本文详细介绍了如何通过多种算法解决在有限区域内放置不同颜色矩形的问题，包括使用数组、堆、排序等数据结构，以及递归、分治策略等方法。文章还提供了代码实现，帮助读者理解并应用这些解决方案。
Shaping Regions
N opaque rectangles (1 <= N <= 1000) of various colors are placed on a white sheet of paper whose size is A wide by B long. The rectangles are put with their sides parallel to the sheet's borders. All rectangles fall within the borders of the sheet so that different figures of different colors will be seen.
The coordinate system has its origin (0,0) at the sheet's lower left corner with axes parallel to the sheet's borders.
PROGRAM NAME: rect1
INPUT FORMAT
The order of the input lines dictates the order of laying down the rectangles. The first input line is a rectangle "on the bottom".
| Line 1: | A, B, and N, space separated (1 <= A,B <= 10,000) |
| Lines 2-N+1: | Five integers: llx, lly, urx, ury, color: the lower left coordinates and upper right coordinates of the rectangle whose color is `color' (1 <= color <= 2500) to be placed on the white sheet. The color 1 is the same color of white as the sheet upon which the rectangles are placed. |
SAMPLE INPUT (file rect1.in)
20 20 3
2 2 18 18 2
0 8 19 19 3
8 0 10 19 4
INPUT EXPLANATION
Note that the rectangle delineated by 0,0 and 2,2 is two units wide and two high. Here's a schematic diagram of the input:11111111111111111111
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
33333333443333333331
11222222442222222211
11222222442222222211
11222222442222222211
11222222442222222211
11222222442222222211
11222222442222222211
11111111441111111111
11111111441111111111The '4's at 8,0 to 10,19 are only two wide, not three (i.e., the grid contains a 4 and 8,0 and a 4 and 8,1 but NOT a 4 and 8,2 since this diagram can't capture what would be shown on graph paper).
OUTPUT FORMAT
The output file should contain a list of all the colors that can be seen along with the total area of each color that can be seen (even if the regions of color are disjoint), ordered by increasing color. Do not display colors with no area.SAMPLE OUTPUT (file rect1.out)
1 91
2 84
3 187
4 38
Analysis by Mathijs Vogelzang
A straightforward approach to this problem would be to make an array which represents the table, and then draw all the rectangles on it. In the end, the program can just count the colors from the array and output them. Unfortunately, the maximum dimensions of this problem are 10,000 x 10,000, which means the program uses 100 million integers. That's too much, so we need another approach.
An approach that does work for such large cases (and it actually is a lot faster too) is to keep track of the rectangles, and delete portions of them when they are covered by other rectangles.
Consider this input set:
0 0 10 10 5
5 0 15 10 10
The program first reads in the first rectangle and puts it in a list. When it reads a new rectangle it checks all items in the list if they overlap with the new rectangle. This is the case, and then it deletes the old rectangle from the list and adds all parts which aren't covered to the list. (So in this case, the program would delete the first rectangle, add 0 0 5 10 5 to the list and then add the second rectangle to the list). ``
If you're unlucky, a new rectangle can create lots of new rectangles (when the new rectangle entirely fits into another one, the program creates four new rectangles which represent the leftover border:
+--------+ +-+--+--+
| | | |2 | |
| | + +--+ |
| +-+ | --> | | | |
| +-+ | |1| |3 |
| | | +--+ |
| | | | 4| |
+--------+ +-+--+--+
This is not a problem however, because there can be only 2500 rectangles and there is plenty of memory, so rectangles have to be cut very much to run out of memory.
Note that with this approach, the only thing that matters is how many rectangles there are and how often they overlap. The maximum dimensions can be as large as you want, it doesn't matter for the running time.
Further Analysis by Tomek Czajka
There is another solution to this problem, which runs in O(n*n*log n) time, but is quite tricky. First, we add one big white rectangle at the bottom - the paper. Then we make two arrays: one containing all vertical edges of the rectangles, and the other the horizontal ones. For each edge we have its coordinates and remember, whether it's the left or right edge (top or bottom). We sort these edges from left to right and from top to bottom. Then we go from left to right (sweep), jumping to every x-coordinate of vertical edges. At each step we update the set of rectangles seen. We also want to update the amount of each color seen so far. So for each x we go from top to bottom, for each y updating the set of rectagles at a point (in the structure described below) and choosing the top one, so that we can update the amounts of colors seen.
