Lempel-Ziv Factorization May Be Harder Than Computing All Runs

The complexity of computing the Lempel-Ziv decomposition 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(n log sigma) time, where n is the length of the input string and sigma is the number of distinct letters in it. We prove an Omega(n log sigma) lower bound on the number of comparisons required to construct the Lempel-Ziv decomposition and thereby conclude that a popular technique of computation of runs using the Lempel-Ziv decomposition cannot achieve 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 the decision tree model the runs problem is easier than the Lempel-Ziv decomposition. Thus we support the conjecture that there is a linear RAM algorithm finding all runs.