Lempel-Ziv Factorization May Be Harder Than Computing All Runs

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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(nσ) time, where n is the length of the input string and σ is the number of distinct letters in it. We prove an Ω(nσ) 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(nσ) 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.