The structure to hold the set of rectangles at a point should allow adding a rectangle (number from 1..1000), deleting a rectangle, and finding the top one. We can implement these operations in O(log n) time if we use a heap. To make adding and deleting run in O(log n) we must also have for each rectangle its position in the heap.
So the total time spent at each point is O(log n). Thus the algorithm works in O(n*n*log n) time.
And a solution from mrsigma:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
FILE *fp,*fo;
struct rect
{
int c;
int x1,y1,x2,y2;
};
int c[2501];
rect r[10001];
int intersect(rect a,const rect &b,rect out[4])
{
/* do they at all intersect? */
if(b.x2<a.x1||b.x1>=a.x2)
return 0;
if(b.y2<a.y1||b.y1>=a.y2)
return 0;
/* they do */
rect t;
if(b.x1<=a.x1&&b.x2>=a.x2&&b.y1<=a.y1&&b.y2>=a.y2)
return -1;
/* cutting `a' down to match b */
int nout=0;
if(b.x1>=a.x1) {
t=a,t.x2=b.x1;
if(t.x1!=t.x2)
out[nout++]=t;
a.x1=b.x1;
}
if(b.x2<a.x2) {
t=a,t.x1=b.x2;
if(t.x1!=t.x2)
out[nout++]=t;
a.x2=b.x2;
}
if(b.y1>=a.y1) {
t=a,t.y2=b.y1;
if(t.y1!=t.y2)
out[nout++]=t;
a.y1=b.y1;
}
if(b.y2<a.y2) {
t=a,t.y1=b.y2;
if(t.y1!=t.y2)
out[nout++]=t;
a.y2=b.y2;
}
return nout;
}
int main(void) {
fp=fopen("rect1.in","rt");
fo=fopen("rect1.out","wt");
int a,b,n;
fscanf(fp,"%d %d %d",&a,&b,&n);
r[0].c=1;
r[0].x1=r[0].y1=0;
r[0].x2=a;
r[0].y2=b;
rect t[4];
int i,j,rr=1;
for(i=0;i<n;i++) {
int tmp;
fscanf(fp,"%d %d %d %d %d",&r[rr].x1,&r[rr].y1,&r[rr].x2,&r[rr].y2,&r[rr].c);
if(r[rr].x1>r[rr].x2) {
tmp=r[rr].x1;
r[rr].x1=r[rr].x2;
r[rr].x2=tmp;
}
if(r[rr].y1>r[rr].y2) {
tmp=r[rr].y1;
r[rr].y1=r[rr].y2;
r[rr].y2=tmp;
}
int nr=rr;
rect curr=r[rr++];
for(j=0;j<nr;j++) {
int n=intersect(r[j],curr,t);
if(!n)
continue;
if(n==-1) {
memmove(r+j,r+j+1,sizeof(rect)*(rr-j-1));
j--;
rr--;
nr--;
continue;
}
r[j]=t[--n];
for(;n-->0;)
r[rr++]=t[n];
}
}
for(i=0;i<rr;i++)
c[r[i].c]+=(r[i].x2-r[i].x1)*(r[i].y2-r[i].y1);
for(i=1;i<=2500;i++)
if(c[i])
fprintf(fo,"%d %d\n",i,c[i]);
return 0;
}
And another fast solution from Saber Fadaee:
In Shaping Regions, I changed the whole A*B page into (2*N) * (2*N).
