil(1,cil(0,it))=cil(1,it); cil(0,cil(1,it))=cil(0,it); endif
cil(0:1,it)=-1
return
end subroutine engage_item
subroutine determine_trn_sign
implicit none
integer:: occ(MAX_ITEM),j1,j2,j3,js
js=+1; occ(1:ni)=0; j1=1; j2=0; j3=0
do
if(occ(j1).eq.0) then
j3=j3+1; j2=j2+1; occ(j1)=1; j1=trn(j1)
else
if(mod(j2,2).eq.0) js=-js
if(j3.eq.ni) exit
j2=0; j1=1; do while(occ(j1).ne.0); j1=j1+1; enddo
endif
enddo
trn(0)=js
return
end subroutine determine_trn_sign
end subroutine gpgen
!-----------------------------------------
subroutine tr_cycle(ni,trn,nc,cyc)
!This subroutine extracts permutation cycles and the sign from a given permutation.
!INPUT:
! - ni - number of indices;
! - trn(0:ni) - permutation, in which trn(0) is the sign returned;
!OUTPUT:
! - trn(0) - sign of the permutation;
! - nc - number of permutation cycles;
! - cyc(0:1,1:ni) - permutation cycles: cyc(1,:) is an index of a cycle; cyc(0,:) is the number of the cycle the index belongs to.
!NOTE:
! - nc=-666 - means ERROR.
! - permutation index values must lie within the range [1..ni].
implicit none
integer, intent(in):: ni
integer, intent(inout):: trn(0:*)
integer, intent(out):: nc,cyc(0:1,*)
integer, parameter:: MAX_IN_MEM=1024
integer i,j,k,l,m,n,k1,k2,k3,k4,k5,k6,ks,kf,ierr
integer, target:: ibuss(MAX_IN_MEM)
integer, allocatable, target:: ibusa(:)
integer, pointer, contiguous:: ibus(:)
nc=0
if(ni.gt.0) then
if(ni.gt.MAX_IN_MEM) then; allocate(ibusa(1:ni)); ibus=>ibusa; else; ibus=>ibuss; endif
if(trn_ok()) then
! ibus(1:ni)=0 !busy flags
trn(0)=+1; n=0; m=0
do while(n.lt.ni)
i=m+1; do while(ibus(i).ne.0); i=i+1; enddo; m=i
nc=nc+1; l=i; j=0
do
n=n+1; j=j+1; cyc(0:1,n)=(/nc,trn(i)/); ibus(i)=nc
if(trn(i).eq.l) then; exit; else; i=trn(i); endif
enddo
if(mod(j,2).eq.0) trn(0)=-trn(0)
enddo
else
trn(0)=-667; nc=-666
endif
nullify(ibus); if(ni.gt.MAX_IN_MEM) deallocate(ibusa)
else
trn(0)=-666; nc=-666
endif
return
contains
logical function trn_ok()
integer:: j1
ibus(1:ni)=0
do j1=1,ni
if(trn(j1).le.0.or.trn(j1).gt.ni) then; trn_ok=.false.; return; endif
if(ibus(trn(j1)).ne.0) then; trn_ok=.false.; return; endif
ibus(trn(j1))=j1
enddo
ibus(1:ni)=0
trn_ok=.true.
return
end function trn_ok
end subroutine tr_cycle
!--------------------------------------------
logical function perm_trivial(ni,trn)
!Checks whether the given permutation trn(0:ni) is trivial or not.
implicit none
integer, intent(in):: ni,trn(0:*)
integer:: i
perm_trivial=.true.
do i=1,ni; if(trn(i).ne.i) then; perm_trivial=.false.; exit; endif; enddo
return
end function perm_trivial
!-------------------------------------------------
logical function perm_trivial_int8(ni,trn)
!Checks whether the given permutation trn(0:ni) is trivial or not.
implicit none
integer(8), intent(in):: ni,trn(0:*)
integer(8):: i
perm_trivial_int8=.true.
do i=1_8,ni; if(trn(i).ne.i) then; perm_trivial_int8=.false.; exit; endif; enddo
return
end function perm_trivial_int8
!---------------------------------------
logical function perm_ok(ni,trn)
!Checks whether the given permutation trn(0:ni) is legitimate or not.
