In this paper we study the adaptive prefix coding problem in cases where the size of the input alphabet is large. We present an online prefix coding algorithm that uses $O(σ^{1 / λ+ ε}) $ bits of space for any constants $\eps>0$, $λ>1$, and encodes the string of symbols in $O(\log \log σ)$ time per symbol \emph{in the worst case}, where $σ$ is the size of the alphabet. The upper bound on the encoding length is $λn H (s) +(λ\ln 2 + 2 + ε) n + O (σ^{1 / λ} \log^2 σ)$ bits.
Describes a near-linear-time algorithm for a variant of Huffman coding, in which the letters may have non-uniform lengths (as in Morse code), but with the restriction that each word to be encoded has equal probability. [See also ``Huffman Coding with Unequal Letter Costs'' (2002).]
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 quasia…
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,…
Due to the emergence of new network applications, current IP lookup engines must support high-bandwidth, low lookup latency and the ongoing growth of IPv6 networks. However, existing solutions are not designed to address jointly those three requirements. This paper introduces SHIP, an IPv6 lookup algorithm that exploits prefix characteristics to build a two-level data structure designed to meet future application requirements. Using both prefix length distribution and prefix density, SHIP first clusters prefixes into groups sharing similar characteristics, then it builds a hybrid trie-tree for…
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…
The longest common prefix (LCP) array is a versatile auxiliary data structure in indexed string matching. It can be used to speed up searching using the suffix array (SA) and provides an implicit representation of the topology of an underlying suffix tree. The LCP array of a string of length $n$ can be represented as an array of length $n$ words, or, in the presence of the SA, as a bit vector of $2n$ bits plus asymptotically negligible support data structures. External memory construction algorithms for the LCP array have been proposed, but those proposed so far have a space requirement of $O(…
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 paper we study the Prefix Sum problem introduced by Fredman. We show that it is possible to perform both update and retrieval in O(1) time simultaneously under a memory model in which individual bits may be shared by several words. We also show that two variants (generalizations) of the problem can be solved optimally in $Θ(\lg N)$ time under the comparison based model of computation.
In this report an entropy bound on the memory size is given for a compression method of leaf-labeled trees. The compression converts the tree into a Directed Acyclic Graph (DAG) by merging isomorphic subtrees.
High-throughput sequencing technologies have led to explosive growth of genomic databases; one of which will soon reach hundreds of terabytes. For many applications we want to build and store indexes of these databases but constructing such indexes is a challenge. Fortunately, many of these genomic databases are highly-repetitive---a characteristic that can be exploited to ease the computation of the Burrows-Wheeler Transform (BWT), which underlies many popular indexes. In this paper, we introduce a preprocessing algorithm, referred to as {\em prefix-free parsing}, that takes a text $T$ as inp…
Due to the increased availability of large datasets of biological sequences, the tools for sequence comparison are now relying on efficient alignment-free approaches to a greater extent. Most of the alignment-free approaches require the computation of statistics of the sequences in the dataset. Such computations become impractical in internal memory when very large collections of long sequences are considered. In this paper, we present a new conceptual data structure, the colored longest common prefix array (cLCP), that allows to efficiently tackle several problems with an alignment-free appro…
We propose an external memory algorithm for the computation of the BWT and LCP array for a collection of sequences. Our algorithm takes the amount of available memory as an input parameter, and tries to make the best use of it by splitting the input collection into subcollections sufficiently small that it can compute their BWT in RAM using an optimal linear time algorithm. Next, it merges the partial BWTs in external memory and in the process it also computes the LCP values. We prove that our algorithm performs O(n AveLcp) sequential I/Os, where n is the total length of the collection, and Av…
We present a work-efficient parallel level-synchronous Breadth First Search (BFS) algorithm for shared-memory architectures which achieves the theoretical lower bound on parallel running time. The optimality holds regardless of the shape of the graph. We also demonstrate the implication of this optimality for the energy consumption of the program empirically. The key idea is never to use more processing cores than necessary to complete the work in any computation step efficiently. We keep the rest of the cores idle to save energy and to reduce other resource contentions (e.g., bandwidth, share…
Classically, coding theory has been concerned with the problem of transmitting a single message in a format which is robust to noise. Recently, researchers have turned their attention to designing coding schemes to make two-way conversations robust to noise. That is, given an interactive communication protocol $Π$, an \emph{interactive coding scheme} converts $Π$ into another communication protocol $Π'$ such that, even if errors are introduced during the execution of $Π'$, the parties are able to determine what the outcome of running $Π$ would be in a noise-free setting. We consider the proble…
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 prove that to store n bits x so that each prefix-sum query Sum(i) := sum_{k < i} x_k can be answered by non-adaptively probing q cells of log n bits, one needs memory > n + n/log^{O(q)} n. Our bound matches a recent upper bound of n + n/log^{Omega(q)} n by Patrascu (FOCS 2008), also non-adaptive. We also obtain a n + n/log^{2^{O(q)}} n lower bound for storing a string of balanced brackets so that each Match(i) query can be answered by non-adaptively probing q cells. To obtain these bounds we show that a too efficient data structure allows us to break the correlations between query answers.
Random Linear Network Coding (RLNC) has emerged as a powerful tool for robust high-throughput multicast. Projection analysis - a recently introduced technique - shows that the distributed packetized RLNC protocol achieves (order) optimal and perfectly pipelined information dissemination in many settings. In the original approach to RNLC intermediate nodes code together all available information. This requires intermediate nodes to keep considerable data available for coding. Moreover, it results in a coding complexity that grows linearly with the size of this data. While this has been identifi…
The problem of variable length and fixed-distortion universal source coding (or D-semifaithful source coding) for stationary and memoryless sources on countably infinite alphabets ($\infty$-alphabets) is addressed in this paper. The main results of this work offer a set of sufficient conditions (from weaker to stronger) to obtain weak minimax universality, strong minimax universality, and corresponding achievable rates of convergences for the worse-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on $…
We resolve the question of optimality for a well-studied packetized implementation of random linear network coding, called PNC. In PNC, in contrast to the classical memoryless setting, nodes store received information in memory to later produce coded packets that reflect this information. PNC is known to achieve order optimal stopping times for the many-to-all multicast problem in many settings. We give a reduction that captures exactly how PNC and other network coding protocols use the memory of the nodes. More precisely, we show that any such protocol implementation induces a transformation …
Sorting extremely large datasets is a frequently occuring task in practice. These datasets are usually much larger than the computer's main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter (1988), are often used. The complexity of comparison based external memory sorting has been understood for decades by now, however the situation remains elusive if we assume the keys to be sorted are integers. In internal memory, one can sort a set of $n$ integer keys of $Θ(\lg n)$ bits each in $O(n)$ time using the classic Radix Sort algorithm, however in external mem…
