Fast and Simple Jumbled Indexing for Binary RLE Strings

Important papers have appeared recently on the problem of indexing binary strings for jumbled pattern matching, and further lowering the time bounds in terms of the input size would now be a breakthrough with broad implications. We can still make progress on the problem, however, by considering other natural parameters. Badkobeh et al.\ (IPL, 2013) and Amir et al.\ (TCS, 2016) gave algorithms that index a binary string in $O (n + \rho^2 \log \rho)$ time, where $n$ is the length and $\rho$ is the number of runs, and Giaquinta and Grabowski (IPL, 2013) gave one that runs in $O (n + \rho^2)$ time. In this paper we propose a new and very simple algorithm that also runs in $O(n + \rho^2)$ time and can be extended either so that the index returns the position of a match (if there is one), or so that the algorithm uses only $O (n)$ bits of space.