!NOTE: keep the permutation fit into the stack!
implicit none
integer, intent(in):: ni,trn(0:*)
integer:: i,j,ibus(1:ni)
ibus(1:ni)=0
do i=1,ni
j=trn(i)
if(j.le.0.or.j.gt.ni) then; perm_ok=.false.; return; endif
if(ibus(j).ne.0) then; perm_ok=.false.; return; else; ibus(j)=i; endif
enddo
perm_ok=.true.
return
end function perm_ok
!---------------------------------------------------
subroutine perm2trans(ni,trn1,trn2,ntrp,trp)
!This subroutine creates a sequence of elementary pair transpositions
!which connect one permutation (old) with another one (new).
!The sign of the second (new) permutation is changed accordingly.
!Schematically: TRN1 ---list_of_transpositions---> TRN2 with a modified sign.
!INPUT:
! - ni - number of indices;
! - trn1(0:ni) - old permutation (&0 - sign);
! - trn2(0:ni) - new permutation (&0 - sign);
!OUTPUT:
! - ntrp - number of elementary transpositions;
! - trp(1:2,1:ntrp) - elementary transpositions in terms of old index numbers.
! Each transposition interchanges two old indices according to their original numbers;
! - trn2(0) - sign of the permutation TRN2 with respect to the permutation TRN1.
!NOTES:
! - Permutation index values must span the range [1..ni].
! - Because AUTOMATIC arrays are employed, the number of indices cannot be too large.
! Switch to pointers if you want to process larger permutations.
implicit none
integer, intent(in):: ni,trn1(0:*),trn2(0:*)
integer, intent(out):: ntrp,trp(2,*)
integer:: i,j,k,l,m,n,k1,k2,k3,k4,k5,k6,ks,kf,ierr
integer:: trn(1:ni),ipos(1:ni),isgn
ntrp=0
if(ni.gt.1) then
if(trn_ok()) then
isgn=+1; do j=1,ni; trn(j)=trn1(j); enddo; do j=1,ni; ipos(trn1(j))=j; enddo
do i=1,ni
if(trn(i).ne.trn2(i)) then
ntrp=ntrp+1; trp(1:2,ntrp)=(/trn(i),trn2(i)/)
j=trn(i); trn(i)=trn2(i); trn(ipos(trn2(i)))=j
ipos(j)=ipos(trn2(i)); ipos(trn2(i))=i
isgn=-isgn
endif
enddo
else
write(*,*)'ERROR(combinatoric:perm2trans): invalid input permutation: ',ni,trn1(1:ni),trn2(1:ni)
stop
endif
endif
return
contains
logical function trn_ok()
integer:: j1
trn_ok=.true.
do j1=1,ni
if(trn1(j1).lt.1.or.trn1(j1).gt.ni.or.trn2(j1).lt.1.or.trn2(j1).gt.ni) then; trn_ok=.false.; exit; endif
enddo
return
end function trn_ok
end subroutine perm2trans
!-----------------------------------------------------------
subroutine permutation_converter(seq2pos,ni,n2o,o2n)
!This subroutine converts between two permutation representations:
!(n2o) new_to_old: a sequence of original item numbers (position --> ID);
!(o2n) old_to_new: new positions of items (ID --> new position).
!Briefly: n2o(POSITION)=ID; o2n(ID)=POSITION. They are inverse permutations.
!INPUT:
! - seq2pos - .TRUE. means a conversion from n2o to o2n, .FALSE. o2n to n2o;
! - ni - number of items;
! - n2o(0:ni)/o2n(0:ni);
!OUTPUT:
! - n2o(0:ni)/o2n(0:ni).
!NOTES:
! - The elements of n2o(1:ni)/o2n(1:ni) must span the range [1..ni] (no argument-validity check is done here!).
implicit none
logical, intent(in):: seq2pos
integer, intent(in):: ni
integer, intent(inout):: n2o(0:ni),o2n(0:ni)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
if(seq2pos) then
o2n(0)=n2o(0); do i=1,ni; o2n(n2o(i))=i; enddo !loop over positions
else
n2o(0)=o2n(0); do i=1,ni; n2o(o2n(i))=i; enddo !loop over item IDs
endif
return
end subroutine permutation_converter
!-------------------------------------------------------
integer function hash_arr_int(hash_range,ni,arr)
!This function returns a hash-mask for a given integer array.
