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/* PBN, Paint-By-Numbers Puzzle */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* A paint-by-number puzzle consists of an m*n grid of pixels (the
canvas) together with m+n cluster-size sequences, one for each row
and column. The goal is to paint the canvas with a picture that
satisfies the following constraints:
1. Each pixel must be blank or white.
2. If a row or column has cluster-size sequence s1, s2, ..., sk,
then it must contain k clusters of black pixels - the first with
s1 black pixels, the second with s2 black pixels, and so on.
It should be noted that "first" means "leftmost" for rows and
"topmost" for columns, and that rows and columns need not begin or
end with black pixels.
Example:
1 1
1 1
2 1 1 1 1 1 2 3
3 2 1 2 1 2 3 4 8 9
3 6 # # # . # # # # # #
1 4 # . . . . . # # # #
1 1 3 . . # . # . . # # #
2 . . . . . . . . # #
3 3 . . # # # . . # # #
1 4 # . . . . . # # # #
2 5 # # . . . # # # # #
2 5 # # . . . # # # # #
1 1 . . . # . . . . . #
3 . . # # # . . . . .
(In Russia this sort of puzzles is known as "Japanese crossword".)
References:
Robert A. Bosch, "Painting by Numbers", 2000.
<http://www.oberlin.edu/~math/faculty/bosch/pbn-page.html> */
param m, integer, >= 1;
/* the number of rows */
param n, integer, >= 1;
/* the number of columns */
param row{i in 1..m, 1..n div 2}, integer, >= 0, default 0;
/* the cluster-size sequence for row i (raw data) */
param col{j in 1..n, 1..m div 2}, integer, >= 0, default 0;
/* the cluster-size sequence for column j (raw data) */
param kr{i in 1..m} := sum{t in 1..n div 2: row[i,t] > 0} 1;
/* the number of clusters in row i */
param kc{j in 1..n} := sum{t in 1..m div 2: col[j,t] > 0} 1;
/* the number of clusters in column j */
param sr{i in 1..m, t in 1..kr[i]} := row[i,t], integer, >= 1;
/* the cluster-size sequence for row i */
param sc{j in 1..n, t in 1..kc[j]} := col[j,t], integer, >= 1;
/* the cluster-size sequence for column j */
check{i in 1..m}: sum{t in 1..kr[i]} sr[i,t] <= n - (kr[i] - 1);
/* check that the sum of the cluster sizes in each row is valid */
check{j in 1..n}: sum{t in 1..kc[j]} sc[j,t] <= m - (kc[j] - 1);
/* check that the sum of the cluster sizes in each column is valid */
check: sum{i in 1..m, t in 1..kr[i]} sr[i,t] =
sum{j in 1..n, t in 1..kc[j]} sc[j,t];
/* check that the sum of the cluster sizes in all rows is equal to the
sum of the cluster sizes in all columns */
param er{i in 1..m, t in 1..kr[i]} :=
if t = 1 then 1 else er[i,t-1] + sr[i,t-1] + 1;
/* the smallest value of j such that row i's t-th cluster can be
placed in row i with its leftmost pixel occupying pixel j */
param lr{i in 1..m, t in 1..kr[i]} :=
if t = kr[i] then n + 1 - sr[i,t] else lr[i,t+1] - sr[i,t] - 1;
/* the largest value of j such that row i's t-th cluster can be
placed in row i with its leftmost pixel occupying pixel j */
param ec{j in 1..n, t in 1..kc[j]} :=
if t = 1 then 1 else ec[j,t-1] + sc[j,t-1] + 1;
/* the smallest value of i such that column j's t-th cluster can be
