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### Counting Distinct Strings

D. Moore, W. F. Smyth, D. Miller
1999 Algorithmica
This paper discusses how to count and generate strings that are \distinct" in two senses: p-distinct and b-distinct.  ...  W e p r e s e n t algorithms to compute, in constant time per string, all p-distinct (respectively, b-distinct) strings of length n formed using exactly k letters, and we a l s o s h o w h o w to compute  ...  DISTINCT BORDER ARRAYS In this section we consider how to generate and how to count b-distinct strings.  ...

### Counting suffix arrays and strings

Klaus-Bernd Schürmann, Jens Stoye
2008 Theoretical Computer Science
For each such equivalence class, we count the number of strings contained in it. We also give exact formulas for computing the number of equivalence classes.  ...  For a fixed alphabet size and string length, we divide the set of all strings into equivalence classes of strings that share the same suffix array.  ...  Theorem 4 gave the number of string-distinct strings and to count the order-distinct strings we just sum up over all possible κ.  ...

### The Number of Distinct Subsequences of a Random Binary String [article]

Michael J. Collins
2013 arXiv   pre-print
We determine the average number of distinct subsequences in a random binary string, and derive an estimate for the average number of distinct subsequences of a particular length.  ...  For any string s, let σ m (s) be the number of distinct subsequences of s of length m, and let σ(s) be the total number of distinct subsequences.  ...  Subsequences of length m If a string s begins with a run of length k, then we can assume with no loss of generality -since we are only counting distinct subsequences -that every nonempty subsequence starts  ...

### Algorithms for subsequence combinatorics

Cees Elzinga, Sven Rahmann, Hui Wang
2008 Theoretical Computer Science
We present theorems that lead to efficient dynamic programming algorithms to count (1) distinct subsequences in a string, (2) distinct common subsequences of two strings, (3) matching joint embeddings  ...  in two strings, (4) distinct subsequences with a given minimum span, and (5) sequences generated by a string allowing characters to come in runs of a length that is bounded from above.  ...  Counting distinct subsequences In this section, we present an efficient solution to the problem of counting all distinct subsequences of length k ≥ 0 of a given string, and study strings that have maximally  ...

### Counting Colours in Compressed Strings [article]

Travis Gagie, Juha Kärkkäinen
2010 arXiv   pre-print
Suppose we are asked to preprocess a string s [1..n] such that later, given a substring's endpoints, we can quickly count how many distinct characters it contains.  ...  Storing these points is equivalent to storing a string s [1. .n] over an alphabet whose size σ is the number of distinct colours, such that later, given a substring's endpoints, we can quickly count how  ...  Given a string s[1. .n], we can build a data structure that takes (1 + o(1))(n log σ + n log log n) bits such that later, given a substring's endpoints, in O(log n) time we can count how many distinct  ...

### Counting Suffix Arrays and Strings [chapter]

Klaus-Bernd Schürmann, Jens Stoye
2005 Lecture Notes in Computer Science
Strings sharing the same suffix array are counted in Section 4 and distinct suffix arrays in Section 5.  ...  For fixed alphabet size and string length we count the number of strings sharing the same suffix array and the number of such suffix arrays.  ...  Theorem 4.5 gave the number of distinct strings, and for the order-distinct strings we just sum up over all possible k. Counting suffix arrays for strings with fixed alphabet.  ...

### Repetitions in strings: Algorithms and combinatorics

Maxime Crochemore, Lucian Ilie, Wojciech Rytter
2009 Theoretical Computer Science
For example we know that the number of (distinct) primitivelyrooted squares in a string of length n is no more than 2n − Θ(log n), conjecture to be n, and that their number of occurrences can be Θ(n log  ...  For these repetitions, we distinguish between the factors (sometimes qualified as distinct) and their occurrences (also called positioned factors).  ...  As we shall see, counting distinct squares, i.e. squares that are distinct strings, or squares occurrences gives very different results.  ...

