1     2     2     4   % top left corner 1 2 - bottom right corner 2 4
 1     3     1     1   % top left corner 1 1 - bottom right corner 3 1

%% PROBLEM
%
%  let P be a matrix whose elements are zeros and ones
%  find the best(*) combination of non-overlapping submatrices of P
%  so that each submatrix respect these properties:
%   - contains at least L zeros and L ones (min area=2*L)
%   - contains at most H elements (max area=H)
%
%  (*) the best is the one which maximize the total number of elements in all the submatrices
%
%  notices: the best combination could be not unique
%           is not always possibile to cover all the elements of P with the submatrices of the best combination
%
%% INPUT
P=round(rand(8,8)); L=1; H=5;
%P=dlmread('small.txt'); L=1; H=5;  % small can be found here https://pastebin.com/RTc5L8We
%P=dlmread('medium.txt'); L=2; H=8; % medium can be found here https://pastebin.com/qXJEiZTX
%P=dlmread('big.txt'); L=4; H=12;   % big can be found here https://pastebin.com/kBFFYg3K
%P=[0 0 0 0 0 1;0 0 0 0 0 1;0 1 0 1 0 1;0 0 0 0 0 0;0 0 0 0 0 0]; L=1; H=6;
P=[0 0 0 0 0;0 1 1 1 0;0 0 0 0 0]; L=1; H=6;
%P=[1,0,0,0,0;1,1,1,1,1;1,0,0,0,0]; L=1; H=5;

%% FIND ALL THE SUBMATRICES OF AREA >= 2*L & <= H
%
%  conv2(input_matrix,shape_matrix,'valid')
%  creates a matrix, where each element is the sum of all the elements contained in
%  the submatrix (contained in input_matrix and with the shape given by shape_matrix)
%  having its top left corner at said element
%
%  ex.  conv2([0,1,2;3,4,5;6,7,8],ones(2,2),'valid')
%       ans =
%             8    12
%            20    24
%       where 8=0+1+3+4 12=1+2+4+5  20=3+4+6+7  24=4+5+7+8
%
s=[]; % will contain the indexes and the area of each submatrix
      % i.e.  1 3 2 5 9  is the submatrix with area 9 and corners in 1 2 and in 3 5
for sH = H:-1:2*L
    div_sH = divisors(sH);
    fprintf('_________AREA %d_________n',sH)
    for k = 1:length(div_sH)
        a = div_sH(k);
        b = div_sH(end-k+1);
        convP = conv2(P,ones(a,b),'valid');
        [i,j] = find((convP >= L) & (convP <= sH-L));
        if ~isempty([i,j])
            if size([i,j],1) ~= 1
%                        rows     columns           area
                s = [s;[i,i-1+a , j,j-1+b , a*b*ones(numel(i),1)]];
            else
                s = [s;[i',i'-1+a,j',j'-1+b,a*b*ones(numel(i),1)]];
            end
            fprintf('[%dx%d] submatrices: %dn',a,b,size(s,1))
        end
    end
end
fprintf('n')
s(:,6)=1:size(s,1);

%% FIND THE BEST COMBINATION
tic
[R,C]=size(P); % rows and columns of P
no_ones=sum(P(:)); % counts how many ones are in P
% a combination of submatrices cannot contain more than max_no_subm submatrices
if no_ones <= R*C-no_ones
    max_no_subm=floor(no_ones/L);
else
    max_no_subm=floor(R*C-no_ones/L);
end
comb(2,1)=R*C+1; % will contain the best combination
s_copy=s; % save a copy of s
[comb,perfect]=recursion(s,s_copy,comb,0,0,R,C,0,false,H,[],size(s,1),false,[0,0,0],0,0,0,0,0,0,max_no_subm);
fprintf('ntime: %2.2fsnn',toc)
if perfect
    disp('***********************************')
    disp('***  PERFECT COMBINATION FOUND  ***')
    disp('***********************************')
end

%% PRINT RESULTS
if (R < 12 && C < 12)
    for i = 1:length(find(comb(2,3:end)))
        optimal_comb_slices(i,:)=s(comb(2,i+2),:);
    end
    optimal_comb_slices(:,1:5)
    P
end

