Related papers: The length of a typical Huffman codeword
Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many…
It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical…
A guessing wiretapper's performance on a Shannon cipher system is analyzed for a source with memory. Close relationships between guessing functions and length functions are first established. Subsequently, asymptotically optimal encryption…
We consider the task of deriving a key with high HILL entropy from an unpredictable source. Previous to this work, the only known way to transform unpredictability into a key that was $\eps$ indistinguishable from having min-entropy was via…
In this study, the output of large language models (LLM) is considered an information source generating an unlimited sequence of symbols drawn from a finite alphabet. Given the probabilistic nature of modern LLMs, we assume a probabilistic…
In many applications of natural language processing (NLP) it is necessary to determine the likelihood of a given word combination. For example, a speech recognizer may need to determine which of the two word combinations ``eat a peach'' and…
We present methods for calculating a measure of phonotactic complexity---bits per phoneme---that permits a straightforward cross-linguistic comparison. When given a word, represented as a sequence of phonemic segments such as symbols in the…
Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$ is called the Cunningham chain generated by $p$ if all elements are prime number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of the…
In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we…
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…
Utilizing the hyperspace of noise-based logic, we show two string verification methods with low communication complexity. One of them is based on continuum noise-based logic. The other one utilizes noise-based logic with random telegraph…
We consider the communication complexity of the Hamming distance of two strings. Bille et al. [SPIRE 2018] considered the communication complexity of the longest common prefix (LCP) problem in the setting where the two parties have their…
We use large language models (LLMs) to uncover long-ranged structure in English texts from a variety of sources. The conditional entropy or code length in many cases continues to decrease with context length at least to $N\sim 10^4$…
A loss-less compression technique is proposed which uses a variable length Region formation technique to divide the input file into a number of variable length regions. Huffman codes are obtained for entire file after formation of regions.…
The index of codivisibility of a set of integers is the size of its largest subset with a common prime divisor. For large random samples of integers, the index of codivisibility is approximately normal.
In a recent article by Chapuy and Perarnau, it was shown that a uniformly chosen automaton on $n$ states with a $2$-letter alphabet has a synchronizing word of length $O(\sqrt{n}\log n)$ with high probability. In this note, we improve this…
Forgetting is removing variables from a logical formula while preserving the constraints on the other variables. In spite of being a form of reduction, it does not always decrease the size of the formula and may sometimes increase it. This…
We show that the integers in the HMM LLL HNF algorithm have bit length O(m.log(m.B)), where m is the number of rows and B is the maximum square length of a row of the input matrix. This is only a little worse than the estimate O(m.log(B))…
Hilberg (1990) supposed that finite-order excess entropy of a random human text is proportional to the square root of the text length. Assuming that Hilberg's hypothesis is true, we derive Guiraud's law, which states that the number of word…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…