Hardness of comparing two run-length encoded strings |
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Authors: | Kuan-Yu Chen Ping-Hui Hsu Kun-Mao Chao |
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Institution: | 1. Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan;2. Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University, Taipei 106, Taiwan;3. Graduate Institute of Networking and Multimedia, National Taiwan University, Taipei 106, Taiwan |
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Abstract: | In this paper, we consider a commonly used compression scheme called run-length encoding. We provide both lower and upper bounds for the problems of comparing two run-length encoded strings. Specifically, we prove the 3sum-hardness for both the wildcard matching problem and the k-mismatch problem with run-length compressed inputs. Given two run-length encoded strings of m and n runs, such a result implies that it is very unlikely to devise an o(mn)-time algorithm for either of them. We then present an inplace algorithm running in O(mnlogm) time for their combined problem, i.e. k-mismatch with wildcards. We further demonstrate that if the aim is to report the positions of all the occurrences, there exists a stronger barrier of Ω(mnlogm)-time, matching the running time of our algorithm. Moreover, our algorithm can be easily generalized to a two-dimensional setting without impairing the time and space complexity. |
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Keywords: | Compressed pattern matching Run-length encoding Sequence comparison |
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