function [comb,perfect,counter,area,v,hold_on,ijk,printed,first_for_i,second_for_i,third_for_i] = recursion(s,s_copy,comb,counter,area,R,C,k,hold_on,H,v,size_s,perfect,ijk,size_s_ovrlppd,size_s_ovrlppd2,printed,third_for_i,second_for_i,first_for_i,max_no_subm)
%
% OUTPUT (that is actually going to be used in the main script)
% comb [matrix] a matrix of two rows, the first one contains the current combination
%        the second row contains the best combination found
% perfect [boolean] says if the combination found is perfect (a combination is perfect if
%           the submatrices cover all the elements in P and if it is made up with
%           the minimum number of submatrices possible)
%
% OUTPUT (only needed in the function itself)
% counter [integer] int that keeps track of how many submatrices are in the current combination
% area [integer] area covered with all the submatrices of the current combination
% v [vector] keeps track of the for loops that are about to end
% hold_on [boolean] helps v to remove submatrices from the current combination
%
% OUTPUT (only needed to print results)
% ijk [vector] contains the indexes of the first three nested for loops (useful to see where the function is working)
% printed [boolean] used to print text on different lines
% first_for_i second_for_i third_for_i [integers] used by ijk
%
%
% INPUT
% s [matrix] the set of all the submatrices of P
% s_copy [matrix] the set of all the submatrices that don't overlap the ones in the current combination
%                 (is equal to s when the function is called for the first time)
% R,C [integers] rows and columns of P
% k [integer] area of the current submatrix
% H [integer] maximum number of cells that a submatrix can contains
% size_s [integer] number of rows of s i.e. number of submatrices in s
% size_s_ovrlppd [integer] used by ijk
% size_s_ovrlppd2 [integer] used by ijk
% max_no_subm [integer] maximum number of submatrices contained in a combination
%
%
%  the function starts considering the first submatrix (call it sub1) in the set 's' of all the submatrices
%  and adds it to the combination
%  then it finds 's_ovrlppd' i.e. the set of all the submatrices that don't overlap sub1
%  and the function calls itself considering the first submatrix (call it sub2) in the set 's_ovrlppd'
%  and adds it to the combination
%  then it finds the set of all the submatrices that don't overlap sub2 and
%  so on until there are no more non-overlapping submatrices
%  then it replaces the last submatrix in the combination with the second one of the last set of non-overlapping
%  submatrices and saves the combination which covers more elements in P
%  and so on for all the submatrices of the set 's'
%
%  DOWNSIDE OF THIS METHOD
%    if 's' contains thousands of submatrices, the function will create hundreds of nested for loops
%    so both time and space complexities can be really high and the computer might freeze
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%   SAVE AND RESET COMBINATIONS   %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   s_copy is empty when no more submatrices can be added to the current
%   combination, in this case we have to check if this combination is
%   better the best combination previosly found, if so then we overwrite it
%
%   then we have to remove one or more submatrices from the combination (depending on
%   how many nested for loops are about to be closed)
%   and compute another combination
%   to 'remove one or more submatrices from the combination' it is necessary to do these things:
%    - reduce the area
%    - reduce the combination
%    - reduce the counter
%
    if isempty(s_copy)
        comb(1,2)=counter;  % final no of submatrices in the combination
        comb(1,1)=R*C-area; % no. of cells remained in P after removing the cells contained in the submatrices of the combination
%       if the combination just found is better than the previous overwrite it
        if comb(1,1)<comb(2,1) || (comb(1,1)==comb(2,1) && comb(1,2)<comb(2,2))
            comb(2,:)=comb(1,:);
            disp(['[area_left] ', num2str(comb(2,1)), ' [slices] ', num2str(comb(2,2))])
            printed=true;
        end

        area=area-k; % tot area of the combination excluding the last sumatrix
        if ~isempty(v) && ~hold_on % more than one submatrix must be removed
            i=size(v,2);
            if i>length(find(v)) % if v contains at least one 0
                while v(i)==1 % find the index of the last 0
                    i=i-1;
                end
                last_i_counter=size(v(i+1:end),2); % no. of consecutive for loop that are about to end
                v=v(1:i-1);
            else
                last_i_counter=i;
                v=[];
            end
            for i=1:last_i_counter
                area=area-s(comb(1,2+counter-i),5); % reduce the area
            end
            comb(1,2+counter-last_i_counter:2+counter)=0; % remove submatrices from the combination
            counter=counter-(last_i_counter+1); % reduce the counter
            hold_on=true;
        else % exactly one submatrix must be removed
            comb(1,2+counter)=0;
            counter=counter-1;
        end
% USED TO SHOW THAT WITHOUT s_ovrlppd(s_ovrlppd(:,6)<pos,:)=[]; THERE ARE
% LOT OF REPEATING COMBINATIONS
%     if isempty(s_copy)
%         comb(size(comb,1)+1,:)=comb(1,:);
%         comb(size(comb,1),2)=counter; % final no of slices in the combination
%         comb(size(comb,1),1)=R*C-area; % no. of cells remained in P after removing the cells contained in the slices of the combination
%
%         area=area-k;
%         if first_for_i == 34
%             comb(1:2,:)=[];
%         end
%         if ~isempty(v) && ~hold_on
%             i=size(v,2);
%             if i>length(find(v))
%                 while v(i)==1
%                     i=i-1;
%                 end
%                 last_i_counter=size(v(i+1:end),2);
%                 v=v(1:i-1);
%             else
%                 last_i_counter=i;
%                 v=[];
%             end
%             for i=1:last_i_counter
%                 area=area-s(comb(1,2+counter-i),5);
%             end
%             comb(1,2+counter-last_i_counter:2+counter)=0;
%             counter=counter-(last_i_counter+1);
%             hold_on=true;
%         else
%             comb(1,2+counter)=0;
%             counter=counter-1;
%         end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%   FIND COMBINATIONS   %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    else
        for i = 1:size(s_copy,1) % fix the i-th submatrix