!INPUT:
! - hash_range - defines the range of this function [0..hash_range-1];
! - ni - number of items in the array;
! - arr(0:ni-1) - items;
!OUTPUT:
! - hash_arr_int - hash-mask.
implicit none
integer, intent(in):: hash_range,ni
integer, intent(in):: arr(0:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
hash_arr_int=0
if(hash_range.gt.0.and.ni.ge.0) then
do i=0,ni-1
hash_arr_int=hash_arr_int+mod(arr(i),hash_range)
hash_arr_int=mod(hash_arr_int,hash_range)
enddo
else
write(*,*)'ERROR(combinatoric:hash_arr_int): invalid arguments: ',hash_range,ni
stop
endif
return
end function hash_arr_int
!--------------------------------------------------------
integer function hash_arr_int8(hash_range,ni,arr)
!This function returns a hash-mask for a given integer*8 array.
!INPUT:
! - hash_range - defines the range of this function [0..hash_range-1];
! - ni - number of items in the array;
! - arr(0:ni-1) - items (integer*8);
!OUTPUT:
! - hash_arr_int - hash-mask.
implicit none
integer, intent(in):: hash_range,ni
integer(8), intent(in):: arr(0:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
integer(8):: hr
hash_arr_int8=0
if(hash_range.gt.0.and.ni.ge.0) then
hr=int(hash_range,8)
do i=0,ni-1
hash_arr_int8=hash_arr_int8+int(mod(arr(i),hr),4)
hash_arr_int8=mod(hash_arr_int8,hash_range)
enddo
else
write(*,*)'ERROR(combinatoric:hash_arr_int8): invalid arguments: ',hash_range,ni
stop
endif
return
end function hash_arr_int8
!---------------------------------------------------
integer function cmp_multinds(ml1,m1,ml2,m2)
!This function compares two integer multiindices.
!INPUT:
! - m1(1:ml1) - 1st multiindex;
! - m2(1:ml2) - 2nd multiindex.
!OUTPUT:
! - cmp_multinds: -X (1<=X<=ml1,ml1=ml2): first difference in two multiindices occurs at position X (left-to-right) and m1(X)<m2(X);
! +X (1<=X<=ml1,ml1=ml2): first difference in two multiindices occurs at position X (left-to-right) and m1(X)>m2(X);
! -X (X>ml1,X>ml2): first multiindex is shorter than the second one (ml1<ml2);
! +X (X>ml1,X>ml2): first multiindex is longer than the second one (ml1>ml2);
! 0 (ml1=ml2): the two multiindices are equal.
implicit none
integer, intent(in):: ml1,ml2,m1(1:*),m2(1:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
if(ml1.ge.0.and.ml2.ge.0) then
if(ml1.eq.ml2) then
cmp_multinds=0
do i=1,ml1
if(m1(i).ne.m2(i)) then; cmp_multinds=sign(i,m1(i)-m2(i)); exit; endif
enddo
elseif(ml1.lt.ml2) then
cmp_multinds=-(max(ml1,ml2)+1)
else
cmp_multinds=+(max(ml1,ml2)+1)
endif
else
write(*,'("ERROR(combinatoric:cmp_multinds): negative multiindex length passed: ",i10,1x,i10)') ml1,ml2
stop
endif
return
end function cmp_multinds
!--------------------------------------------------------
integer function cmp_multinds_int8(ml1,m1,ml2,m2)
!This function compares two integer*8 multiindices.
!INPUT:
! - m1(1:ml1) - 1st multiindex (integer*8);
! - m2(1:ml2) - 2nd multiindex (integer*8).
