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Documents authored by Rizzo, Nicola

Document
Research
DOI: 10.4230/OASIcs.Grossi.2
On the Construction of Elastic Degenerate Strings

Authors: Nicola Rizzo, Veli Mäkinen, and Nadia Pisanti

Published in: OASIcs, Volume 132, From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday (2025)

Abstract
An elastic degenerate string (EDS) is a sequence of sets of strings. In the context of bioinformatics, EDSes can be used to represent the variations observed in a population from its consensus genome. Pattern matching and comparison problems on EDSes have been widely studied in the literature, but their construction has been largely omitted. We fill this gap by showing how algorithms originally developed for related problems of founder reconstruction can be adapted to minimize the total cardinality of the EDS sets and total length of the EDS strings in linear time, given suitable multiple alignments representing the input data.
Cite as

Nicola Rizzo, Veli Mäkinen, and Nadia Pisanti. On the Construction of Elastic Degenerate Strings. In From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 132, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rizzo_et_al:OASIcs.Grossi.2,
  author =	{Rizzo, Nicola and M\"{a}kinen, Veli and Pisanti, Nadia},
  title =	{{On the Construction of Elastic Degenerate Strings}},
  booktitle =	{From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday},
  pages =	{2:1--2:13},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-391-1},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{132},
  editor =	{Conte, Alessio and Marino, Andrea and Rosone, Giovanna and Vitter, Jeffrey Scott},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Grossi.2},
  URN =		{urn:nbn:de:0030-drops-238014},
  doi =		{10.4230/OASIcs.Grossi.2},
  annote =	{Keywords: multiple sequence alignment, pattern matching, data structures, segmentation algorithms, founder reconstruction, dynamic programming, semi-dynamic range minimum queries, positional Burrows-Wheeler transform}
}
Document
DOI: 10.4230/OASIcs.Manzini.13
Graph Indexing Beyond Wheeler Graphs

Authors: Jarno N. Alanko, Elena Biagi, Massimo Equi, Veli Mäkinen, Simon J. Puglisi, Nicola Rizzo, Kunihiko Sadakane, and Jouni Sirén

Published in: OASIcs, Volume 131, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday (2025)

Abstract
After the discovery of the FM index, which linked the Burrows-Wheeler transform (BWT) to pattern matching on strings, several contemporaneous strands of research began on indexing more complex structures with the BWT, such as tries, finite languages, de Bruijn graphs, and aligned sequences. These directions can now be viewed as culminating in the theory of Wheeler Graphs, but sometimes they go beyond. This chapter reviews the significant body of "proto Wheeler Graph" indexes, many of which exploit characteristics of their specific case to outperform Wheeler graphs, especially in practice.
Cite as

Jarno N. Alanko, Elena Biagi, Massimo Equi, Veli Mäkinen, Simon J. Puglisi, Nicola Rizzo, Kunihiko Sadakane, and Jouni Sirén. Graph Indexing Beyond Wheeler Graphs. In The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 131, pp. 13:1-13:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alanko_et_al:OASIcs.Manzini.13,
  author =	{Alanko, Jarno N. and Biagi, Elena and Equi, Massimo and M\"{a}kinen, Veli and Puglisi, Simon J. and Rizzo, Nicola and Sadakane, Kunihiko and Sir\'{e}n, Jouni},
  title =	{{Graph Indexing Beyond Wheeler Graphs}},
  booktitle =	{The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday},
  pages =	{13:1--13:29},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-390-4},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{131},
  editor =	{Ferragina, Paolo and Gagie, Travis and Navarro, Gonzalo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Manzini.13},
  URN =		{urn:nbn:de:0030-drops-239215},
  doi =		{10.4230/OASIcs.Manzini.13},
  annote =	{Keywords: indexing, compression, compressed data structures, string algorithms, pattern matching}
}
Document
DOI: 10.4230/LIPIcs.WABI.2023.10
Finding Maximal Exact Matches in Graphs

Authors: Nicola Rizzo, Manuel Cáceres, and Veli Mäkinen

Published in: LIPIcs, Volume 273, 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)

