Label (space separated): matlab combination algorithm mapping
The generation of matlab combination sequence, bijection function of sequence and sequence number
1.1 generating combined sequences
% Functions of the system
>> nchoosek(1:4, 3)
ans =
1 2 3
1 2 4
1 3 4
2 3 4
% Functions I wrote
>> myChoose(4, 3)
ans =
1 2 3
1 2 4
1 3 4
2 3 4
The nchoosek() and myChoose() functions of the system can generate all combinations, but the chooseContinue() function called by myChoose() can generate combination sequences one by one, which can save memory. For example:
>> n=4; r=3;
seq =[1:r]
[seq,loopJudge]=chooseContinue(n,r,seq)
seq =
1 2 3
seq =
1 2 4
loopJudge =
1
>> [seq,loopJudge]=chooseContinue(n,r,seq)
seq =
1 3 4
loopJudge =
1
>> [seq,loopJudge]=chooseContinue(n,r,seq)
seq =
2 3 4
loopJudge =
1
>> [seq,loopJudge]=chooseContinue(n,r,seq)
seq =
[]
loopJudge =
0
1.2 mapping of composite sequences to ordinals
>> r=3;
>> sequence2num(r,[1 2 3])
ans =
1
>> sequence2num(r,[1 2 4])
ans =
2
>> sequence2num(r,[1 3 4])
ans =
3
>> sequence2num(r,[2 3 4])
ans =
4
1.3 ordinal mapping to composite sequences
>> n=4; r=3;
>> seq= num2sequence(n,r,1)
seq =
1 2 3
>> seq= num2sequence(n,r,2)
seq =
1 2 4
>> seq= num2sequence(n,r,3)
seq =
1 3 4
>> seq= num2sequence(n,r,4)
seq =
2 3 4
1.4 source code
1.4.1 myChoose()
function seqs=myChoose(n,r)
%Generate the main function of [generate all combinations one by one]
%% Original algorithm
%Time complexity is O(C(n,r)*r),The space complexity is O(C(n,r)*r)
%However, when it is only necessary to return the combined sequences one by one (usually satisfied), the spatial complexity is O(r)
%Time: 2013/12/28 15:28 Name: Deng nengcai
%%
seqs=zeros(nchoosek(n,r),r);
loopJudge=true; i=1;
seqs(i,:)=[1:r];
while true
[seqTemp,loopJudge]=chooseContinue(n,r,seqs(i,:));
if loopJudge
seqs(i+1,:)=seqTemp;
i=i+1;
else break; end
end
end
1.4.2 chooseContinue()
function [seq,loopJudge]=chooseContinue(n,r,seq)
% Subfunction: generate next combination
%Time: 2013/12/28 15:28 Name: Deng nengcai
%Find the child that can move forward
i=1;
while i+1<=r
if seq(i+1)==seq(i)+1
i=i+1;
else break; end
end
if i==r && seq(r)==n %When the last sequence has been reached
seq=[]; loopJudge=false;
else
seq(i)=seq(i)+1; %Section i Move forward
for j=1:i-1, seq(j)=j; end %Move back to home position
loopJudge=true;
end
end
1.4.3 sequence2num()
function num= sequence2num(r,seq)
%Map [combination sequence] to C(n,r)Of all combinations of
%% Original algorithm
%Time: 2013/12/27 22:33 Name: Deng nengcai
%%
num=0;
for i=r:-1:1
if seq(i)==i, break; end
num=num+nchoosek(seq(i)-1,i);
end
num=num+1;
end
% % % %Theoretical basis: Combinatorial Mathematics((U.S.A)Richard A.Brualdi)
% % % num=nchoosek(n,r);
% % % seq
% % % for i=1:r
% % % ni=n-seq(i),ri=r-i+1
% % % if ni~=0
% % % num=num-nchoosek(ni,ri);
% % % else
% % % break;
% % % end
% % % end
% % % end [Not using the theory]
1.4.4 num2sequence()
function seq= num2sequence(n,r,num)
%take C(n,r)The sequence number of a combination of is mapped to the combination sequence
%% Original algorithm
%When [pre calculated C(j,i)Matrix,
%Time complexity is O(n*r),The space complexity is O(n*r)
%Time: 2013/12/28 15:28 Name: Deng nengcai
%%
seq=[1:r];
lastPoint=n;
num=num-1;
for i=r:-1:1
% '##loop i'
if num==0, break; end
for j=lastPoint:-1:1
% j_i=[j,i] % ( i==j ||
if num<nchoosek(j,i) && num>=nchoosek(j-1,i)
seq(i)=j; lastPoint=j-1; break;
end
end
num=num-nchoosek(j-1,i);
end
end
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end