Generalized Straight-Line Programs

TL;DR

The paper extends grammar-balancing techniques from SLPs to Generalized SLPs (GSLPs), introducing Iterated SLPs (ISLPs) and Run-Length SLPs (RLSLPs). It proves that balanced GSLPs exist with height O(log n) and linear-size overhead, and shows ISLPs can break the substring-complexity bound δ on certain text families while still supporting direct access and various substring queries in polylogarithmic time. The work also develops data-structures for navigating ISLPs, enabling direct access, substring extraction, and composable queries such as RMQ and NSV/PSV, with near-optimal time/space trade-offs, especially when restricting to low-d ISLPs or RLSLPs. A key application is efficient Karp-Rabin fingerprint computation on RLSLP-compressed texts, achieved via a general composable-function framework. Overall, the results advance practical grammar-compression-based indexing for highly repetitive texts.

Abstract

It was recently proved that any Straight-Line Program (SLP) generating a given string can be transformed in linear time into an equivalent balanced SLP of the same asymptotic size. We generalize this proof to a general class of grammars we call Generalized SLPs (GSLPs), which allow rules of the form $A \rightarrow x$ where $x$ is any Turing-complete representation (of size $|x|$) of a sequence of symbols (potentially much longer than $|x|$). We then specialize GSLPs to so-called Iterated SLPs (ISLPs), which allow rules of the form $A \rightarrow Π_{i=k_1}^{k_2} B_1^{i^{c_1}}\cdots B_t^{i^{c_t}}$ of size $2t+2$. We prove that ISLPs break, for some text families, the measure $δ$ based on substring complexity, a lower bound for most measures and compressors exploiting repetitiveness. Further, ISLPs can extract any substring of length $λ$, from the represented text $T[1.. n]$, in time $O(λ+ \log^2 n\log\log n)$. This is the first compressed representation for repetitive texts breaking $δ$ while, at the same time, supporting direct access to arbitrary text symbols in polylogarithmic time. We also show how to compute some substring queries, like range minima and next/previous smaller value, in time $O(\log^2 n \log\log n)$. Finally, we further specialize the grammars to Run-Length SLPs (RLSLPs), which restrict the rules allowed by ISLPs to the form $A \rightarrow B^t$. Apart from inheriting all the previous results with the term $\log^2 n \log\log n$ reduced to the near-optimal $\log n$, we show that RLSLPs can exploit balance to efficiently compute a wide class of substring queries we call ``composable'' -- i.e., $f(X \cdot Y)$ can be obtained from $f(X)$ and $f(Y)$...

Figures (2)

• Figure 1: The DAG and SC-decomposition of an unfolded RLSLP generating the string $\texttt{0}(\texttt{0}(\texttt{0}\texttt{1})^6\texttt{1}^2)^6(\texttt{0}\texttt{1})^5\texttt{1}^3$. The value to the left of a node is the number of paths from the root to that node, and the value to the right is the number of paths from the node to sink nodes. Red edges belong to the SC-decomposition of the DAG. Blue (resp. green) edges branch from an SC-path to the left (resp. to the right).
• Figure 2: Data structures built for the ISLP rule $A \rightarrow \prod_{i=1}^5B^{i}C^{i^2}D^{i}EEE^iB^{i^2}C^{i^3}D$, with $|\mathtt{exp}(B)| = 2$, $|\mathtt{exp}(C)| = 3$, $|\mathtt{exp}(D)| = 4$, and $|\mathtt{exp}(E)| = 7$. We show some of the polynomials to be simulated with these data structures.

Theorems & Definitions (44)

• Definition 1
• Definition 2
• Lemma 1
• Definition 3
• Lemma 2
• Theorem 3
• proof
• Definition 4
• Definition 5
• Proposition 4