Program rrect1;
Var
Inf,Outf : Text;
A,B,N,I,Z,Middle,J : Longint;
Color : Array [1..2500] of Longint;
D : Array [1..5000,1..5000] of boolean;
Xar,Yar : Array [0..2500] of Longint;
Col : Array [1..10000] of Record
x1 : Longint;
x2 : Longint;
y1 : Longint;
y2 : Longint;
c : Longint;
End;
Function Find (K1 : integer) : Longint;
Var
Pointer,N1,N2 : Longint;
Begin
N1 := 1;
N2 := N + N;
While N1 > 0 Do Begin
Pointer := (N1 + N2) Div 2;
If Xar[Pointer] = K1 then Begin
Find := Pointer;
Exit;
End;
If Xar[Pointer] > K1 Then
N2 := Pointer - 1;
If Xar[Pointer] < K1 Then
N1 := Pointer + 1;
End;
End;
Function Find1 (K2 : Longint) : Longint;
Var
Pointer,N1,N2 : Longint;
Begin
N1 := 1;
N2 := N + N;
While N1 > 0 Do Begin
Pointer := (N1 + N2) Div 2;
If Yar[Pointer] = K2 then Begin
Find1 := Pointer;
Exit;
End;
If Yar[Pointer] > K2 Then
N2 := Pointer - 1;
If Yar[Pointer] < K2 Then
N1 := Pointer + 1;
End;
End;
Procedure Partition1 ( Lf , Rg : Longint );
Var
Pivot,L,R,Temp : Longint;
Begin
Pivot := Yar[Lf];
L := Lf;
R := Rg;
While L < R Do Begin
While (Yar[L] <= Pivot) and (L <= R) Do Inc(L);
While (Yar[R] > Pivot) And (R >= L) Do Dec(R);
If L < R Then
begin
Temp := Yar[L];
Yar[L] := Yar[R];
Yar[R] := Temp;
end;
End;
Middle := R;
Temp := Yar[Lf];
Yar[Lf] := Yar[R];
Yar[R] := Temp;
End;
Procedure QSort1 ( Left , Right : Longint );
Begin
if Left < Right Then
Begin
Partition1 (Left,Right);
QSort1 (Left,Middle-1);
QSort1 (Middle + 1, Right);
End;
End;
Procedure Partition ( Lf , Rg : Longint );
Var
Pivot,L,R,Temp : Longint;
Begin
Pivot := Xar[Lf];
L := Lf;
R := Rg;
While L < R Do
Begin
While (Xar[L] <= Pivot) and (L <= R) Do Inc(L);
While (Xar[R] > Pivot) And (R >= L) Do Dec(R);
If L < R Then
begin
Temp := Xar[L];
Xar[L] := Xar[R];
Xar[R] := Temp;
end;
End;
Middle := R;
Temp := Xar[Lf];
Xar[Lf] := Xar[R];
Xar[R] := Temp;
End;
Procedure QSort ( Left , Right : Longint );
Begin
if Left < Right Then
Begin
Partition (Left,Right);
QSort (Left,Middle-1);
QSort (Middle + 1, Right);
End;
End;
Begin
Assign (Inf ,'rect1.in');
Reset (Inf);
Readln (Inf,A,B,N);
For I := 1 To N Do
Readln (Inf , Col[I].x1,col[i].y1,col[i].x2,col[i].y2,col[i].c);
Close (Inf);
For I := 1 to 2500 do Color[I] := 0;
Color[1] := A * B;
For I := 0 to n do
For J := 0 to n do
D[I,J] := False;
Xar[0] := 0;
Xar[n+n+1] := a;
For I := 1 To N do Xar[i] := Col[i].x1;
For I := N + 1 To N + N do Xar[i] := Col[i - n].x2;
Qsort (1,N + N);
Yar[0] := 0;
Yar[n+n+1] := b;
For I := 1 To N Do Yar[i] := Col[i].y1;
For I := N + 1 To N + N do Yar[i] := col[i - n].y2;
Qsort1 (1,N + N);
For I := N Downto 1 Do
For J := find(Col[i].x1) + 1 to find(col[i].x2) do
For Z := find1(col[i].y1) + 1 to find1(col[i].y2) do
If not D[J,Z] then
Begin
If col[i].c > 1 then
Begin
Middle := (Xar[j] - Xar[j-1]) * (Yar[z] - Yar[z-1]);
Color[Col[i].c] := color[Col[i].c] + Middle;
Color[1] := Color[1] - Middle;
End;
D[J,Z] := True;
End;
Assign (Outf, 'rect1.out');
Rewrite (Outf);
For I := 1 To 2500 Do
if color[i] > 0 then
Writeln (Outf,i,' ', Color[i]);
Close(Outf);
End.
And, finally, an unbelievably compact recursive solution from Christoph Roick:
This solution uses recursion. We start with painting the last rectangle and go through to the first (the white paper). We save the edges of every rectangle, because we shouldn't paint over the rectangles above it. Now we have to divide the rectangles in 0 to 4 pieces around the rectangles covering it. In the end we have a lot of small rectangles and the colors can be added to an array by calculating the areas of the rectangle. So, we won't get any problems concerning too less memory.
