ASCII Exact Cover with Rectangles

Question

Challenge

Given a rectangular area arrange a group of rectangles such that they cover the rectangular area entirely.

Input

• An integer denoting the height.

• An integer denoting the width.

• The dimensions of the rectangles consisting of the following form: axb,cxd,... where a,b,c, and d are integers - any reasonable format is acceptable.

Output

An exact cover of the rectangular area.

Rectangles used to cover the area are represented in the following way:

2x3

00
00
00

or

000
000

1x3

0
0
0

or

000

Each rectangle is represented using a character 0-9 where 0 is used for the first rectangle inputted 1 for the second and so on. The max number of rectangles given in input is therefore 10.

Test Cases

Input 1

5
7
4x2,3x2,3x2,5x2,5x1

Output 1

0000111
0000111
2233333
2233333
2244444

Input 2

4
4
3x2,2x1,2x2,2x2

Output 2

0001
0001
2233
2233

Input 3

2
10
4x2,3x2,3x2

Output 3

0000111222
0000111222

Clarifications

• The dimensions of the rectangles used to cover the region are interchangeable e.g. for axb a could be the height or width same goes for b.
• If there are multiple solutions any of them is acceptable (just display 1).
• Assume all inputs are valid.
• You can have a full program or a function (inputs may be treated broadly as 3 separate arguments).
• Output to stdout (or something similar).
• This is code-golf so shortest answer in bytes wins.

Answer 1

Python 3, 262 bytes

from numpy import*
def g(h,w,t,m,i):
 if[]==t:print(m);return
 u,v=t[0]
 for _ in'12':
  for r,c in argwhere(m<0):
   if(m[r:r+u,c:c+v]<0).all()and u+r<=h and v+c<=w:e=copy(m);e[r:r+u,c:c+v]=i;g(h,w,t[1:],e,i+1)
  u,v=v,u
f=lambda h,w,t:g(h,w,t,zeros((h,w))-1,0)

Called like: f(4,4,[(3,2),(2,1),(2,2),(2,2)]). It uses a brute force, recursive function that prints all solutions (uses fewer bytes).