Normal, Abby Normal, Prefix Normal

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 results about the number $pnw(n)$ of prefix normal words of length $n$, showing that $pnw(n) =\Omega\left(2^{n - c\sqrt{n\ln n}}\right)$ for some $c$ and $pnw(n) = O \left(\frac{2^n (\ln n)^2}{n}\right)$. We introduce efficient algorithms for testing the prefix normal property and a "mechanical algorithm" for computing prefix normal forms. We also include games which can be played with prefix normal words. In these games Alice wishes to stay normal but Bob wants to drive her "abnormal" -- we discuss which parameter settings allow Alice to succeed.Comment: Accepted at FUN '1

Culture and Personal Epistemology: U.S. and Middle Eastern Students’ beliefs about Scientific Knowledge and Knowing

Middle Eastern (Omani) and Western (U.S.) students’ beliefs about knowledge and knowing in the sciences were compared on four dimensions of personal epistemology proposed by Hofer and Pintrich ( Review of Educational Research (1997), 67 , 88–140). As predicted, given their experiences with comparatively traditional political and religious institutions, Omani more so than U.S. college students were more likely to accept scientific authorities as the basis of scientific truth. Furthermore, Omani men were more accepting of authorities than were Omani women, but there was no gender difference among U.S. students. Omani more than U.S. students also believed that knowledge in the sciences was simpler and more certain, which is consistent with comparisons between U.S. and Asian students (e.g., Qian & Pan, 2002, A comparision of epistemological beliefs and learning from science text between American and Chinese high school students. In B. K. Hofer & P. R. Pintrich (Eds.), Personal epistomology: The psychology of beliefs about knowledge and knowing (pp. 365–385), Mahwah, NJ: Erlbaum). Students in the two countries did not differ, however, in whether their beliefs were based on personal opinions versus systematic evidence. Suggestions for further research included directly assessing experiences with, and attitudes toward, authorities in academic and other areas of students’ lives.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43812/1/11218_2005_Article_1826.pd

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))

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].

Indexes for Jumbled Pattern Matching in Strings, Trees and Graphs

We consider how to index strings, trees and graphs for jumbled pattern matching when we are asked to return a match if one exists. For example, we show how, given a tree containing two colours, we can build a quadratic-space index with which we can find a match in time proportional to the size of the match. We also show how we need only linear space if we are content with approximate matches