||November 11, 2010
||Atsuyoshi Nakamura, Associate Professor, Graduate School of Information Science and Technology Hokkaido University
||Algorithms for Finding a Minimum Repetition Representation of a String
A string with many repetitions can be written compactly
by replacing h-fold contiguous repetitions of substring r with (r)h.
We refer to such a compact representation as a
repetition representation string or RRS,
by which a set of disjoint or nested tandem arrays can be compacted.
In this paper, we study the problem of finding a minimum RRS or MRRS,
where the size of an RRS is defined to be the sum of its component
and the sizes needed to describe the repetitions (・)h which are
defined as wR(h) using a repetition weight function wR.
We develop two dynamic programming algorithms to solve the problem.
One is CMR that works for any repetition weight function,
and the other is CMR-C that is faster but can be applied only when
the repetition weight function is constant.
CMR-C is an O(w(n+z)) - time algorithm using
O(n+z) space for a given string with length n,
where w and z are the number of distinct primitive
tandem repeats and the number of their occurrences, respectively.
Since w=O(n) and z=O(n log n) in the worst case,
CMR-C is an O(n2 log n)-time O(n log n)-space algorithm,
which is faster than CMR by ((log n)/n)-factor.