A Suffix Tree Or Not a Suffix Tree?

In this paper we study the structure of suffix trees. Given an unlabeled tree \(\tau \) on n nodes and suffix links of its internal nodes, we ask the question “Is \(\tau \) a suffix tree?", i.e., is there a string S whose suffix tree has the same topological structure as \(\tau \)? We place no restrictions on S, in particular we do not require that S ends with a unique symbol. This corresponds to considering the more general definition of implicit or extended suffix trees. Such general suffix trees have many applications and are for example needed to allow efficient updates when suffix trees are built online. We prove that \(\tau \) is a suffix tree if and only if it is realized by a string S of length \(n-1\), and we give a linear-time algorithm for inferring S when the first letter on each edge is known. This generalizes the work of I et al. [Discrete Appl. Math. 163, 2014].

[1]  Maxime Crochemore,et al.  Cover Array String Reconstruction , 2010, CPM.

[2]  W. F. Smyth,et al.  Verifying a border array in linear time , 1999 .

[3]  Peter Weiner,et al.  Linear Pattern Matching Algorithms , 1973, SWAT.

[4]  Esko Ukkonen,et al.  On-line construction of suffix trees , 1995, Algorithmica.

[5]  Hideo Bannai,et al.  Counting Parameterized Border Arrays for a Binary Alphabet , 2009, LATA.

[6]  Maxime Crochemore,et al.  Forty Years of Text Indexing , 2013, CPM.

[7]  Eric Rivals,et al.  Reverse engineering of compact suffix trees and links: A novel algorithm , 2014, J. Discrete Algorithms.

[8]  Ayumi Shinohara,et al.  Inferring Strings from Graphs and Arrays , 2003, MFCS.

[9]  Gui-Rong Liu,et al.  Differential quadrature solutions of eighth-order boundary-value differential equations , 2002 .

[10]  Gregory Kucherov,et al.  On the combinatorics of suffix arrays , 2012, Inf. Process. Lett..

[11]  Arnaud Lefebvre,et al.  Words over an ordered alphabet and suffix permutations , 2002, RAIRO Theor. Informatics Appl..

[12]  Ramesh Hariharan,et al.  Optimal Parallel Construction of Minimal Suffix and Factor Automata , 1996, Parallel Process. Lett..

[13]  Hideo Bannai,et al.  Verifying a Parameterized Border Array in O(n1.5) Time , 2010, CPM.

[14]  Hideo Bannai,et al.  Inferring strings from suffix trees and links on a binary alphabet , 2011, Discret. Appl. Math..

[15]  Hideo Bannai,et al.  Verifying and enumerating parameterized border arrays , 2011, Theor. Comput. Sci..

[16]  Arnaud Lefebvre,et al.  Efficient validation and construction of border arrays and validation of string matching automata , 2009, RAIRO Theor. Informatics Appl..

[17]  Giuseppe F. Italiano,et al.  On suffix extensions in suffix trees , 2011, Theor. Comput. Sci..

[18]  Artur Jez,et al.  Validating the Knuth-Morris-Pratt Failure Function, Fast and Online , 2010, Theory of Computing Systems.