!OUTPUT:
! - cmp_multinds: -X (1<=X<=ml1,ml1=ml2): first difference in two multiindices occurs at position X (left-to-right) and m1(X)<m2(X);
! +X (1<=X<=ml1,ml1=ml2): first difference in two multiindices occurs at position X (left-to-right) and m1(X)>m2(X);
! -X (X>ml1,X>ml2): first multiindex is shorter than the second one (ml1<ml2);
! +X (X>ml1,X>ml2): first multiindex is longer than the second one (ml1>ml2);
! 0 (ml1=ml2): the two multiindices are equal.
implicit none
integer, intent(in):: ml1,ml2
integer(8), intent(in):: m1(1:*),m2(1:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
if(ml1.ge.0.and.ml2.ge.0) then
if(ml1.eq.ml2) then
cmp_multinds_int8=0
do i=1,ml1
if(m1(i).ne.m2(i)) then; cmp_multinds_int8=int(sign(int(i,8),m1(i)-m2(i)),4); exit; endif
enddo
elseif(ml1.lt.ml2) then
cmp_multinds_int8=-(max(ml1,ml2)+1)
else
cmp_multinds_int8=+(max(ml1,ml2)+1)
endif
else
write(*,'("ERROR(combinatoric:cmp_multinds_int8): negative multiindex length passed: ",i10,1x,i10)') ml1,ml2
stop
endif
return
end function cmp_multinds_int8
!-----------------------------------------------------------------
function multindx_cmp(ml1,m1,ml2,m2,prm1,prm2) result(cmp)
!Compares two multi-indices. Optional permutations specify index priorities for both multi-indices:
!prmX(1) is the first index to compare, prmX(2) is the second, and so on.
implicit none
integer:: cmp !out: comparison result: -1:m1<m2, 0:m1=m2, +1:m1>m2
integer(4), intent(in):: ml1 !in: length of multi-index 1
integer(8), intent(in):: m1(1:*) !in: multi-index 1
integer(4), intent(in):: ml2 !in: length of multi-index 2
integer(8), intent(in):: m2(1:*) !in: multi-index 2
integer(4), intent(in), optional:: prm1(1:ml1) !in: permutation for multi-index 1
integer(4), intent(in), optional:: prm2(1:ml2) !in: permutation for multi-index 2
integer(4):: i
cmp=0
if(present(prm1)) then
if(present(prm2)) then
if(ml1.lt.ml2) then
cmp=-1
elseif(ml1.gt.ml2) then
cmp=+1
else
do i=1,ml1
if(m1(prm1(i)).ne.m2(prm2(i))) then; cmp=int(sign(1_8,m1(prm1(i))-m2(prm2(i))),4); exit; endif
enddo
endif
else
if(ml1.lt.ml2) then
cmp=-1
elseif(ml1.gt.ml2) then
cmp=+1
else
do i=1,ml1
if(m1(prm1(i)).ne.m2(i)) then; cmp=int(sign(1_8,m1(prm1(i))-m2(i)),4); exit; endif
enddo
endif
endif
elseif(present(prm2)) then
if(ml1.lt.ml2) then
cmp=-1
elseif(ml1.gt.ml2) then
cmp=+1
else
do i=1,ml1
if(m1(i).ne.m2(prm2(i))) then; cmp=int(sign(1_8,m1(i)-m2(prm2(i))),4); exit; endif
enddo
endif
else
if(ml1.lt.ml2) then
cmp=-1
elseif(ml1.gt.ml2) then
cmp=+1
else
do i=1,ml1
if(m1(i).ne.m2(i)) then; cmp=int(sign(1_8,m1(i)-m2(i)),4); exit; endif
enddo
endif
endif
return
end function multindx_cmp
!----------------------------------------------------------------
subroutine multindx_merge(ml1,m1,ml2,m2,mlr,mr,sign_corr)
!This subroutine merges two multiindices (with index reodering).
!INPUT:
! - ml1 - length of the 1st multiindex;
! - m1(1:ml1) - 1st multiindex (left);
! - ml2 - length of the 2nd multiindex;
! - m2(1:ml2) - 2nd multiindex (right);
!OUTPUT:
! - mlr - length of the multiindex-result (ml1+ml2);
! - mr(1:mlr) - multiindex-result;
! - sign_corr - sign correction (+1/-1/0): 0 when a repeated index is present.
