function [board, orientation] = solver(tiles, boardSize)
[board,orientation,solved,nrow,ncol,ntiles,nElem] = board_perfect_snake(boardSize,tiles);
if solved;return;end
cs=hankel(1:4,0:3);
played = tiles;
numUnplayed = ntiles-nElem;
sums = sum(tiles,2);
if numUnplayed > 0
[sorted,si] = sort(sums);
numUnplayedp1=numUnplayed+1;
equals = find(sorted(numUnplayedp1)==sorted);
if nnz(sorted(numUnplayedp1) == sorted(numUnplayedp1:end)) ~= length(equals)
temp_si = si(equals);
[trash,mi] = sort(max(tiles(temp_si,:),[],2));
si(equals(end:-1:1))= temp_si(mi);
end
played(si(1:numUnplayed),:) = Inf(numUnplayed,4);
ntiles=nElem;
height=nrow;
else
% height=max(min(ceil(sqrt(ntiles)), nrow),ceil(ntiles/ncol));
height=ceil(max(min(sqrt(ntiles), nrow),ntiles/ncol));
end
[y,x]=ind2sub([height,ncol],1:ntiles);
ind = sub2ind(boardSize,y,x);
board(ind) = -1;
last = x(end);
b =repmat(board,[1,1,18]);
o = ones(size(tiles,1), 18);
NZ0 = prod(played,2);
heightn1=height-1;
[b(:,:,1),o(:,1),list] = place([1 height],1:last,b(:,:,1),played,played,o(:,1),ntiles, cs,nrow,ncol,false,NZ0);
[b(:,:,1),o(:,1),list] = place(height:-1:1,[last 1],b(:,:,1),played,list,o(:,1),ntiles, cs,nrow,ncol,false,NZ0);
newList(:,:,1) = list;
[b(:,:,2),o(:,2),list] = place(height:-1:1,last,b(:,:,2),played,played,o(:,2),ntiles, cs,nrow,ncol,false,NZ0);
[b(:,:,2),o(:,2),list] = place(height,1:last,b(:,:,2),played,list,o(:,2),ntiles, cs,nrow,ncol,false,NZ0);
[b(:,:,2),o(:,2),list] = place(1:heightn1,1,b(:,:,2),played,list,o(:,2),ntiles, cs,nrow,ncol,false,NZ0);
[b(:,:,2),o(:,2),list] = place(1,2:last,b(:,:,2),played,list,o(:,2),ntiles, cs,nrow,ncol,false,NZ0);
newList(:,:,2) = list;
ncoln1=ncol-1;
for i=[2 5]
c = floor(i/5) + 1;
[b(:,:,i+1),o(:,i+1)] = place(heightn1:-1:2,ncoln1:-1:2,b(:,:,c),played,newList(:,:,c),o(:,c),ntiles , cs,nrow,ncol,true,NZ0);
[b(:,:,i+2),o(:,i+2)] = place(2:heightn1,2:ncoln1,b(:,:,c),played,newList(:,:,c),o(:,c),ntiles, cs,nrow,ncol,true,NZ0);
[b(:,:,i+3),o(:,i+3)] = place(2:heightn1,ncoln1:-1:2,b(:,:,c),played,newList(:,:,c),o(:,c),ntiles, cs,nrow,ncol,true,NZ0);
end
[b(:,:,9),o(:,9)] = place(heightn1:-1:2,2:ncoln1,b(:,:,2),played,newList(:,:,2),o(:,2),ntiles, cs,nrow,ncol,true,NZ0);
[b(:,:,1),o(:,1)] = place(height:-1:1,1:ncol,b(:,:,12),played,played,o(:,12),ntiles,cs,nrow,ncol,-1,NZ0);
[b(:,:,2 ), o(:,2 )] = calibrating_pure_werner(played, boardSize,nrow,ncol,nrow,true);
if nElem < 1000
for Sii = 10:17
[b(:,:,Sii),o(:,Sii)] = BiswasMichaelC(height,last, played, played, o(:,Sii), ntiles, cs,nrow,ncol);
end
S=17;
if nElem < 200
S = 18;
[b(:,:,S ), o(:,S )] = calibrating_pure_werner(tiles, boardSize,nrow,ncol,1,false);
end
end
overall = zeros(S,1);
parfor i=1:S
overall(i) = overallScore(b(:,:,i),nrow,ncol,o(:,i),tiles, cs);
end
[overall ord]=sort(overall);
b=b(:,:,ord);
o=o(:,ord);
for q=1:2
[b(:,:,q),o(:,q)]=localopt(b(:,:,q), o(:,q),tiles,cs,nrow,ncol);
overall(q) = overallScore(b(:,:,q),nrow,ncol,o(:,q),tiles, cs);
end
[trash,mo] = min(overall);
board = b(:,:,mo);
orientation = o(:,mo);
end
function [board,orientation,gain]=localopt(board, orientation,tiles,cs,r,c)
gain=0;
n = size(tiles,1);
didx = [3;4;1;2];
uidx = [1;2;3;4];
lidx = [4;1;2;3];
ridx = [2;3;4;1];
bp = board>0;
bt = board(bp);
te = [zeros(1,4);tiles];
oe = [1;orientation];
ub = [zeros(1,c);board(1:end-1,:)];
db = [board(2:end,:);zeros(1,c)];
lb = [zeros(r,1) board(:,1:end-1)];
rb = [board(:,2:end) zeros(r,1)];
ut = ub(bp)+1;
dt = db(bp)+1;
lt = lb(bp)+1;
rt = rb(bp)+1;
np1=n+1;
nv = [te(ut(:)+(didx(oe(ut))-1)*(np1)),...