Abstract
We study the problem of finding maximal exact matches (MEMs) between a query string Q and a labeled graph G. MEMs are an important class of seeds, often used in seed-chain-extend type of practical alignment methods because of their strong connections to classical metrics. A principled way to speed up chaining is to limit the number of MEMs by considering only MEMs of length at least κ (κ-MEMs). However, on arbitrary input graphs, the problem of finding MEMs cannot be solved in truly sub-quadratic time under SETH (Equi et al., ICALP 2019) even on acyclic graphs. In this paper we show an O(n⋅ L ⋅ d^{L-1} + m + M_{κ,L})-time algorithm finding all κ-MEMs between Q and G spanning exactly L nodes in G, where n is the total length of node labels, d is the maximum degree of a node in G, m = |Q|, and M_{κ,L} is the number of output MEMs. We use this algorithm to develop a κ-MEM finding solution on indexable Elastic Founder Graphs (Equi et al., Algorithmica 2022) running in time O(nH² + m + M_κ), where H is the maximum number of nodes in a block, and M_κ is the total number of κ-MEMs. Our results generalize to the analysis of multiple query strings (MEMs between G and any of the strings). Additionally, we provide some preliminary experimental results showing that the number of graph MEMs is an order of magnitude smaller than the number of string MEMs of the corresponding concatenated collection.
Cite as

Nicola Rizzo, Manuel Cáceres, and Veli Mäkinen. Finding Maximal Exact Matches in Graphs. In 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 273, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rizzo_et_al:LIPIcs.WABI.2023.10,
  author =	{Rizzo, Nicola and C\'{a}ceres, Manuel and M\"{a}kinen, Veli},
  title =	{{Finding Maximal Exact Matches in Graphs}},
  booktitle =	{23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-294-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{273},
  editor =	{Belazzougui, Djamal and Ouangraoua, A\"{i}da},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2023.10},
  URN =		{urn:nbn:de:0030-drops-186364},
  doi =		{10.4230/LIPIcs.WABI.2023.10},
  annote =	{Keywords: Sequence to graph alignment, bidirectional BWT, r-index, suffix tree, founder graphs}
}
Document
DOI: 10.4230/LIPIcs.CPM.2022.19
Indexable Elastic Founder Graphs of Minimum Height

Authors: Nicola Rizzo and Veli Mäkinen

Published in: LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Abstract
Indexable elastic founder graphs have been recently proposed as a data structure for genomics applications supporting fast pattern matching queries. Consider segmenting a multiple sequence alignment MSA[1..m,1..n] into b blocks MSA[1..m,1..j₁], MSA[1..m,j₁+1..j₂], …, MSA[1..m,j_{b-1}+1..n]. The resulting elastic founder graph (EFG) is obtained by merging in each block the strings that are equivalent after the removal of gap symbols, taking the strings as the nodes of the block and the original MSA connections as edges. We call an elastic founder graph indexable if a node label occurs as a prefix of only those paths that start from a node of the same block. Equi et al. (ISAAC 2021) showed that such EFGs support fast pattern matching and studied their construction maximizing the number of blocks and minimizing the maximum length of a block, but left open the case of minimizing the maximum number of distinct strings in a block that we call graph height. For the simplified gapless setting, we give an O(mn) time algorithm to find a segmentation of an MSA minimizing the height of the resulting indexable founder graph, by combining previous results in segmentation algorithms and founder graphs. For the general setting, the known techniques yield a linear-time parameterized solution on constant alphabet Σ, taking time O(m n² log|Σ|) in the worst case, so we study the refined measure of prefix-aware height, that omits counting strings that are prefixes of another considered string. The indexable EFG minimizing the maximum prefix-aware height provides a lower bound for the original height: by exploiting exploiting suffix trees built from the MSA rows and the data structure answering weighted ancestor queries in constant time of Belazzougui et al. (CPM 2021), we give an O(mn)-time algorithm for the optimal EFG under this alternative height.
Cite as

Nicola Rizzo and Veli Mäkinen. Indexable Elastic Founder Graphs of Minimum Height. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rizzo_et_al:LIPIcs.CPM.2022.19,
  author =	{Rizzo, Nicola and M\"{a}kinen, Veli},
  title =	{{Indexable Elastic Founder Graphs of Minimum Height}},
  booktitle =	{33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-234-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{223},
  editor =	{Bannai, Hideo and Holub, Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.19},
  URN =		{urn:nbn:de:0030-drops-161467},
  doi =		{10.4230/LIPIcs.CPM.2022.19},
  annote =	{Keywords: multiple sequence alignment, pattern matching, data structures, segmentation algorithms, dynamic programming, suffix tree}
}
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