Program rrect1;
Var
Inf,Outf : Text;
A,B,N,I,Z,Middle,J : Longint;
Color : Array [1..2500] of Longint;
D : Array [1..5000,1..5000] of boolean;
Xar,Yar : Array [0..2500] of Longint;
Col : Array [1..10000] of Record
x1 : Longint;
x2 : Longint;
y1 : Longint;
y2 : Longint;
c : Longint;
End;
Function Find (K1 : integer) : Longint;
Var
Pointer,N1,N2 : Longint;
Begin
N1 := 1;
N2 := N + N;
While N1 > 0 Do Begin
Pointer := (N1 + N2) Div 2;
If Xar[Pointer] = K1 then Begin
Find := Pointer;
Exit;
End;
If Xar[Pointer] > K1 Then
N2 := Pointer - 1;
If Xar[Pointer] < K1 Then
N1 := Pointer + 1;
End;
End;
Function Find1 (K2 : Longint) : Longint;
Var
Pointer,N1,N2 : Longint;
Begin
N1 := 1;
N2 := N + N;
While N1 > 0 Do Begin
Pointer := (N1 + N2) Div 2;
If Yar[Pointer] = K2 then Begin
Find1 := Pointer;
Exit;
End;
If Yar[Pointer] > K2 Then
N2 := Pointer - 1;
If Yar[Pointer] < K2 Then
N1 := Pointer + 1;
End;
End;
Procedure Partition1 ( Lf , Rg : Longint );
Var
Pivot,L,R,Temp : Longint;
Begin
Pivot := Yar[Lf];
L := Lf;
R := Rg;
While L < R Do Begin
While (Yar[L] <= Pivot) and (L <= R) Do Inc(L);
While (Yar[R] > Pivot) And (R >= L) Do Dec(R);
If L < R Then
begin
Temp := Yar[L];
Yar[L] := Yar[R];
Yar[R] := Temp;
end;
End;
Middle := R;
Temp := Yar[Lf];
Yar[Lf] := Yar[R];
Yar[R] := Temp;
End;
Procedure QSort1 ( Left , Right : Longint );
Begin
if Left < Right Then
Begin
Partition1 (Left,Right);
QSort1 (Left,Middle-1);
QSort1 (Middle + 1, Right);
End;
End;
Procedure Partition ( Lf , Rg : Longint );
Var
Pivot,L,R,Temp : Longint;
Begin
Pivot := Xar[Lf];
L := Lf;
R := Rg;
While L < R Do
Begin
While (Xar[L] <= Pivot) and (L <= R) Do Inc(L);
While (Xar[R] > Pivot) And (R >= L) Do Dec(R);
If L < R Then
begin
Temp := Xar[L];
Xar[L] := Xar[R];
Xar[R] := Temp;
end;
End;
Middle := R;
Temp := Xar[Lf];
Xar[Lf] := Xar[R];
Xar[R] := Temp;
End;
Procedure QSort ( Left , Right : Longint );
Begin
if Left < Right Then
Begin
Partition (Left,Right);
QSort (Left,Middle-1);
QSort (Middle + 1, Right);
End;
End;
Begin
Assign (Inf ,'rect1.in');
Reset (Inf);
Readln (Inf,A,B,N);
For I := 1 To N Do
Readln (Inf , Col[I].x1,col[i].y1,col[i].x2,col[i].y2,col[i].c);
Close (Inf);
For I := 1 to 2500 do Color[I] := 0;
Color[1] := A * B;
For I := 0 to n do
For J := 0 to n do
D[I,J] := False;
Xar[0] := 0;
Xar[n+n+1] := a;
For I := 1 To N do Xar[i] := Col[i].x1;
For I := N + 1 To N + N do Xar[i] := Col[i - n].x2;
Qsort (1,N + N);
Yar[0] := 0;
Yar[n+n+1] := b;
For I := 1 To N Do Yar[i] := Col[i].y1;
For I := N + 1 To N + N do Yar[i] := col[i - n].y2;
Qsort1 (1,N + N);
For I := N Downto 1 Do
For J := find(Col[i].x1) + 1 to find(col[i].x2) do
For Z := find1(col[i].y1) + 1 to find1(col[i].y2) do
If not D[J,Z] then
Begin
If col[i].c > 1 then
Begin
Middle := (Xar[j] - Xar[j-1]) * (Yar[z] - Yar[z-1]);
Color[Col[i].c] := color[Col[i].c] + Middle;
Color[1] := Color[1] - Middle;
End;
D[J,Z] := True;
End;
Assign (Outf, 'rect1.out');
Rewrite (Outf);
For I := 1 To 2500 Do
if color[i] > 0 then
Writeln (Outf,i,' ', Color[i]);
Close(Outf);
End.
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