2

I'm looking for some sort of algorithm so I can quickly identify similar matrices, the matrices are not stored permanently so I'd need a way of mapping each matrix to an easily stored values, after which I can use the same mapping on future matrices and quickly compare against the calculated values.

The way I'm defining a similar matrix is simply whether or not they are identical when mirrored by vertical line, horizontal line, major diagonal and minor diagonal.

For example

1 2 3
4 5 6
7 8 9

is identified as the same as all the following matrices

3 2 1
6 5 4
9 8 7
-----
7 8 9
4 5 6
1 2 3
-----
1 4 7
2 5 8
3 6 9
-----
9 6 3
8 5 2
7 4 1

I've been stumped on this problem for days now, the matrices aren't stored permanently therefore comparing them directly isn't possible. Could anyone point me in the right direction? Thanks

rhysw
  • 21

4 Answers4

1

Calculate a hash value for each of the eight symmetric matrixes and keep only the smallest of them - that should do the trick. This gives you the same value for each matrix of an equivalence class.

Doc Brown
  • 218,378
0

Instead of storing one hash value, store 5 hash values 

M0= 1 2 3
    4 5 6
    7 8 9

is identified as the same as all the following matrices

Operations which creates M1 to M4 are written below

M1= 3 2 1  (vertical mirror  M1:= VM(M0) )
    6 5 4
    9 8 7
-----
M2= 7 8 9 (horizontal mirror M2:= HM(M0) )
    4 5 6
    1 2 3
-----
M3= 1 4 7 (transpose M3= T(M0) )
    2 5 8
    3 6 9
-----
M4=     9 6 3 (M4= VM(T(MO) )
    8 5 2
    7 4 1

all your equivalent matrices (total 7 in addition to the original) can be written as combinations of VM(), HM(), and T().

so if you hold 8 hash values for a matrix, you can compare your hash values.

To calculate a hash value of the matrix M0 (numbers are only digits in your case), you can convert it to the string "123456789" and use a hash function.

I hope this helps.

Umut Kahramankaptan
  • 129
0

You can use mathematics to identify similarity. I have assumed each example you have given after the initial matrix for similarity as condition.

For the matrix in your example - 

Let Ux = 1 2 3
         4 5 6
         7 8 9

Let Ix = 0 0 1
         0 1 0
         1 0 0

If you perform the matrix multiplication of Ux and Ix you would obtain - 

Ux . Ix = 3 2 1
          6 5 4
          9 8 7

Which is condition 1.

If you do

Ix . Ux = 7 8 9
          4 5 6
          1 2 3

Which is condition 2. So, you could use Ix to identify two conditions of similarity you have defined.

For the condition 3 it is Transpose(Ux).

For the condition 4 it is Trasnpose(Ix . Ux . Ix).

Hope this helps!

Siva Senthil
  • 206
0

Based on some back-of-the-envelope thinking, I would suggest the following approach: If you sum each individual row, each column, each diagonal, and the 4 corners, you will obtain a set of 9 numbers, which can then be sorted. This sorted list of numbers will be identical for all the cases you listed and their rotations, but should be unique (i.e., not collide with any other arrangements). If this analysis is true, you could use a trivial hash function of that 9-number list as your stored value.

(Of course, if rotations are excluded from your similarity classification, then I will have to go back to the drawing board.)

kmote
  • 3,372