Efficient Data Structures for Range Shortest Unique Substring Queries

• Main Authors: Paniz Abedin, Arnab Ganguly, Solon P. Pissis, Sharma V. Thankachan
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Algorithms
• Subjects: shortest unique substring; suffix tree; heavy-light decomposition; range queries; geometric data structures
• Online Access: https://www.mdpi.com/1999-4893/13/11/276

Let <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">T</mi><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></mrow></semantics></math></inline-formula> be a string of length <i>n</i> and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">T</mi><mo>[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>]</mo></mrow></semantics></math></inline-formula> be the substring of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> starting at position <i>i</i> and ending at position <i>j</i>. A substring <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">T</mi><mo>[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>]</mo></mrow></semantics></math></inline-formula> of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> is a repeat if it occurs more than once in <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula>; otherwise, it is a unique substring of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula>. Repeats and unique substrings are of great interest in computational biology and information retrieval. Given string <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> as input, the Shortest Unique Substring problem is to find a shortest substring of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> that does not occur elsewhere in <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula>. In this paper, we introduce the range variant of this problem, which we call the Range Shortest Unique Substring problem. The task is to construct a data structure over <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> answering the following type of online queries efficiently. Given a range <inline-formula><math display="inline"><semantics><mrow><mo>[</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>]</mo></mrow></semantics></math></inline-formula>, return a shortest substring <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">T</mi><mo>[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>]</mo></mrow></semantics></math></inline-formula> of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> with exactly one occurrence in <inline-formula><math display="inline"><semantics><mrow><mo>[</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>]</mo></mrow></semantics></math></inline-formula>. We present an <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>n</mi><mo form="prefix">log</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-word data structure with <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msub><mo form="prefix">log</mo><mi>w</mi></msub><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> query time, where <inline-formula><math display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><mo form="prefix">log</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the word size. Our construction is based on a non-trivial reduction allowing for us to apply a recently introduced optimal geometric data structure [Chan et al., ICALP 2018]. Additionally, we present an <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-word data structure with <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msqrt><mi>n</mi></msqrt><msup><mo form="prefix">log</mo><mi>ϵ</mi></msup><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> query time, where <inline-formula><math display="inline"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is an arbitrarily small constant. The latter data structure relies heavily on another geometric data structure [Nekrich and Navarro, SWAT 2012].