Reconstruction of Trees from Jumbled and Weighted Subtrees

Let T be an edge-labeled graph, where the labels are from a finite alphabet Sigma. For a subtree U of T the Parikh vector of U is a vector of length |Sigma| which specifies the multiplicity of each label in U. We ask when T can be reconstructed from the multiset of Parikh vectors of all its subtrees, or all of its paths, or all of its maximal paths. We consider the analogous problems for weighted trees. We show how several well-known reconstruction problems on labeled strings, weighted strings and point sets on a line can be included in this framework. We present reconstruction algorithms and non-reconstructibility results, and extend the polynomial method, previously applied to jumbled strings [Acharya et al., SIAM J. on Discr. Math, 2015] and weighted strings [Bansal et al., CPM 2004], to deal with general trees and special tree classes

On the Parikh-de-Bruijn grid

We introduce the Parikh-de-Bruijn grid, a graph whose vertices arefixed-order Parikh vectors, and whose edges are given by a simple shiftoperation. This graph gives structural insight into the nature of sets ofParikh vectors as well as that of the Parikh set of a given string. We show itsutility by proving some results on Parikh-de-Bruijn strings, the abelian analogof de-Bruijn sequences

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in O(log2⁡n) amortized time per word, while the best generation algorithm hitherto has O(n) running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages

Binary Jumbled Pattern Matching on Trees and Tree-Like Structures

Abstract. Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of length i and has exactly j 1-bits. This prob-lem naturally generalizes to vertex-labeled trees and graphs by replacing “substring ” with “connected subgraph”. In this paper, we give an O(n2 / log2 n)-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an O(g2/3n4/3/(logn)4/3)-time solution for strings that are compressed by a grammar of size g. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixed-parameter tractable with respect to the treewidth w of the graph, even for a constant number of different vertex-labels, thus improving the previous best nO(w) algorithm [ICALP’07].

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in O(log2 n) amortized time per word, while the best generation algorithm hitherto has O(n) running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages

On prefix normal words and prefix normal forms

A 1-prefix normal word is a binary word with the property that no factor has more 1s than the prefix of the same length; a 0-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of 1s and 0s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors. We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We give enumeration results on pnw(n), the number of prefix normal words of length n, showing that, for sufficiently large n, 2n 124nlg\u2061n 64pnw(n) 642n 12lg\u2061n+1. For fixed density (number of 1s), we show that the ordinary generating function of the number of prefix normal words of length n and density d is a rational function. Finally, we give experimental results on pnw(n), discuss further properties, and state open problem

On Combinatorial Generation of Prefix Normal Words

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present an efficient algorithm for exhaustively listing the prefix normal words with a fixed length. The algorithm is based on the fact that the language of prefix normal words is a bubble language, a class of binary languages with the property that, for any word w in the language, exchanging the first occurrence of 01 by 10 in w results in another word in the language. We prove that each prefix normal word is produced in O(n) amortized time, and conjecture, based on experimental evidence, that the true amortized running time is O(log(n))

On Combinatorial Generation of Prefix Normal Words

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present an efficient algorithm for exhaustively listing the prefix normal words with a fixed length. The algorithm is based on the fact that the language of prefix normal words is a bubble language, a class of binary languages with the property that, for any word w in the language, exchanging the first occurrence of 01 by 10 in w results in another word in the language. We prove that each prefix normal word is produced in O(n) amortized time, and conjecture, based on experimental evidence, that the true amortized running time is O(log(n))