Towards optimal packed string matching

In the packed string matching problem, it is assumed that each machine word can accommodate up to α characters, thus an n-character string occupies n/α memory words. (a) We extend the Crochemore-Perrin constant-spaceO(n)-time string matching algorithm to run in optimal O(n/α) time and even in real-time, achieving a factor α speedup over traditional algorithms that examine each character individually. Our macro-level algorithm only uses the standard AC instructions of the word-RAM model (i.e. no integer multiplication) plus two specialized micro-level AC word-size packed string instructions. The main word-size stringmatching instruction wssm is available in contemporary commodity processors. The other word-size maximum-suffix instruction wslm is only required during the pattern preprocessing. Benchmarks show that our solution can be efficiently implemented, unlike some prior theoretical packed string matching work. (b) We also consider the complexity of the packed string matching problem in the classical word-RAM model in the absence of the specialized micro-level instructions wssm and wslm. We propose micro-level algorithms for the theoretically efficient emulation using parallel algorithms techniques to emulate wssm and using the Four-Russians technique to emulate wslm. Surprisingly, our bit-parallel emulation of wssm also leads to a new simplified parallel random access machine string matching algorithm. As a byproduct to facilitate our results we develop a new algorithm for finding the leftmost (most significant) 1 bits in consecutive non-overlapping blocks of uniform size inside a word. This latter problem is not known to be reducible to finding the rightmost 1, which can be easily solved, since we do not know how to reverse the bits of a word in O(1) time. Interestingly, the pattern pre-processing steps of our macro-level algorithm, our micro-level algorithm, and our parallel random access machine algorithm are all less efficient than their corresponding text processing, leaving gaps on the complexities of these three problems. ∗Preliminary versions of this work were presented at FSTTCS 2011 [9] and at CPM 2012 [16]. †Intel Research and Development Center, Haifa, Israel. ‡Technical University of Denmark, Copenhagen, Denmark. §Caesarea Rothschild Institute for Interdisciplinary Applications of Computer Science, University of Haifa, Haifa, Israel. Partially supported by the European Research Council (ERC) Project SFEROT and by the Israeli Science Foundation Grants 686/07, 347/09 and 864/11. ¶University of Liverpool, Liverpool, United Kingdom. ‖Dipartimento di Informatica, Università di Pisa, Pisa, Italy. Partially supported by Italian project PRIN AlgoDEEP (2008TFBWL4) of MIUR. ∗∗Computer Science Department, University of Haifa, Haifa, Israel.