!NOTES:
! - No error checks.
implicit none
integer, intent(in):: ml1,m1(1:*),ml2,m2(1:*)
integer, intent(out):: mlr,mr(1:*),sign_corr
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
mlr=0; sign_corr=+1
!merge:
if(ml1.gt.0.and.ml2.gt.0) then
k1=1; k2=1
mloop: do
mlr=mlr+1
if(m1(k1).le.m2(k2)) then
mr(mlr)=m1(k1)
if(k1.lt.ml1) then
k1=k1+1
else
l=ml2-k2+1; mr(mlr+1:mlr+l)=m2(k2:ml2); mlr=mlr+l
exit mloop
endif
else
mr(mlr)=m2(k2); sign_corr=sign_corr*(-1+2*mod(ml1-k1,2))
if(k2.lt.ml2) then
k2=k2+1
else
l=ml1-k1+1; mr(mlr+1:mlr+l)=m1(k1:ml1); mlr=mlr+l
exit mloop
endif
endif
enddo mloop
elseif(ml1.gt.0.and.ml2.le.0) then
mlr=ml1; mr(1:ml1)=m1(1:ml1)
elseif(ml1.le.0.and.ml2.gt.0) then
mlr=ml2; mr(1:ml2)=m2(1:ml2)
endif
!check index repeats:
do i=1,mlr-1; if(mr(i).eq.mr(i+1)) then; sign_corr=0; exit; endif; enddo
return
end subroutine multindx_merge
!------------------------------------------------------------------
integer function cmp_arrays_int(preorder,ml1,m1,ml2,m2,trn)
!This function compares two integer arrays with an optional preodering.
!INPUT:
! - preorder - if .true., both arrays will be ordered before the comparison;
! - ml1 - length of the 1st array;
! - m1(1:ml1) - 1st array;
! - ml2 - length of the 2nd array;
! - m2(1:ml2) - 2nd array;
!OUTPUT:
! - cmp_arrays_int: -X (1<=X<=ml1,ml1=ml2): first difference in two arrays occurs at position X (left-to-right) and m1(X)<m2(X);
! +X (1<=X<=ml1,ml1=ml2): first difference in two arrays occurs at position X (left-to-right) and m1(X)>m2(X);
! -X (X>ml1,X>ml2): first array is shorter than the second one (ml1<ml2);
! +X (X>ml1,X>ml2): first array is longer than the second one (ml1>ml2);
! 0 (ml1=ml2): the two arrays are equal.
! - trn(0:) - if preorder=.true. and the arrays are equal (cmp_arrays_int=0),
! this (optional) output array will contain the permutation matching the two arrays:
! A new order of old elements of the 2nd array (N2O) that matches the 1st array (both with the original ordering).
implicit none
logical, intent(in):: preorder
integer, intent(in):: ml1,ml2
integer, intent(in):: m1(1:ml1),m2(1:ml2)
integer, intent(out), optional:: trn(0:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
integer, allocatable:: prm1(:),prm2(:)
if(ml1.ge.0.and.ml2.ge.0) then
if(ml1.lt.ml2) then
cmp_arrays_int=-(max(ml1,ml2)+1)
elseif(ml1.gt.ml2) then
cmp_arrays_int=(max(ml1,ml2)+1)
else !ml1=ml2
cmp_arrays_int=0
if(preorder) then
allocate(prm1(0:ml1),prm2(0:ml2),STAT=ierr)
if(ierr.ne.0) then; write(*,*)'ERROR(combinatoric:cmp_arrays_int): memory allocation failed!'; stop; endif
prm1(0)=+1; do i=1,ml1; prm1(i)=i; enddo
prm2(0)=+1; do i=1,ml2; prm2(i)=i; enddo
call merge_sort_key_int(ml1,m1,prm1) !prm1 is N2O
call merge_sort_key_int(ml2,m2,prm2) !prm2 is N2O
do i=1,ml1
if(m1(prm1(i)).ne.m2(prm2(i))) then; cmp_arrays_int=sign(i,m1(prm1(i))-m2(prm2(i))); exit; endif
enddo
if(cmp_arrays_int.eq.0.and.present(trn)) then
trn(0)=prm1(0)*prm2(0); do i=1,ml1; trn(prm1(i))=prm2(i); enddo
endif
deallocate(prm1,prm2)
else
do i=1,ml1
if(m1(i).ne.m2(i)) then; cmp_arrays_int=sign(i,m1(i)-m2(i)); exit; endif
enddo
endif
endif
else !invalid arguments
write(*,*)'ERROR(combinatoric:cmp_arrays_int): invalid arguments: ',ml1,ml2
stop
endif
return
end function cmp_arrays_int
!-------------------------------------------------------------------
integer function cmp_arrays_int8(preorder,ml1,m1,ml2,m2,trn)
!This function compares two integer(8) arrays with an optional preodering.