te(rt(:)+(lidx(oe(rt))-1)*(np1)),...
te(dt(:)+(uidx(oe(dt))-1)*(np1)),...
te(lt(:)+(ridx(oe(lt))-1)*(np1))];
bt=bt(:);
bv = tiles(bt(:,ones(1,4))+(cs(orientation(bt),:)-1)*n);
zeros401=zeros(4,1);
zeros1016=zeros(1,16);
zeros404=zeros(4);
for i=1:2
pv = sum(abs(bv-nv),2);
[mpv,id]=max(pv);
while mpv
bid=bt(id);
bid1=bid+1;
pv(id)=0;
obid = orientation(bid);
bx = tiles(bid,:);
p0 = sum(abs(bx(cs)-nv(id+zeros401,:)),2);
[pmin,ido]=min(p0);
gain=gain+(p0(obid)-pmin);
if p0(obid)>pmin
bv(id,:)=bx(cs(ido,:));
orientation(bid)=ido;
u1=ut==bid1;
d1=dt==bid1;
l1=lt==bid1;
r1=rt==bid1;
nv(u1,1) = tiles(bid,didx(ido));
nv(d1,3) = tiles(bid,uidx(ido));
nv(l1,4) = tiles(bid,ridx(ido));
nv(r1,2) = tiles(bid,lidx(ido));
L=u1|d1|l1|r1;
pv(L) = sum(abs(bv(L,:)-nv(L,:)),2);
end
[mpv,id]=max(pv);
end
pv = sum(abs(bv-nv),2);
[mpv,id]=max(pv);
N=numel(pv);
tv = tiles(bt,:);
k=0;
while mpv>0
bid = bt(id);
bid1=bid+1;
nvid = zeros1016;
nvid(:) = nv(id+zeros401,:);
u1=ut==bid1;
d1=dt==bid1;
l1=lt==bid1;
r1=rt==bid1;
L=~(u1|d1|l1|r1);
L(id)=false;
qv = Inf(N,4);
qv(L,:) = reshape(sum(reshape(abs(tv(L,cs)-nvid(ones(sum(L),1),:)),[],4),2),[],4);
[iv,to] = min(qv(:));
jd = mod(to-1,N)+1;
bjd = bt(jd);
bjd1=bjd+1;
% objd = orientation(bjd);
jo = ceil(to/N);
nvjd = nv(jd+zeros401,:);
bvid = zeros404;
bvid(:) = tiles(bid,cs);
[jv,io]=min(sum(abs(bvid - nvjd),2));
jpv = sum(abs(bv(jd,:) - nv(jd,:)));
if iv+jv < mpv+jpv
bt(jd) = bid;
bt(id) = bjd;
orientation(bid) = io;
orientation(bjd) = jo;
board(bp)=bt;
bv(id,:) = tv(jd,cs(jo,:));
bv(jd,:) = tv(id,cs(io,:));
bx = tv(id,:);
tv(id,:) = tv(jd,:);
tv(jd,:) = bx;
nv(u1,1) = tiles(bjd,didx(jo));
nv(d1,3) = tiles(bjd,uidx(jo));
nv(l1,4) = tiles(bjd,ridx(jo));
nv(r1,2) = tiles(bjd,lidx(jo));
u2=ut==bjd1;
d2=dt==bjd1;
l2=lt==bjd1;
r2=rt==bjd1;
nv(u2,1) = tiles(bid,didx(io));
nv(d2,3) = tiles(bid,uidx(io));
nv(l2,4) = tiles(bid,ridx(io));
nv(r2,2) = tiles(bid,lidx(io));
L=u1|d1|l1|r1|u2|d2|l2|r2;
pv(L) = sum(abs(bv(L,:)-nv(L,:)),2);
uid = ut==bid1;
ujd = ut==bjd1;
ut(uid) = bjd1;
ut(ujd) = bid1;
did = dt==bid1;
djd = dt==bjd1;
dt(did) = bjd1;
dt(djd) = bid1;
lid = lt==bid1;
ljd = lt==bjd1;
lt(lid) = bjd1;
lt(ljd) = bid1;
rid = rt==bid1;
rjd = rt==bjd1;
rt(rid) = bjd1;
rt(rjd) = bid1;
k=k+1;
end
pv(id)=0;
pv(jd)=0;
[mpv,id]=max(pv);
end
end
end
function score = overallScore(board,nrow,ncol, orientation, tiles, cs)
tile = zeros(nrow*4,ncol);
tmp = tiles.';
for y = 1:nrow
for x = 1:ncol
if board(y,x)
tile((y-1)*4+1:y*4,x)=tmp(cs(orientation(board(y,x)),:),board(y,x));
end
end
end
out_tiles = 1:size(tiles,1);
in_tiles = board(board>0);
out_tiles(in_tiles) = [];
score = sum(sum(abs(tile(5:4:end,1:end)-tile(3:4:end-4,1:end))))+ ...
