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matrix power
By bayram, 11 years ago, In English

Can anyone explain O(log(n)) solution for this problem with matrix power.
Thanks for your helping!

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11 years ago, # |
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s[i] = (a, b, c, d)
=>
s[i + 1].a = s[i].b + s[i].c + s[i].d
s[i + 1].b = s[i].a + s[i].c + s[i].d
s[i + 1].c = s[i].a + s[i].b + s[i].d
s[i + 1].d = s[i].a + s[i].b + s[i].c
=>
s[i + 1]  = s[i] * ( 0 1 1 1 ) = s[i] * M
                   ( 1 0 1 1 )
                   ( 1 1 0 1 )
                   ( 1 1 1 0 )
=>
s[i + k] = s[i] * (M pow k)
=>
s[0].d = 1;
s[n] = s[0] * (M pow n)
ans = s[n].d
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11 years ago, # |
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The fastest matrix multiplication algorithm is Strassen_algorithm then you have to apply Binary Exponentiation .

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11 years ago, # |
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Tetrahedron can be represented as a graph with 4 vertices and problem is to count number of ways with fixed length K. There is an explanation of the solution with matrix exponentiation: here

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