Periodic properties of matrices

Transcription

1 Periodic properties of matrices Technická univerzita Košice

2 References 1 E. Draženská, M. Molnárová, Periods of Monge matrices with zero-weight cycles, Proc. of the Conf. Informatics and Algorithms, Prešov (1998), M. Molnárová, Periods of matrices with zero-weight cycles in max-algebra, Tatra Mountains Math. Publ. 16 (1999), M. Molnárová, Computational complexity of Nachtigall s representation, Optimization 52 (2003), M. Molnárová, J. Pribiš, Matrix period in max-algebra, Discrete Appl. Math. 103 (2000), M. Molnárová Generalized matrix period in max-plus algebra, Linear Algebra and its Applications 404 (2005),

3 Content 1 Periodic properties of matrices Periodic matrices with zero weight cycles Sufficient and necessary condition for matrix periodicity Linear periodic matrices

4 Periodic matrices with zero weight cycles Periodic behaviour of matrix - Example 1 A = A 2 = A 3 = ε 1 ε ε ε 1 0 ε ε ε ε 0 1 ε ε ε 1 ε 0 ε ε ε 0 ε ε ε 0 = A 4 = A 5 = A 6.

5 Periodic matrices with zero weight cycles Periodic behaviour of matrix - Example 2 A= ε 1 ε ε ε ε ε ε 0 A 2 = 1 1 ε... A 4 = ε A 5 = = A A 6 = A 7 = = A 9 = A 10 ε

6 Periodic matrices with zero weight cycles Periodic behaviour of matrix - Example 3 A= ε 1 ε ε ε ε A 2 = A 5 = ε ε ε ε A 4 = = A 6 =... ε

7 Periodic matrices with zero weight cycles Periodic matrices - Definition Definition Let A G (n, n). We say, that A is almost periodic, if for all i, j is the sequence a ij = ( a (r) ij ; r N + ) almost periodic, i. e. ( p > 0) ( R) ( r > R) a (r+p) ij = a (r) ij. The smallest number p with above property is the period of aij with notation per(aij ). The period of A is defined as per(a) = lcm{ per(a ij); i, j N }.

8 Periodic matrices with zero weight cycles Zero-weight matrices Definition Let A G (n, n). We say that A fulfills the cycle-condition, if the weight of any cycle in the digraph G(A) is equal to zero, in notation w(c) = 0. A = ε 2 ε ε ε ε ε ε 1 ε ε ε 1 1 ε 5 ε ε ε ε ε ε 2 ε ε ε ε 2 ε 8 ε ε ε ε ε ε

9 Periodic matrices with zero weight cycles Period of strongly connected component - Definition Definition Let A G (n, n). The set of all non-trivial strongly connected components of G(A) is denoted by SCC (G(A)) and the set of all strongly connected components by SCC(G(A)). We denote by λ(k) the maximum mean weight of the cycles in K SCC(G(A)). We define the period of K SCC (G(A)) as per(k) = gcd { c ; c je cyklus v K, c > 0 }. If K is trivial, then per(k) = 0. A 11 = ε 2 ε ε ε ε per(k 1 ) = gcd { 2, 3 } = 1

10 Periodic matrices with zero weight cycles Sufficient condition for matrix periodicity Theorem Let A G (n, n) fulfill the cycle-condition. Let d N. Then the following assertions are equivalent (i) per(a) d, (ii) ( K SCC (G)) per(k) d. Theorem Let A G (n, n) fulfill the cycle-condition. Then per(a) = lcm { per(k); K SCC (G) }

11 Periodic matrices with zero weight cycles Algorithm for checking the sufficient condition for matrix periodicity Theorem Let A G (n, n).the cycle-condition can be verified in O(n 3 ) time. 1 find (A) 2 check δ ii 0, for all i 3 find (A ) (conjugated matrix) 4 check δ ii 0, for all i 0 min max 0 = equalities hold

12 Periodic matrices with zero weight cycles Algorithm for computing the matrix period Theorem Let A G (n, n) fulfill the cycle-condition. Then period per(a) can be computed in O(n 3 ) time. 1 find all strongly connected components in O(n 3 ) time by (A) (δ ij and δ ji finite) 2 compute the period of each s.c.c. in O(n 2 ) time by Balcer-Veinott algorithm 3 compute per(a) in O(n log n) time as the least common multiple of the periods of all non-trivial s.c.c. by Euclid algorithm for computation of the greatest common divisor gcd{a, b} ab lcm{a, b} = gcd{a, b}

13 Periodic matrices with zero weight cycles Period of matrix - Example Example: Check the cycle-condition and compute the period of the given matrix in positive case. ε 2 ε ε ε ε ε ε 1 ε ε ε A = 1 1 ε 5 ε ε ε ε ε ε 2 ε ε ε ε 2 ε 8 ε ε ε ε ε ε Solution: K 1 = {1, 2, 3} per(k 1 ) = gcd { 2, 3 } = 1 K 2 = {4, 5} per(k 2 ) = gcd { 2 } = 2 per(a) = lcm { per(k); K SCC (G) } = lcm { 2, 1 } = 2

14 Periodic matrices with zero weight cycles Period of matrix - Example A 10 = ɛ ɛ ε 0 ε 10 A 11 = ε ε ε ε 2 ε ε ε ε ε 0 ε ε ε ε 2 ε 8 ε ε ε ε ε ε ε ε ε ε ε ε A 12 = ε ε ε 0 ε 10 A 13 = ε ε ε ε 2 ε ε ε ε ε 0 ε ε ε ε 2 ε 8 ε ε ε ε ε ε ε ε ε ε ε ε

