Selling Land
As you may know, the country of Absurdistan is full of abnormalities. For example, the whole
country can be divided into unit squares that are either grass or swamp. Also, the country
is famous for its incapable bureaucrats. If you want to buy a piece of land (called a parcel),
you can only buy a rectangular area, because they cannot handle other shapes. The price of
the parcel is determined by them and is proportional to the perimeter of the parcel, since the
bureaucrats are unable to multiply integers and thus cannot calculate the area of the parcel.
Per owns a parcel in Absurdistan surrounded by swamp and he wants to sell it, possibly in
parts, to some buyers. When he sells a rectangular part of his land, he is obliged to announce
this to the local bureaucrats. They will first tell him the price he is supposed to sell it for. Then
they will write down the name of the new owner and the coordinates of the south-east corner
of the parcel being sold. If somebody else already owns a parcel with a south-east corner at
the same spot, the bureaucrats will deny the change of ownership.
Per realizes that he can easily trick the system. He can sell overlapping areas, because
bureaucrats only check whether the south-east corners are identical. However, nobody wants
to buy a parcel containing swamp.
In this example, dark squares represent swamp. Per may, for example, sell three overlapping grey
areas, with dimensions 2 × 1, 2 × 4 and 4 × 1 respectively. The total perimeter is 6 + 12 + 10 = 28.
Note that he can get more money by selling even more land. This figure corresponds to the case in
the sample input.
Now Per would like to know how many parcels of each perimeter he needs to sell in order
to maximize his profit. Can you help him? You may assume that he can always find a buyer
for each piece of land, as long as it doesn’t contain any swamps. Also, Per is sure that no
square within his parcel is owned by somebody else.
Input
On the first line a positive integer: the number of test cases, at most 100. After that per test
case:
• One line with two integers n and m (1 ≤ n, m ≤ 1 000): the dimensions of Per’s parcel.
• n lines, each with m characters. Each character is either ‘#’ or ‘.’. The j-th character
on the i-th line is a ‘#’ if position (i, j) is a swamp, and ‘.’ if it is grass. The north-west
corner of Per’s parcel has coordinates (1, 1), and the south-east corner has coordinates
(n, m).
Output
Per test case:
• Zero or more lines containing a complete list of how many parcels of each perimeter
Per needs to sell in order to maximize his profit. More specifically, if Per should sell pi
parcels of perimeter i in the optimal solution, output a single line “pi x i”. The lines
should be sorted in increasing order of i. No two lines should have the same value of i,
and you should not output lines with pi = 0.
Sample in- and output
Input
1
6 5
..#.#
.#...
#..##
...#.
#....
#..#.
Output
6 x 4
5 x 6
5 x 8
3 x 10
1 x 12
比赛时就有思路，可惜时间不够了，最后阶段在冲刺另一题——可惜也是赛后几分钟才解决。。。
1 #include <stdio.h>
2 #include <string.h>
3
4 #define L 1009
5 #define T (L*4)
6
7 int n, m, cant[ L ][ L ], width[ L ][ L ], height[ L ][ L ];
8
9 void input() {
10 int i, j;
11 char tmp[ L ];
12 scanf( "%d%d", &n, &m );
13 gets( tmp );
14 for ( i = 1; i <= n; ++i ) {
15 gets( tmp );
16 for ( j = 0; j < m; ++j ) {
17 cant[ i ][ j+1 ] = ( tmp[ j ] == '#' );
18 }
19 }
20 }
21
22 void solve() {
23 int i, j, k, most, up[ L ], left[ L ], right[ L ];
24 memset( width, 0, sizeof(width) );
25 memset( height, 0, sizeof(height) );
26 for ( j = 1; j <= m; ++j ) {
27 up[ j ] = 1;
28 left[ j ] = 1;
29 right[ j ] = m;
30 }
31 for ( i = 1; i <= n; ++i ) {
32 most = 1;
33 for ( j = 1; j <= m; ++j ) {
34 if ( cant[ i ][ j ] ) {
35 up[ j ] = i + 1;
36 left[ j ] = 1;
37 most = j + 1;
38 }
39 else if ( most > left[ j ] ) {
40 left[ j ] = most;
41 }
42 }
43
44 most = m;
45 for ( j = m; j >= 1; --j ) {
46 if ( cant[ i ][ j ] ) {
47 most = j - 1;
48 right[ j ] = m;
49 }
50 else if ( right[ j ] > most ) {
51 right[ j ] = most;
52 }
53 }
54
55 up[ 0 ] = up[ 1 ] - 1;
56 for ( j = 1; j <= m; ++j ) {
57 if ( ( cant[ i ][ j ] ) || ( up[ j ] == up[ j - 1 ] ) ) {
58 continue;
59 }
60 for ( k = left[ j ]; k <= right[ j ]; ++k ) {
61 if ( (k-left[j]+1) + (i-up[j]+1) > width[i][k] + height[i][k] ) {
62 width[ i ][ k ] = k - left[ j ] + 1;
63 height[ i ][ k ] = i - up[ j ] + 1;
64 }
65 }
66 }
67 }
68 }
69
70 void output() {
71 int cnt[ T ], i, j;
72 memset( cnt, 0, sizeof(cnt) );
73 for ( i = 1; i <= n; ++i ) {
74 for ( j = 1; j <= m; ++j ) {
75 ++cnt[ ( width[i][j] + height[i][j] ) << 1 ];
76 }
77 }
78 for ( i = 1; i < T; ++i ) {
79 if ( cnt[ i ] > 0 ) {
80 printf( "%d x %d\n", cnt[ i ], i );
81 }
82 }
83 }
84
85 int main() {
86 int td;
87 scanf( "%d", &td );
88 while ( td-- > 0 ) {
89 input();
90 solve();
91 output();
92 }
93 return 0;
94 }
95