### Counting Lyndon Subsequences [article]

Ryo Hirakawa, Yuto Nakashima, Shunsuke Inenaga, Masayuki Takeda
2021 arXiv   pre-print
We show (1) the maximum total number of Lyndon subsequences in a string, (2) the expected total number of Lyndon subsequences in a string, (3) the expected number of distinct Lyndon subsequences in a string  ...  In this paper, we consider a more general problem "counting Lyndon subsequences".  ...  of distinct Lyndon subsequences in all strings of length n over an alphabet Σ of size σ.  ...

### RDFStats - An Extensible RDF Statistics Generator and Library

Andreas Langegger, Wolfram Woss
2009 2009 20th International Workshop on Database and Expert Systems Application
The first one uses a histogram bin for all distinct values and counts the occurrences of equal strings.  ...  Fig. 1 . 1 |u| = COUNT(*) { [] a c } for (c : classes) more c get list of properties used with c props = DISTINCT { [] a c ; ?  ...

### Distinctive patterns in the first movement of Brahms' String Quartet in C minor

Darrell Conklin
2010 Journal of Mathematics and Music - Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance
The method is applied to the first movement of Brahms' string quartet No. 1 in C Minor, with the first movements of string quartets Nos. 2 and 3 used as an anticorpus.  ...  In this paper a computational inductive method for maximally general distinctive pattern discovery in music is developed.  ...  Each node contains a pattern, its total count, and its distinctiveness.  ...

### PPDC: A Privacy-Preserving Distinct Counting Scheme for Mobile Sensing

Xiaochen Yang, Ming Xu, Shaojing Fu, Yuchuan Luo
2019 Applied Sciences
In this paper, we propose a Privacy-Preserving Distinct Counting scheme (PPDC) for mobile sensing.  ...  Among major mobile sensing applications, the distinct counting problem aiming to find the number of distinct elements in a data stream with repeated elements, is extremely important for avoiding waste  ...  Distinct Counting On the other hand, distinct counting is also of interest to a lot of researches.  ...

### Lower Bounds for the Number of Repetitions in 2D Strings [article]

Paweł Gawrychowski, Samah Ghazawi, Gad M. Landau
2021 arXiv   pre-print
First, we construct an infinite family of n× n 2D strings with Ω(n^3) distinct tandems. Second, we construct an infinite family of n× n 2D strings with Ω(n^2log n) distinct quartics.  ...  We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs.  ...  Of course, any fragment of the string a n is a palindrome, therefore we would like to count distinct palindromes.  ...

### Page 79 of Computational Linguistics Vol. 30, Issue 1 [page]

2004 Computational Linguistics
In fact, the three- character string |] #U ¥ has three distinct oie “S”, T,% (“S” denotes the start of a sentence), and four distinct suffixes, 3, , {k, J (“E” denotes the termination of a sentence).  ...  Therefore, “S” and “E” will be counted multiple times, and we regard % are probably very rarely other characters prec i. or succeeding it.  ...

### Counting Distinct Patterns in Internal Dictionary Matching

Panagiotis Charalampopoulos, Tomasz Kociumaka, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, Juliusz Straszyński, Tomasz Waleń, Wiktor Zuba, Oren Weimann, Inge Li Gørtz
2020 Annual Symposium on Combinatorial Pattern Matching
We design an 𝒪(n log² n)-size data structure that allows us to count distinct squares in a text fragment T[i..j] in 𝒪(log n) time.  ...  We also consider the special case when the dictionary consists of all square factors of the string.  ...  Counting Distinct Patterns in Internal Dictionary Matching We now proceed to np-squares.  ...

### Counting Distinct Patterns in Internal Dictionary Matching [article]

Panagiotis Charalampopoulos, Tomasz Kociumaka, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, Juliusz Straszyński, Tomasz Waleń, Wiktor Zuba
2020 arXiv   pre-print
We design an O(n log^2 n)-size data structure that allows us to count distinct squares in a text fragment T[i . . j] in O(log n) time.  ...  We also consider the special case when the dictionary consists of all square factors of the string.  ...  Query: CountDistinct(i, j): Count all distinct patterns P ∈ D that occur in T [i . . j]. Observe that the input size is n + d, while the total length of strings in D could be Θ(n · d).  ...
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