15 Sufficient and necessary condition for matrix periodicity Highly connected components A = ε 2 ε ε ε ε 2 ε 3 ε ε ε ε ε ε 0 ε ε ε ε 1 ε 1 0 ε ε 1 ε ε ε ε ε ε 1 ε ε G(A)

16 Sufficient and necessary condition for matrix periodicity Period of highly connected component - Definition Definition Let A G (n, n). We say that two nodes i, j G(A) are highly connected, in notation: i h j, if i, j are contained in a cycle c with maximal cycle mean value w(c) = λ(a). The subdigraphs induced by the equivalence classes of the reflexive hull of h are called highly connected components in G(A), the set of all such components is denoted by HCC(G(A)). A component K HCC(G(A)) is called trivial, if K contains no cycle of positive length with cycle mean value equal to λ(a). The set of all non-trivial components K HCC(G(A)) is denoted by HCC (G(A)). For any K HCC(G(A)), the high period of K is defined as hper(k) = gcd { c ; c is a cycle in K, c > 0, w(c) = λ(a) }. If K is trivial, then hper(k) = 0.

17 Sufficient and necessary condition for matrix periodicity Highly connected components A 22 = ε 0 ε ε 1 ε ε ε ε ε 1 ε ε G(A 22 )

18 Sufficient and necessary condition for matrix periodicity Sufficient and necessary condition for matrix periodicity Theorem Let A G (n, n), then the statements (i) A is almost periodic, (ii) ( K SCC (G(A)) ) λ(k) = 0 are equivalent. Theorem Let A G (n, n)is almost periodic. Then per(a) = lcm { hper(k); K HCC (G(A)) }

19 Sufficient and necessary condition for matrix periodicity Algorithm for checking periodicity and computing matrix period Theorem There is an O(n 3 ) algorithm A, which decides for a given matrix A G (n, n) whether A is almost periodic and computes per(a) in the positive case. 1 find all strongly connected components in O(n 3 ) time by (A) (δ ij and δ ji finite) 2 compute λ(k) for all non-trivial strongly connected components in O(n 3 ) time by Karp algorithm 3 find all highly connected components and their periods in O(n 3 ) time by Gavalec algorithm

20 Sufficient and necessary condition for matrix periodicity Algorithm for checking periodicity and computing matrix period 4 compute per(a) in O(n log n) time as the least common multiple of the periods of all non-trivial s.c.c. by Euclid algorithm for computation of the greatest common divisor gcd{a, b} ab lcm{a, b} = gcd{a, b}

21 Sufficient and necessary condition for matrix periodicity Gavalec algorithm for finding HCC and the high periods Theorem Let A G (n, n). Let K HCC (G(A)). If an arc e E(K) is not contained in a zero-cycle, then hper(k) = hper(k e). Theorem Let A G (n, n). Let K HCC (G(A)). If every arc e E(K) is contained in a zero-cycle, then the mean value of any cycle in K is zero and hper(k) = per(k). D: a ij + δ ji 0

22 Sufficient and necessary condition for matrix periodicity Gavalec algorithm for finding HCC and the high periods

23 Sufficient and necessary condition for matrix periodicity Period of matrix - Example Example: Check the periodicity and compute the period of the given matrix in positive case. ε 2 ε ε ε ε 2 ε 3 ε ε ε A = ε ε ε 0 ε ε ε ε 1 ε 1 0 ε ε 1 ε ε ε ε ε ε 1 ε ε Solution: K 1 = {1, 2} hper(k 1 ) = 2 K 2 = {3, 4, 5} hper(k 2 ) = 3 per(a) = lcm { hper(k); K HCC (G(A)) } = lcm { 2, 3 } = 6

24 Sufficient and necessary condition for matrix periodicity Period of matrix - Example ε ε A 9 = ε ε ε ε = A 15 =... ε ε ε ε ε ε A 10 = ε ε ε ε = A 16 =... ε ε ε ε

25 Linear periodic matrices Linear period of matrix - Example A = ε 0 ε ε ε ε 4 ε 5 ε ε ε ε ε ε 2 ε ε ε ε 1 ε 3 2 ε ε 1 ε ε ε ε ε ε 1 ε ε G(A)

26 Linear periodic matrices Linear period of matrix - Example ε ε A 9 = ε ε ε ε A 15 = ε ε ε ε ε ε A 10 = ε ε ε ε A 16 = ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε

27 Linear periodic matrices Linear period of matrix - Definition Definition Let A G (n, n). We say, that A is almost linear periodic, if for all i, j is the sequence a ij = ( a (r) ij ; r N + ) almost linear periodic, i. e. ( p > 0) ( R ij ) ( q ij R ) ( r > R ij ) a (r+p) ij = a (r) ij + p q ij. The smallest number p with above property is the linear period of aij with notation lper(aij ). The element q ij is the linear factor of aij with notation lfac(aij ). The smallest number R ij is the linear defect of aij with notation ldef(aij ). The linear period of A is defined as lper(a) = lcm{ lper(a ij); i, j N }. The matrix lfac(a) = (lfac(aij )) is called the linear factor matrix of A. The number ldef(a) = max{r ij } is called the linear defect of A.

28 Linear periodic matrices Sufficient condition for matrix linear periodicity Theorem Let A G (n, n). Let ( K SCC (G(A)) ) be λ(k) = λ(a). Then (i) A is almost linear periodic with lfac(a) = Q, q ij = λ(a), i, j, (ii) for linear periodicity of A holds lper(a) = lcm { hper(k); K HCC (G(A)) }. Theorem There is an O(n 3 ) algorithm A, which decides for a given matrix A G (n, n) whether A is almost linear periodic with a constant linear factor matrix and computes lper(a) in the positive case.

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