Huffman coding finds an optimal prefix code for a given probability mass function. Consider situations in which one wishes to find an optimal code with the restriction that all codewords have lengths that lie in a user-specified set of lengths (or, equivalently, no codewords have lengths that lie in a complementary set). This paper introduces a polynomial-time dynamic programming algorithm that finds optimal codes for this reserved-length prefix coding problem. This has applications to quickly encoding and decoding lossless codes. In addition, one modification of the approach solves any quasiarithmetic prefix coding problem, while another finds optimal codes restricted to the set of codes with g codeword lengths for user-specified g (e.g., g=2).
Given a probability distribution over a set of n words to be transmitted, the Huffman Coding problem is to find a minimal-cost prefix free code for transmitting those words. The basic Huffman coding problem can be solved in O(n log n) time but variations are more difficult. One of the standard techniques for solving these variations utilizes a top-down dynamic programming approach. In this paper we show that this approach is amenable to dynamic programming speedup techniques, permitting a speedup of an order of magnitude for many algorithms in the literature for such variations as mixed radix,…
There is a large literature devoted to the problem of finding an optimal (min-cost) prefix-free code with an unequal letter-cost encoding alphabet of size. While there is no known polynomial time algorithm for solving it optimally there are many good heuristics that all provide additive errors to optimal. The additive error in these algorithms usually depends linearly upon the largest encoding letter size. This paper was motivated by the problem of finding optimal codes when the encoding alphabet is infinite. Because the largest letter cost is infinite, the previous analyses could give infinit…
A common complaint about adaptive prefix coding is that it is much slower than static prefix coding. Karpinski and Nekrich recently took an important step towards resolving this: they gave an adaptive Shannon coding algorithm that encodes each character in (O (1)) amortized time and decodes it in (O (\log H)) amortized time, where $H$ is the empirical entropy of the input string $s$. For comparison, Gagie's adaptive Shannon coder and both Knuth's and Vitter's adaptive Huffman coders all use (Θ(H)) amortized time for each character. In this paper we give an adaptive Shannon coder that both enco…
In prefix coding over an infinite alphabet, methods that consider specific distributions generally consider those that decline more quickly than a power law (e.g., Golomb coding). Particular power-law distributions, however, model many random variables encountered in practice. For such random variables, compression performance is judged via estimates of expected bits per input symbol. This correspondence introduces a family of prefix codes with an eye towards near-optimal coding of known distributions. Compression performance is precisely estimated for well-known probability distributions usin…
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 practical applications, is one such variant, for which codes are restricted to the set of codes in which none of the $n$ codewords is longer than a given length, $l_{\max}$. Binary length-limited coding can be done in $O(n l_{\max})$ time and O(n) space via the widely used Package-Merge algorithm and with even smaller asymptotic complexity using a lesser-known …
In this work we consider discrete-time multiple-input multiple-output (MIMO) linear-quadratic-Gaussian (LQG) control where the feedback consists of variable length binary codewords. To simplify the decoder architecture, we enforce a strict prefix constraint on the codewords. We develop a data compression architecture that provably achieves a near minimum time-average expected bitrate for a fixed constraint on the LQG performance. The architecture conforms to the strict prefix constraint and does not require time-varying lossless source coding, in contrast to the prior art.
Universal fixed-to-variable lossless source coding for memoryless sources is studied in the finite blocklength and higher-order asymptotics regimes. Optimal third-order coding rates are derived for general fixed-to-variable codes and for prefix codes. It is shown that the non-prefix Type Size code, in which codeword lengths are chosen in ascending order of type class size, achieves the optimal third-order rate and outperforms classical Two-Stage codes. Converse results are proved making use of a result on the distribution of the empirical entropy and Laplace's approximation. Finally, the fixed…
We develop a general framework to prove Kraft-type inequalities for prefix-free permutation codes for source coding with various notions of permutation code and prefix. We also show that the McMillan-type converse theorem in most of these cases does not hold, and give a general form of a counterexample. Our approach is more general and works for other structures besides permutation codes. The classical Kraft inequality for prefix-free codes as well as results about permutation codes follow as corollaries of our main theorem and main counterexample.
