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The complexity of computing the Lempel-Ziv factorization and the set of allruns (= maximal repetitions) is studied in the decision tree model ofcomputation over ordered alphabet. It is known that both these problems can besolved by RAM algorithms in $O(n\log\sigma)$ time, where $n$ is the length ofthe input string and $\sigma$ is the number of distinct letters in it. We provean $\Omega(n\log\sigma)$ lower bound on the number of comparisons required toconstruct the Lempel-Ziv factorization and thereby conclude that a populartechnique of computation of runs using the Lempel-Ziv factorization cannotachieve an $o(n\log\sigma)$ time bound. In contrast with this, we exhibit an$O(n)$ decision tree algorithm finding all runs in a string. Therefore, in thedecision tree model the runs problem is easier than the Lempel-Zivfactorization. Thus we support the conjecture that there is a linear RAMalgorithm finding all runs.

Lempel-Ziv Factorization May Be Harder Than Computing All Runs