sum(sum(abs(tile(2:4:end,1:end-1)-tile(4:4:end,2:end))))+ ...
sum(tile(4:4:end,1))+sum(tile(2:4:end,end))+sum(tile(end-1,1:end))+ ...
sum(tile(1,1:end))+sum(sum(tiles(out_tiles,:)));
end
function [board, ort, lst] = BiswasMichaelC(height, width, tiles, lst, ort, ntiles, cs,nrow,ncol)
[cols,rows]=meshgrid(1:width,1:height);
center_x = width/2 + 0.5;
center_y = height/2 + 0.5;
dist_center = ((cols - center_x).^2 + (rows - center_y).^2).^(1/2.3);
dist_center = dist_center + rand(size(dist_center))*0.03; %param
[trash, sort_index] = sort(dist_center(:));
loop_order = flipud(sort_index)';
board = zeros(nrow,ncol);
brd = zeros(height,width)-1;
if ntiles < height*width
loop_order(~ismember(loop_order,1:ntiles)) = [];
brd(ntiles+1:end) = 0;
end
score = inf(size(lst,1),1,4);
mst = ~isinf(lst(:,1));
for ii = loop_order
y = rows(ii);
x = cols(ii);
if (brd(y,x)==-1)
[n,e,s,w] = findBoundaries(brd, tiles, ort, y, x, cs,height,width);
nesw = cat(3, [n e s w], [w n e s], [s w n e],[e s w n]);
score=score+inf;
score(mst,:,:) = sum(abs(int32(bsxfun(@minus,lst(mst,:),nesw))),2);
[ms, o] = find(score==min(score(:)));
which = round(rand*(length(ms)-1)+1);
t = ms(which);
ort(t) = o(which);
brd(y,x) = t;
% lst(t,:) = inf(1,4);
mst(t) = false;
end
lst(~mst,:) = inf;
end
board(1:height,1:width) = brd;
end
function [brd, ort, lst] = place(rowRange, colRange, brd, tiles, lst, ort, ntiles, cs,nrow,ncol,flag,NZ)
k = 1;
score = inf(size(lst,1),1,4);
mst = ~isinf(lst(:,1));
for y=rowRange
for x=colRange
if (brd(y,x)==-1)
[n,e,s,w] = findBoundaries(brd, tiles, ort, y, x, cs,nrow,ncol);
nesw = cat(3, [n e s w], [w n e s], [s w n e],[e s w n]);
score = score+inf;
score(mst,:,:) = sum(abs(int32(bsxfun(@minus,lst(mst,:),nesw))),2);
[TT, O] = find(score==min(score(:)));
%
if flag==-1
idx_t = ceil(length(TT)/1.5);%round(rand*(length(ms)-1)+1);
elseif flag
[trash,idx_t] = max(NZ(TT));
else
[trash,idx_t] = min(NZ(TT));
end
t = TT(idx_t);
ort(t) = O(idx_t);
brd(y,x)=t;
% lst(t,:)= inf(1,4);
mst(t) = false;
k = k + 1;
if (k > ntiles)
return;
end
end
end
end
lst(~mst,:)=inf;
end
function [north, east, south, west] = findBoundaries(board, tiles, orientation, row, col, cs,nrow,ncol)
north = boundary(board, tiles, orientation, row-1,col ,3, cs,nrow,ncol );
south = boundary(board, tiles, orientation, row+1,col ,1, cs,nrow,ncol );
west = boundary(board, tiles, orientation, row ,col-1,2, cs,nrow,ncol );
east = boundary(board, tiles, orientation, row ,col+1,4, cs,nrow,ncol );
end
function value = boundary(board, tiles, orientation, row,col,id, cs,nrow,ncol )
if row<1 || col<1 || row>nrow || col>ncol || board(row,col) == 0
value = 0;
