Springer Nature
version
time
Lempel-Ziv compression
complexity
properties
2022-08-04T17:21
binaries
approach
polynomial time
problem
solvability
length
https://scigraph.springernature.com/explorer/license/
length of variables
function
first main result
1998
relation
https://doi.org/10.1007/bfb0055097
LZ
intricate algorithms
encoding
main results
variables
new approach
words
second main result
Application of Lempel-Ziv encodings to the solution of word equations
compression
first solution
exponential function
false
terms
1998-01-01
applications
deterministic time
731-742
One of the most intricate algorithms related to words is Makanin's algorithm solving word equations. The algorithm is very complicated and the complexity of the problem of solving word equations is not well understood. Word equations can be used to define various properties of strings, e.g. general versions of pattern-matching with variables. This paper is devoted to introduce a new approach and to study relations between Lempel-Ziv compressions and word equations. Instead of dealing with very long solutions we propose to deal with their Lempel-Ziv encodings. As our first main result we prove that each minimal solution of a word equation is highly compressible (exponentially compressible for long solutions) in terms of Lempel-Ziv encoding. A simple algorithm for solving word equations is derived. If the length of minimal solution is bounded by a singly exponential function (which is believed to be always true) then LZ encoding of each minimal solution is of a polynomial size (though the solution can be exponentially long) and solvability can be checked in nondeterministic polynomial time. As our second main result we prove that the solvability can be tested in polynomial deterministic time if the lengths of all variables are given in binary. We show also that lexicographically first solution for given lengths of variables is highly compressible in terms of Lempel-Ziv encodings.
size
word equations
results
polynomial size
general version
solution
paper
simple algorithm
minimal solutions
string
nondeterministic polynomial time
long solutions
equations
chapters
chapter
Makanin's algorithm
properties of strings
algorithm
Mathematical Sciences
Sven
Skyum
Winskel
Glynn
Springer Nature - SN SciGraph project
pub.1026190890
dimensions_id
Wojciech
Plandowski
978-3-540-68681-1
978-3-540-64781-2
Automata, Languages and Programming
Department of Computer Science, University of Liverpool, UK
Department of Computer Science, University of Liverpool, UK
Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02-097, Warszawa, Poland
Turku Centre for Computer Science and Department of Mathematics, Turku University, 20 014, Turku, Finland
Turku Centre for Computer Science and Department of Mathematics, Turku University, 20 014, Turku, Finland
Larsen
Kim G.
Pure Mathematics
Rytter
Wojciech
doi
10.1007/bfb0055097