In this paper we consider the use of variable length non prefix-free codes for coding constrained sequences of symbols. We suppose to have a Markov source where some state transitions are impossible, i.e. the stochastic matrix associated with the Markov chain has some null entries. We show that classic Kraft inequality is not a necessary condition, in general, for unique decodability under the above hypothesis and we propose a relaxed necessary inequality condition. This allows, in some cases, the use of non prefix-free codes that can give very good performance, both in terms of compression an…
Algorithms for deriving Huffman codes and the recently developed algorithm for compiling PIFO trees to trees of fixed shape (Mohan et al. 2022) are similar, but work with different underlying algebraic operations. In this paper, we exploit the monadic structure of prefix codes to create a generalized Huffman algorithm that has these two applications as special cases.
This paper describes a new method of data encoding which may be used in various modern digital, computer and telecommunication systems and devices. The method permits the compression of data for storage or transmission, allowing the exact original data to be reconstructed without any loss of content. The method is characterized by the simplicity of implementation, as well as high speed and compression ratio. The method is based on a unique scheme of binary-ternary prefix-free encoding of characters of the original data. This scheme does not require the transmission of the code tables from enco…
In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the {\em soft-length limited Huffman coding} problem. In this version, there is a penalty associated with each of the $n$ characters of the alphabet whose encodings exceed a specified bound $D$($\leq n$), where the penalty increases linear…
We present a new algorithm for dynamic prefix-free coding, based on Shannon coding. We give a simple analysis and prove a better upper bound on the length of the encoding produced than the corresponding bound for dynamic Huffman coding. We show how our algorithm can be modified for efficient length-restricted coding, alphabetic coding and coding with unequal letter costs.
We introduce a new construction for the balancing of non-binary sequences that make use of Gray codes for prefix coding. Our construction provides full encoding and decoding of sequences, including the prefix. This construction is based on a generalization of Knuth's parallel balancing approach, which can handle very long information sequences. However, the overall sequence composed of the information sequence, together with the prefix must be balanced. This is reminiscent of Knuth's serial algorithm. The encoding of our construction does not make use of lookup tables, while the decoding proce…
Many data compressors regularly encode probability distributions for entropy coding - requiring minimal description length type of optimizations. Canonical prefix/Huffman coding usually just writes lengths of bit sequences, this way approximating probabilities with powers-of-2. Operating on more accurate probabilities usually allows for better compression ratios, and is possible e.g. using arithmetic coding and Asymmetric Numeral Systems family. Especially the multiplication-free tabled variant of the latter (tANS) builds automaton often replacing Huffman coding due to better compression at si…
Evolving secret sharing schemes do not require prior knowledge of the number of parties $n$ and $n$ may be infinitely countable. It is known that the evolving $2$-threshold secret sharing scheme and prefix coding of integers have a one-to-one correspondence. However, it is not known what prefix coding of integers to use to construct the scheme better. In this paper, we propose a new metric $K_Σ$ for evolving $2$-threshold secret sharing schemes $Σ$. We prove that the metric $K_Σ\geq 1.5$ and construct a new prefix coding of integers, termed $λ$ code, to achieve the metric $K_Λ=1.59375$. Thus, …
We consider a real-time streaming source coding system in which an encoder observes a sequence of randomly arriving symbols from an i.i.d. source, and feeds binary codewords to a FIFO buffer that outputs one bit per time unit to a decoder. Each source symbol represents a status update by the source, and the timeliness of the system is quantified by the age of information (AoI), defined as the time difference between the present time and the generation time of the most up-to-date symbol at the output of the decoder. When the FIFO buffer is allowed to be empty, we propose an optimal prefix-free …
The optimal prefix-free machine U is a universal decoding algorithm used to define the notion of program-size complexity H(s) for a finite binary string s. Since the set of all halting inputs for U is chosen to form a prefix-free set, the optimal prefix-free machine U can be regarded as an instantaneous code for noiseless source coding scheme. In this paper, we investigate the properties of optimal prefix-free machines as instantaneous codes. In particular, we investigate the properties of the set U^{-1}(s) of codewords associated with a symbol s. Namely, we investigate the number of codewords…