!INPUT:
! - preorder - if .true., both arrays will be ordered before the comparison;
! - ml1 - length of the 1st array;
! - m1(1:ml1) - 1st array;
! - ml2 - length of the 2nd array;
! - m2(1:ml2) - 2nd array;
!OUTPUT:
! - cmp_arrays_int8: -X (1<=X<=ml1,ml1=ml2): first difference in two arrays occurs at position X (left-to-right) and m1(X)<m2(X);
! +X (1<=X<=ml1,ml1=ml2): first difference in two arrays occurs at position X (left-to-right) and m1(X)>m2(X);
! -X (X>ml1,X>ml2): first array is shorter than the second one (ml1<ml2);
! +X (X>ml1,X>ml2): first array is longer than the second one (ml1>ml2);
! 0 (ml1=ml2): the two arrays are equal.
! - trn(0:) - if preorder=.true. and the arrays are equal (cmp_arrays_int8=0),
! this (optional) output array will contain the permutation matching the two arrays:
! A new order of old elements of the 2nd array (N2O) that matches the 1st array (both with the original ordering).
implicit none
logical, intent(in):: preorder
integer, intent(in):: ml1,ml2
integer(8), intent(in):: m1(1:ml1),m2(1:ml2)
integer, intent(out), optional:: trn(0:*)
integer:: i,j,k,l,m,n,k0,k1,k2,k3,ks,kf,ierr
integer(8), allocatable:: prm1(:),prm2(:)
integer(8):: ml
if(ml1.ge.0.and.ml2.ge.0) then
if(ml1.lt.ml2) then
cmp_arrays_int8=-(max(ml1,ml2)+1)
elseif(ml1.gt.ml2) then
cmp_arrays_int8=(max(ml1,ml2)+1)
else !ml1=ml2
cmp_arrays_int8=0; ml=int(ml1,8)
if(preorder) then
allocate(prm1(0:ml),prm2(0:ml),STAT=ierr)
if(ierr.ne.0) then; write(*,*)'ERROR(combinatoric:cmp_arrays_int8): memory allocation failed!'; stop; endif
prm1(0)=+1_8; do i=1,ml1; prm1(i)=int(i,8); enddo
prm2(0)=+1_8; do i=1,ml2; prm2(i)=int(i,8); enddo
call merge_sort_key_int8(ml,m1,prm1) !prm1 is N2O
call merge_sort_key_int8(ml,m2,prm2) !prm2 is N2O
do i=1,ml1
if(m1(prm1(i)).ne.m2(prm2(i))) then; cmp_arrays_int8=int(sign(int(i,8),m1(prm1(i))-m2(prm2(i))),4); exit; endif
enddo
if(cmp_arrays_int8.eq.0.and.present(trn)) then
trn(0)=int(prm1(0)*prm2(0),4); do i=1,ml1; trn(prm1(i))=int(prm2(i),4); enddo
endif
deallocate(prm1,prm2)
else
do i=1,ml1
if(m1(i).ne.m2(i)) then; cmp_arrays_int8=int(sign(int(i,8),m1(i)-m2(i)),4); exit; endif
enddo
endif
endif
else !invalid arguments
write(*,*)'ERROR(combinatoric:cmp_arrays_int8): invalid arguments: ',ml1,ml2
stop
endif
return
end function cmp_arrays_int8
!--------------------------------------------------