Optimum lopsided binary trees
TL;DR: Almost-optimum algorithms for the lopsided case of unbounded searching are obtained and some extensions to nonconstant costs are briefly sketched.
Abstract: Binary search trees with costs a and b, respectively, on the left and right edges (lopsided search trees) are considered. The exact shape, minimum worst-case cost, and minimum average cost of lopsided trees of n internal nodes are determined for nonnegative a and b; the costs are both roughly logp(n + 1) where p is the unique real number in the interval (1. 2] satisfying 1/pa + 1/pb = 1. Search procedures are given that come within a small additive constant of the lower bounds. Almost-optimum algorithms for the lopsided case of unbounded searching are also obtained. Some extensions to nonconstant costs are briefly sketched.
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2,415 citations
TL;DR: This work considers the problem of constructing prefix-free codes of minimum cost when the encoding alphabet contains letters of unequal length and introduces a new dynamic programming solution that optimally encodes n words in O(n/sup C+2/) time.
Abstract: We consider the problem of constructing prefix-free codes of minimum cost when the encoding alphabet contains letters of unequal length. The complexity of this problem has been unclear for thirty years with the only algorithm known for its solution involving a transformation to integer linear programming. We introduce a new dynamic programming solution to the problem. It optimally encodes n words in O(n/sup C+2/) time, if the costs of the letters are integers between 1 and C. While still leaving open the question of whether the general problem is solvable in polynomial time, our algorithm seems to be the first one that runs in polynomial time for fixed letter costs.
95 citations
Cites background from "Optimum lopsided binary trees"
...Since then, different authors have studied various aspects of the problem such as finding bounds on the cost of the solution [1, 13, 19] or solving the special case in which all codewords are equally likely to occur (pi = 1/n for all i) [15, 6, 7, 11, 18, 23, 5], but not much is known about the general problem, not even if it is NP-hard or solvable in polynomial time....
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TL;DR: Questions related to counting and representing code and parse trees are discussed and variants of Huffman coding in which the assignment of 0s and 1s within codewords is significant such as bidirectionality and synchronization are discussed.
Abstract: This paper surveys the theoretical literature on fixed-to-variable-length lossless source code trees, called code trees, and on variable-length-to-fixed lossless source code trees, called parse trees. In particular, the following code tree topics are outlined in this survey: characteristics of the Huffman (1952) code tree; Huffman-type coding for infinite source alphabets and universal coding; the Huffman problem subject to a lexicographic constraint, or the Hu-Tucker (1982) problem; the Huffman problem subject to maximum codeword length constraints; code trees which minimize other functions besides average codeword length; coding for unequal cost code symbols, or the Karp problem, and finite state channels; and variants of Huffman coding in which the assignment of 0s and 1s within codewords is significant such as bidirectionality and synchronization. The literature on parse tree topics is less extensive. Treated here are: variants of Tunstall (1968) parsing; dualities between parsing and coding; dual tree coding in which parsing and coding are combined to yield variable-length-to-variable-length codes; and parsing and random number generation. Finally, questions related to counting and representing code and parse trees are also discussed.
84 citations
TL;DR: Large deviations are used to prove a general theorem on the asymptotic edge-weighted height Hn* of a large class of random trees for which HN* ∼ c log n for some positive constant c.
Abstract: We use large deviations to prove a general theorem on the asymptotic edge-weighted height Hn* of a large class of random trees for which Hn* ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees, random recursive trees and plane oriented trees for instance. New applications include the heights of some random lopsided trees and of the intersection of random trees.
52 citations
Cites background from "Optimum lopsided binary trees"
...New applications include the heights of some random lopsided trees [19] and of the intersection of random trees....
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...Such encodings lead to trees whose edges have non-equal length, lopsided trees [19], [6]....
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19 May 2002
TL;DR: In this paper, a polynomial-time approximation algorithm for the problem was proposed, where the letters of the encoding alphabet may have non-uniform lengths, where cost s is the sum of the lengths of the letters in s.
Abstract: (MATH) In the standard Huffman coding problem, one is given a set of words and for each word a positive frequency. The goal is to encode each word w as a codeword c(w) over a given alphabet. The encoding must be prefix free (no codeword is a prefix of any other) and should minimize the weighted average codeword size Σw freq w, |c(w)|. The problem has a well-known polynomial-time algorithm due to Huffman [15].Here we consider the generalization in which the letters of the encoding alphabet may have non-uniform lengths. The goal is to minimize the weighted average codeword length Σw freq (w) cost(c(w)), where cost s is the sum of the (possibly non-uniform) lengths of the letters in s. Despite much previous work, the problem is not known to be NP-hard, nor was it previously known to have a polynomial-time approximation algorithm. Here we describe a polynomial-time approximation scheme (PTAS) for the problem.
38 citations
References
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TL;DR: This final installment of the paper considers the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now.
Abstract: In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.
65,425 citations
Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
Book•
01 Jan 1968TL;DR: This chapter discusses Coding for Discrete Sources, Techniques for Coding and Decoding, and Source Coding with a Fidelity Criterion.
Abstract: Communication Systems and Information Theory. A Measure of Information. Coding for Discrete Sources. Discrete Memoryless Channels and Capacity. The Noisy-Channel Coding Theorem. Techniques for Coding and Decoding. Memoryless Channels with Discrete Time. Waveform Channels. Source Coding with a Fidelity Criterion. Index.
6,684 citations
"Optimum lopsided binary trees" refers background in this paper
...a generalization of Kraft’s inequality [ 9 ] given in [ 151....
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5,562 citations
Book•
30 Nov 1961
TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.
Abstract: Matrix Properties and Concepts.- Nonnegative Matrices.- Basic Iterative Methods and Comparison Theorems.- Successive Overrelaxation Iterative Methods.- Semi-Iterative Methods.- Derivation and Solution of Elliptic Difference Equations.- Alternating-Direction Implicit Iterative Methods.- Matrix Methods for Parabolic Partial Differential Equations.- Estimation of Acceleration Parameters.
5,317 citations