placed in column j with its topmost pixel occupying pixel i */
param lc{j in 1..n, t in 1..kc[j]} :=
if t = kc[j] then m + 1 - sc[j,t] else lc[j,t+1] - sc[j,t] - 1;
/* the largest value of i such that column j's t-th cluster can be
placed in column j with its topmost pixel occupying pixel i */
var z{i in 1..m, j in 1..n}, binary;
/* z[i,j] = 1, if row i's j-th pixel is painted black
z[i,j] = 0, if row i's j-th pixel is painted white */
var y{i in 1..m, t in 1..kr[i], j in er[i,t]..lr[i,t]}, binary;
/* y[i,t,j] = 1, if row i's t-th cluster is placed in row i with its
leftmost pixel occupying pixel j
y[i,t,j] = 0, if not */
var x{j in 1..n, t in 1..kc[j], i in ec[j,t]..lc[j,t]}, binary;
/* x[j,t,i] = 1, if column j's t-th cluster is placed in column j with
its topmost pixel occupying pixel i
x[j,t,i] = 0, if not */
s.t. fa{i in 1..m, t in 1..kr[i]}:
sum{j in er[i,t]..lr[i,t]} y[i,t,j] = 1;
/* row i's t-th cluster must appear in row i exactly once */
s.t. fb{i in 1..m, t in 1..kr[i]-1, j in er[i,t]..lr[i,t]}:
y[i,t,j] <= sum{jp in j+sr[i,t]+1..lr[i,t+1]} y[i,t+1,jp];
/* row i's (t+1)-th cluster must be placed to the right of its t-th
cluster */
s.t. fc{j in 1..n, t in 1..kc[j]}:
sum{i in ec[j,t]..lc[j,t]} x[j,t,i] = 1;
/* column j's t-th cluster must appear in column j exactly once */
s.t. fd{j in 1..n, t in 1..kc[j]-1, i in ec[j,t]..lc[j,t]}:
x[j,t,i] <= sum{ip in i+sc[j,t]+1..lc[j,t+1]} x[j,t+1,ip];
/* column j's (t+1)-th cluster must be placed below its t-th cluster */
s.t. fe{i in 1..m, j in 1..n}:
z[i,j] <= sum{t in 1..kr[i], jp in er[i,t]..lr[i,t]:
j-sr[i,t]+1 <= jp and jp <= j} y[i,t,jp];
/* the double coverage constraint stating that if row i's j-th pixel
is painted black, then at least one of row i's clusters must be
placed in such a way that it covers row i's j-th pixel */
s.t. ff{i in 1..m, j in 1..n}:
z[i,j] <= sum{t in 1..kc[j], ip in ec[j,t]..lc[j,t]:
i-sc[j,t]+1 <= ip and ip <= i} x[j,t,ip];
/* the double coverage constraint making sure that if row i's j-th
pixel is painted black, then at least one of column j's clusters
covers it */
s.t. fg{i in 1..m, j in 1..n, t in 1..kr[i], jp in er[i,t]..lr[i,t]:
j-sr[i,t]+1 <= jp and jp <= j}: z[i,j] >= y[i,t,jp];
/* the constraint to prevent white pixels from being covered by the
row clusters */
s.t. fh{i in 1..m, j in 1..n, t in 1..kc[j], ip in ec[j,t]..lc[j,t]:
i-sc[j,t]+1 <= ip and ip <= i}: z[i,j] >= x[j,t,ip];
/* the constraint to prevent white pixels from being covered by the
column clusters */
/* there is no need for an objective function here */
solve;
for {i in 1..m}
{ printf{j in 1..n} " %s", if z[i,j] then "#" else ".";
printf "\n";
}
data;
/* These data correspond to the example above. */
param m := 10;
param n := 10;
param row : 1 2 3 4 :=
1 3 6 . .
2 1 4 . .
3 1 1 3 .
4 2 . . .
5 3 3 . .
6 1 4 . .
7 2 5 . .
8 2 5 . .
9 1 1 . .
10 3 . . . ;
param col : 1 2 3 4 :=
1 2 3 . .
2 1 2 . .
3 1 1 1 1
4 1 2 . .
5 1 1 1 1
6 1 2 . .
7 2 3 . .
8 3 4 . .
9 8 . . .
10 9 . . . ;
end;
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