In this letter, we consider a Linear Quadratic Gaussian (LQG) control system where feedback occurs over a noiseless binary channel and derive lower bounds on the minimum communication cost (quantified via the channel bitrate) required to attain a given control performance. We assume that at every time step an encoder can convey a packet containing a variable number of bits over the channel to a decoder at the controller. Our system model provides for the possibility that the encoder and decoder have shared randomness, as is the case in systems using dithered quantizers. We define two extremal …
The problem of reconstructing strings from substring information has found many applications due to its importance in genomic data sequencing and DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry devices. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide upper and lower bounds on the asymptotic rate of the underlying codebooks. Our code co…
This paper presents prefix codes which minimize various criteria constructed as a convex combination of maximum codeword length and average codeword length or maximum redundancy and average redundancy, including a convex combination of the average of an exponential function of the codeword length and the average redundancy. This framework encompasses as a special case several criteria previously investigated in the literature, while relations to universal coding is discussed. The coding algorithm derived is parametric resulting in re-adjusting the initial source probabilities via a weighted pr…
Protecting against link failures in communication networks is essential to increase robustness, accessibility, and reliability of data transmission. Recently, network coding has been proposed as a solution to provide agile and cost efficient network protection against link failures, which does not require data rerouting, or packet retransmission. To achieve this, separate paths have to be provisioned to carry encoded packets, hence requiring either the addition of extra links, or reserving some of the resources for this purpose. In this paper, we propose network protection codes against a sing…
Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter $q$, $0{<}q{<}1$. By encoding pairs of symbols, it is possible to reduce the redundancy penalty of symbol-by-symbol encoding, while preserving the simplicity of the encoding and decoding procedures typical of Golomb codes and their variants. It is shown that optimal codes for these so-called two-dimensional geometric distributions are \emph{singular}, in the sense that a prefix code that is optimal for one value of the parameter $q$ cannot…
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 practical applications, is one such variant, in which codes are restricted to the set of codes in which none of the $n$ codewords is longer than a given length, $l_{\max}$. Binary length-limited coding can be done in $O(n l_{\max})$ time and O(n) space via the widely used Package-Merge algorithm. In this paper the Package-Merge approach is generalized without i…
Non-uniquely decodable codes can be defined as the codes that cannot be uniquely decoded without additional disambiguation information. These are mainly the class of non-prefix-free codes, where a codeword can be a prefix of other(s), and thus, the codeword boundary information is essential for correct decoding. Although the codeword bit stream consumes significantly less space when compared to prefix--free codes, the additional disambiguation information makes it difficult to catch the performance of prefix-free codes in total. Previous studies considered compression with non-prefix-free code…
In this paper, we consider a discrete time linear quadratic Gaussian (LQG) control problem in which state information of the plant is encoded in a variable-length binary codeword at every time step, and a control input is determined based on the codewords generated in the past. We derive a lower bound of the rate achievable by the class of prefix-free codes attaining the required LQG control performance. This lower bound coincides with the infimum of a certain directed information expression, and is computable by semidefinite programming (SDP). Based on a technique by Silva et al., we also pro…
A rateless code-i.e., a rate-compatible family of codes-has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones. A perfect family of such codes is one in which each of the codes in the family is capacity-achieving. We show by construction that perfect rateless codes with low-complexity decoding algorithms exist for additive white Gaussian noise channels. Our construction involves the use of layered encoding and successive decoding, together with repetition using time-varying layer weights. As an illustration of our framework, we design a practical …
We consider a setup in which Alice selects a pdf $f$ from a set of prescribed pdfs $\mathscr{P}$ and sends a prefix-free codeword $W$ to Bob in order to allow him to generate a single instance of the random variable $X\sim f$. We describe a universal coding scheme for this setup and establish an upper bound on the expected codeword length when the pdf $f$ is bounded, orthogonally concave (which includes quasiconcave pdf), and has a finite first absolute moment. A dyadic decomposition scheme is used to express the pdf as a mixture of uniform pdfs over hypercubes. Alice randomly selects a hyperc…
Adaptive coding faces the following problem: given a collection of source classes such that each class in the collection has non-trivial minimax redundancy rate, can we design a single code which is asymptotically minimax over each class in the collection? In particular, adaptive coding makes sense when there is no universal code on the union of classes in the collection. In this paper, we deal with classes of sources over an infinite alphabet, that are characterized by a dominating envelope. We provide asymptotic equivalents for the redundancy of envelope classes enjoying a regular variation …
