We present highly optimized data structures for the dynamic predecessor problem, where the task is to maintain a set $S$ of $w$-bit numbers under insertions, deletions, and predecessor queries (return the largest element in $S$ no larger than a given key). The problem of finding predecessors can be viewed as a generalized form of the membership problem, or as a simple version of the nearest neighbour problem. It lies at the core of various real-world problems such as internet routing. In this work, we engineer (1) a simple implementation of the idea of universe reduction, similar to van-Emde-Boas trees (2) variants of y-fast tries [Willard, IPL'83], and (3) B-trees with different strategies for organizing the keys contained in the nodes, including an implementation of dynamic fusion nodes [Pǎtraşcu and Thorup, FOCS'14]. We implement our data structures for $w=32,40,64$, which covers most typical scenarios. Our data structures finish workloads faster than previous approaches while being significantly more space-efficient, e.g., they clearly outperform standard implementations of the STL by finishing up to four times as fast using less than a third of the memory. Our tests also provide more general insights on data structure design, such as how small sets should be stored and handled and if and when new CPU instructions such as advanced vector extensions pay off.

For a text T[1,n], a Longest Common Extension (LCE) query lce_T(i,j) asks for the length of the longest common prefix of the suffixes T[i,n] and T[j,n] identified by their starting positions 1 ≤ i,j ≤ n. A classic problem in stringology asks to preprocess a static text T[1,n] over an alphabet of size σ so that LCE queries can be efficiently answered on-line. Since its introduction in the 1980’s, this problem has found numerous applications: in suffix sorting, edit distance computation, approximate pattern matching, regularities finding, string mining, and many more. Text-book solutions offer O(n) preprocessing time and O(1) query time, but they employ memory-heavy data structures, such as suffix arrays, in practice several times bigger than the text itself. Very recently, more space efficient solutions using O(nlogσ) bits of total space or even only O(log n) bits of extra space have been proposed: string synchronizing sets [Kempa and Kociumaka, STOC'19, and Birenzwige et al., SODA'20] and in-place fingerprinting [Prezza, SODA'18]. The goal of this article is to present well-engineered implementations of these new solutions and study their practicality on a commonly agreed text corpus. We show that both perform extremely well in practice, with space consumption of only around 10% of the input size for string synchronizing sets (around 20% for highly repetitive texts), and essentially no extra space for fingerprinting. Interestingly, our experiments also show that both solutions become much faster than naive scanning even for finding common prefixes of moderate length, contradicting a common belief that sophisticated data structures for LCE queries are not competitive with naive approaches [Ilie and Tinta, SPIRE'09].