Full-fledged Real-Time Indexing for Constant Size Alphabets

In this paper we describe a data structure that supports pattern matching queries on a dynamically arriving text over an alphabet ofconstant size. Each new symbol can be prepended to $T$ in O(1) worst-case time. At any moment, we can report all occurrences of a pattern $P$ in the current text in $O(|P|+k)$ time, where $|P|$ is the length of $P$ and $k$ is the number of occurrences. This resolves, under assumption of constant-size alphabet, a long-standing open problem of existence of a real-time indexing method for string matching (see \cite{AmirN08})

Four-Dimensional Dominance Range Reporting in Linear Space

In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+?} n + k log^? n) time, where k is the number of reported points, n is the number of points in the data structure, and ? is an arbitrarily small positive constant. Our second data structure uses O(n log^? n) space and answers queries in O(log n+k) time. These are the first data structures for this problem that use linear (resp. O(n log^? n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^? n)-space data structure supports queries in O(n^? + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ? 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+?} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time