TY - CHAP
T1 - How Much Different Are Two Words with Different Shortest Periods
AU - Alzamel, Mai
AU - Crochemore, Maxime
AU - Iliopoulos, Costas S.
AU - Kociumaka, Tomasz
AU - Kundu, Ritu
AU - Radoszewski, Jakub
AU - Rytter, Wojciech
AU - Wale, Tomasz
PY - 2018
Y1 - 2018
N2 - Sometimes the difference between two distinct words of the same length cannot be smaller than a certain minimal amount. In particular if two distinct words of the same length are both periodic or quasiperiodic, then their Hamming distance is at least 2. We study here how the minimum Hamming distance dist (x,y)dist(x,y)between two words x, y of the same length n depends on their periods. Similar problems were considered in [1] in the context of quasiperiodicities. We say that a period p of a word x is primitive if x does not have any smaller period p'ptextasciiacutexwhich divides p. For integers p, n (pbackslashle np≤n) we define backslashmathcal Pp(n)Pp(n)as the set of words of length n with primitive period p. We show several results related to the following functions introduced in this paper for pbackslashne qp≠qand n backslashge backslashmax (p,q)n≥max(p,q). backslashbeginaligned backslashmathcal Dp,q(n) = backslashmin backslash,backslashbackslash, dist (x,y)backslash,:backslash; xbackslashin backslashmathcal Pp(n), backslash,ybackslashin backslashmathcal Pq(n)backslash,backslash, backslashbackslash Np,q(h) = backslashmax backslash,backslashbackslash, n backslash,:backslash; backslashmathcal Dp,q(n)backslashle hbackslash,backslash. backslashqquad backslashqquad backslashendalignedDp,q(n)=mindist(x,y):x∈Pp(n),y∈Pq(n),Np,q(h)=maxn:Dp,q(n)≤h.
AB - Sometimes the difference between two distinct words of the same length cannot be smaller than a certain minimal amount. In particular if two distinct words of the same length are both periodic or quasiperiodic, then their Hamming distance is at least 2. We study here how the minimum Hamming distance dist (x,y)dist(x,y)between two words x, y of the same length n depends on their periods. Similar problems were considered in [1] in the context of quasiperiodicities. We say that a period p of a word x is primitive if x does not have any smaller period p'ptextasciiacutexwhich divides p. For integers p, n (pbackslashle np≤n) we define backslashmathcal Pp(n)Pp(n)as the set of words of length n with primitive period p. We show several results related to the following functions introduced in this paper for pbackslashne qp≠qand n backslashge backslashmax (p,q)n≥max(p,q). backslashbeginaligned backslashmathcal Dp,q(n) = backslashmin backslash,backslashbackslash, dist (x,y)backslash,:backslash; xbackslashin backslashmathcal Pp(n), backslash,ybackslashin backslashmathcal Pq(n)backslash,backslash, backslashbackslash Np,q(h) = backslashmax backslash,backslashbackslash, n backslash,:backslash; backslashmathcal Dp,q(n)backslashle hbackslash,backslash. backslashqquad backslashqquad backslashendalignedDp,q(n)=mindist(x,y):x∈Pp(n),y∈Pq(n),Np,q(h)=maxn:Dp,q(n)≤h.
U2 - 10.1007/978-3-319-92016-0_16
DO - 10.1007/978-3-319-92016-0_16
M3 - Chapter
SN - 9783319920160
SP - 168
EP - 178
BT - Artificial Intelligence Applications and Innovations
A2 - Iliadis, Lazaros
A2 - Maglogiannis, Ilias
A2 - Plagianakos, Vassilis
PB - Springer International Publishing
CY - Cham
ER -