elseif board(row,col) == -1
value = nan;
else
value = tiles(board(row,col),cs(orientation(board(row,col)),id));
end
end
function [board, orientation] = calibrating_pure_werner(tiles, boardSize,nrows,ncols,w,flag)
board = zeros(boardSize,'single');
ntiles = size(tiles, 1);
orientation = ones(ntiles, 1);
tiles = [tiles; tiles(:,[2:4,1]); tiles(:,[3:4,1:2]); tiles(:,[4,1:3])];
tilestaken = false(4*ntiles, 1);
boarde = board;
boarde(w,1) = 4;
[v, ix] = max(boarde(:));
nextrow = mod(ix-1,nrows)+1;
nextcol = ceil(ix/nrows) ;
while v && ~all(tilestaken);
tix = findbesttilefor(nextrow, nextcol,tiles,board,nrows,ncols,tilestaken,flag);
placetile(tix, nextrow, nextcol)
[v, ix] = max(boarde(:));
nextrow = mod(ix-1,nrows)+1;
nextcol = ceil(ix/nrows) ;
end
function placetile(tileix, row, col)
board(row, col) = tileix;
boarde(row, col) = 0;
rowm1=row > 1;
colm1= col > 1;
rowmnrows=row < nrows;
colmncols=col < ncols;
rowp1=row+1;
colp1=col+1;
coln1=col-1;
rown1=row-1;
placetile1(tileix, row, col,rowm1,colm1,rowmnrows,colmncols,rowp1,rown1,colp1,coln1 )
placetile2( rowm1 ,colmncols,colm1, rown1,colp1,coln1)
placetile3( colm1,rowmnrows ,colmncols,rowp1 ,colp1,coln1)
ix = mod(tileix-1, ntiles)+1;
orientation(ix) = ceil(tileix/ntiles);
tilestaken(ix+ntiles*(0:3), :) = true;
function placetile1(tileix, row, col,rowm1,colm1,rowmnrows,colmncols,rowp1,rown1,colp1,coln1 )
if rowm1 && ~board(rown1,col); boarde(rown1, col ) = boarde(rown1, col ) + tiles(tileix, 1); end;
if rowmnrows && ~board(rowp1,col ); boarde(rowp1, col ) = boarde(rowp1, col ) + tiles(tileix, 3); end;
if colm1 && ~board(row ,coln1); boarde(row , coln1) = boarde(row , coln1) + tiles(tileix, 4); end;
if colmncols && ~board(row ,colp1); boarde(row , colp1) = boarde(row , colp1) + tiles(tileix, 2); end;
end
function placetile2( rowm1,colmncols,colm1, rown1,colp1,coln1)
if rowm1 && colm1 && ~board(rown1,coln1); boarde(rown1, coln1) = boarde(rown1, coln1) + 0.1; end;
if rowm1 && colmncols && ~board(rown1,colp1); boarde(rown1, colp1) = boarde(rown1, colp1) + 0.1; end;
end
function placetile3( colm1,rowmnrows,colmncols,rowp1, colp1,coln1)
if rowmnrows && colm1 && ~board(rowp1,coln1); boarde(rowp1, coln1) = boarde(rowp1, coln1) + 0.1; end;
if rowmnrows && colmncols && ~board(rowp1,colp1); boarde(rowp1, colp1) = boarde(rowp1, colp1) + 0.1; end;
end
end
boardzeros = board == 0;
board = mod(board-1, ntiles)+1;
board(boardzeros) = 0;
end
function ret = findbesttilefor(r, c,tiles,board,nrows,ncols,tilestaken,flag)
function myret=spycat(x,y,z,tiles,board)
if board(x,y) > 0; myret = tiles(board(x,y), z);
else myret = nan;
end
end
v = zeros(1,4);
if r > 1;
v(1) = spycat(r-1,c , 3,tiles,board);