%           if the combination cover all elements of P & the no. of submatrices is the minimum possibile
            if comb(2,1)==0 && comb(2,2)==ceil(R*C/H)
                perfect=true;
                return
            end

            pos=s_copy(i,6); % position of current submatrix in s
            comb(1,3+counter)=pos; % add the position to the combination
            k=s(pos,5); % save the area of the current submatrix
            area=area+k; % area covered with all the submatrices of the combination up now
            counter=counter+1;

%           if the area in P covered by the current combination could not
%           be bigger than the best combination found up now, then discard
%           the current combination and consider the next one
            if R*C-area-(max_no_subm-counter)*H > comb(2,1) && i < size(s_copy,1)
%              R*C-area-(max_no_subm-counter)*H can be negative if ceil(R*C/H) < max_no_subm
                counter=counter-1;
                comb(1,3+counter)=0;
                area=area-k;
            else
                s_ovrlppd=s_copy; % initializing the set of non-overlapping submatrices
                s_ovrlppd(s_copy(:,1)<=s_copy(i,2) & s_copy(:,2)>=s_copy(i,1) & s_copy(:,3)<=s_copy(i,4) & s_copy(:,4)>=s_copy(i,3),:)=[]; % delete submatrices that overlap the i-th one
                s_ovrlppd(s_ovrlppd(:,6)<pos,:)=[]; % remove submatrices that will generate combinations already studied
%               KEEP TRACK OF THE NESTED 'FOR' LOOPS ENDS REACHED
                if i==size(s_copy,1) % if i is the last cycle of the current for loop
                    v(size(v,2)+1)=1; % a 1 means that the code entered the last i of a 'for' loop
                    if size(s_ovrlppd,1)~=0 % hold on until an empty s_ovrlppd is found
                        hold_on=true;
                    else
                        hold_on=false;
                    end
                elseif ~isempty(v) && size(s_ovrlppd,1)~=0
                    v(size(v,2)+1)=0; % a 0 means that s_ovrlppd in the last i of a 'for' loop is not empty => a new 'for' loop is created
                end
%%%%%%%%%%%%%%%%%%%%%%%%
%%%   PRINT STATUS   %%%
%%%%%%%%%%%%%%%%%%%%%%%%
                if size(s_copy,1)==size_s
                    ijk(1)=i;
                    ijk(2:3)=0;
%                     if ~printed && i~=1
%                         fprintf(repmat('b',1,numel(num2str(first_for_i))+numel(num2str(second_for_i))+numel(num2str(third_for_i))+2+2+1)) % [] ,, return
%                    else
%                        printed=false;
%                    end
                    fprintf('[%d,%d,%d]n',ijk)
                    size_s_ovrlppd=size(s_ovrlppd,1);
                    first_for_i=i;
                    second_for_i=0;
                elseif size(s_copy,1)==size_s_ovrlppd
                    ijk(2)=i;
                    ijk(3)=0;
                    if ~printed
                        fprintf(repmat('b',1,numel(num2str(first_for_i))+numel(num2str(second_for_i))+numel(num2str(third_for_i))+2+2+1)) % [] ,, return
                    else
                        printed=false;
                    end
                    fprintf('[%d,%d,%d]n',ijk)
                    size_s_ovrlppd2=size(s_ovrlppd,1);
                    second_for_i=i;
                    third_for_i=0;
                elseif size(s_copy,1)==size_s_ovrlppd2
                    ijk(3)=i;
                    if ~printed
                        fprintf(repmat('b',1,numel(num2str(first_for_i))+numel(num2str(second_for_i))+numel(num2str(third_for_i))+2+2+1))
                    else
                        printed=false;
                    end
                    fprintf('[%d,%d,%d]n',ijk)
                    third_for_i=i;
                end
                [comb,perfect,counter,area,v,hold_on,ijk,printed,first_for_i,second_for_i,third_for_i]=recursion(s,s_ovrlppd,comb,counter,area,R,C,k,hold_on,H,v,size_s,perfect,ijk,size_s_ovrlppd,size_s_ovrlppd2,printed,third_for_i,second_for_i,first_for_i,max_no_subm);
            end
        end
    end
end