This paper considers insertion and deletion channels with the additional assumption that the channel input sequence is implicitly divided into segments such that at most one edit can occur within a segment. No segment markers are available in the received sequence. We propose code constructions for the segmented deletion, segmented insertion, and segmented insertion-deletion channels based on subsets of Varshamov-Tenengolts codes chosen with pre-determined prefixes and/or suffixes. The proposed codes, constructed for any finite alphabet, are zero-error and can be decoded segment-by-segment. We…
Huffman coding finds a prefix code that minimizes mean codeword length for a given probability distribution over a finite number of items. Campbell generalized the Huffman problem to a family of problems in which the goal is to minimize not mean codeword length but rather a generalized mean known as a quasiarithmetic or quasilinear mean. Such generalized means have a number of diverse applications, including applications in queueing. Several quasiarithmetic-mean problems have novel simple redundancy bounds in terms of a generalized entropy. A related property involves the existence of optimal …
In entropy coding, universal coding of integers~(UCI) is a binary universal prefix code, such that the ratio of the expected codeword length to $\max\{1, H(P)\}$ is less than or equal to a constant expansion factor $K_{\mathcal{C}}$ for any probability distribution $P$, where $H(P)$ is the Shannon entropy of $P$. $K_{\mathcal{C}}^{*}$ is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest $K_{\mathcal{C}}^{*}$. Based on prior research, the range of $K_{\mathcal{C}}^{*}$ for the optimal UCI is $2\leq K_{\mathcal{C}}^{*}\leq 2.75$. Cu…
Symmetric fix-free codes are prefix condition codes in which each codeword is required to be a palindrome. Their study is motivated by the topic of joint source-channel coding. Although they have been considered by a few communities they are not well understood. In earlier work we used a collection of instances of Boolean satisfiability problems as a tool in the generation of all optimal binary symmetric fix-free codes with n codewords and observed that the number of different optimal codelength sequences grows slowly compared with the corresponding number for prefix condition codes. We demons…
Let $A$ be a finite or countable alphabet and let $θ$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $θ$ ($θ$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $θ$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $θ$-invariant codes such as prefix (bifix) codes, codes with a finite dec…
This paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikan's bounds, they are expressed in terms of the Arimoto-Rényi conditional entropy. Although Arikan's bounds are asymptotically tight, the improvement of the bounds in this paper is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probabili…
This paper presents lossless prefix codes optimized with respect to a pay-off criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new coding algorithm which transforms the initial source probability vector into a new probability vector according to a merging rule. The coding algorithm is equivalent to a partition of the source alphabet into disjoint sets on which a new transformed probability vector is defined as a function of the initial source probability vector and a scalar parameter. The p…
Canonical Huffman code is an optimal prefix-free compression code whose codewords enumerated in the lexicographical order form a list of binary words in non-decreasing lengths. Gagie et al. (2015) gave a representation of this coding capable to encode or decode a symbol in constant worst case time. It uses $σ\lg \ell_{\text{max}} + o(σ) + O(\ell_{\text{max}}^2)$ bits of space, where $σ$ and $\ell_{\text{max}}$ are the alphabet size and maximum codeword length, respectively. We refine their representation to reduce the space complexity to $σ\lg \ell_{\text{max}} (1 + o(1))$ bits while preservin…
This paper presents new lower and upper bounds for the compression rate of binary prefix codes optimized over memoryless sources according to various nonlinear codeword length objectives. Like the most well-known redundancy bounds for minimum average redundancy coding - Huffman coding - these are in terms of a form of entropy and/or the probability of an input symbol, often the most probable one. The bounds here, some of which are tight, improve on known bounds of the form L in [H,H+1), where H is some form of entropy in bits (or, in the case of redundancy objectives, 0) and L is the length ob…
Motivated by polymer-based data-storage platforms that use chains of binary synthetic polymers as the recording media and read the content via tandem mass spectrometers, we propose a new family of codes that allows for both unique string reconstruction and correction of multiple mass errors. We consider two approaches: The first approach pertains to asymmetric errors and it is based on introducing redundancy that scales linearly with the number of errors and logarithmically with the length of the string. The construction allows for the string to be uniquely reconstructed based only on its erro…