end
if c < ncols;
v(2)= spycat(r,c+1,4,tiles,board) ;
end
if r < nrows;
v(3) = spycat(r+1,c,1,tiles,board);
end
if c > 1;
v(4)=spycat(r,c-1,2,tiles,board) ;
end
mask = ~isnan(v);
tilecosts = sum(abs(bsxfun(@minus, tiles(:,mask), v(mask))), 2);
tilecosts(tilestaken) = inf;
if flag
[trash, ret] = min(tilecosts - 1e-10*sum(tiles(:, isnan(v)), 2));
else
[trash, ret] = min(tilecosts-sum(tiles,2).*1e-6);
end
end
% Richard routines
function [board,orientation,solved,nr,nc,numtiles,nrnc]=board_perfect_snake(boardSize,tiles)
% check for perfect unique board
solved=false;
board = zeros(boardSize,'single');
numtiles=size(tiles,1);
orientation = ones(numtiles, 1);
[nr nc]=size(board);
nrnc=nr*nc;
if numtiles~=nrnc,return;end
% minimum required #tiles with a zero
if size(find( tiles==0),1)<2*(nr+nc-2),return;end % No solution
otiles=tiles;
[tv,orientation,otiles,valid]= ...
find_corner(tiles,otiles,orientation);
if ~valid,return;end
board(1)=tv; % 2 pts
for i=1:nr
if i>1
tile=board(i-1);
[board,orientation,otiles,valid]= ...
find_tiles(board,orientation,tiles,otiles,i,1,nrnc,tile,3);
% [board,orientation,otiles,valid]= ...
% first_column(board,orientation,otiles,tiles,i,j,nelem,tile,edge);
if ~valid,return;end
end % i>1
for j=2:nc
tile=board(i,j-1);
[board,orientation,otiles,valid]= ...
find_tiles(board,orientation,tiles,otiles,i,j,nrnc,tile,2);
if ~valid,return;end
end % j 2:nc
end % i 1:nr
solved=true;
end
function [tv,orientation,otiles,valid]= ...
find_corner(tiles,otiles,orientation)
valid=false;
cs=[4 1 2 3]; % 4 pts define outside
for i=1:4
adj= tiles(:,i) + tiles(:,cs(i) );
tv=find(adj==0,1); % Corner found
if ~isempty(tv)
orientation(tv)=i;
otiles(tv,:)=rot_tile(i,tiles(tv,:)); % or to i 2 pts
valid=true;
return; % corner found and rotated to position
end
end
% No corner found
end % find corner
function [one_tile]=rot_tile(rot,tile)
one_tile=circshift(tile,[0 1-rot]);
end
function [board,orientation,otiles,valid]= ...
find_tiles(board,orientation,tiles,otiles,i,j,nrnc,tile,ul)
% [board,orientation,otiles,valid]= ...
% first_column(board,orientation,otiles,tiles,i,j,nrnc,tile,UL);
valid=false; % could remove valid=false from returns
tvec=find(otiles==otiles(tile,ul));
if size(tvec,1)~=2,return;end % Non Unique solutions
tvecn1=tvec-1;
tnum=mod(tvecn1,nrnc)+1;
trot=floor((tvecn1)/nrnc)+1;
if tnum(1)~=tile % use
tv=tnum(1);
rot=trot(1);
else % use 2nd
tv=tnum(2);
rot=trot(2);
end
% for found pushed to L(4) 4:1,3:4,2:3,1:2
if j>1 % doing rows
rot = mod(rot,4)+1;
end
board(i,j)=tv;
otiles(tv,:)=rot_tile(rot,tiles(tv,:));
orientation(tv)=rot;
valid=true;
end
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