The complexity of computing the Lempel-Ziv factorization and the set of all
runs (= maximal repetitions) is studied in the decision tree model of
computation over ordered alphabet. It is known that both these problems can be
solved by RAM algorithms in O(nlogσ) time, where n is the length of
the input string and σ is the number of distinct letters in it. We prove
an Ω(nlogσ) lower bound on the number of comparisons required to
construct the Lempel-Ziv factorization and thereby conclude that a popular
technique of computation of runs using the Lempel-Ziv factorization cannot
achieve an o(nlogσ) time bound. In contrast with this, we exhibit an
O(n) decision tree algorithm finding all runs in a string. Therefore, in the
decision tree model the runs problem is easier than the Lempel-Ziv
factorization. Thus we support the conjecture that there is a linear RAM
algorithm finding all runs.Comment: 12 pages, 3 figures, submitte