The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

We consider the Windows Scheduling (WS) problem, which is a restricted version of Unit-Fractions Bin Packing, and it is also called Inventory Replenishment in the context of Supply Chain. In brief, WS problem is to schedule the use of communication channels to clients. Each client c i is characterized by an active cycle and a window w i . During the period of time that any given client c i is active, there must be at least one transmission from c i scheduled in any w i consecutive time slots, but at most one transmission can be carried out in each channel per time slot. The goal is to minimize the number of channels used. We extend previous online models, where decisions are permanent, assuming that clients may be reallocated at some cost. We assume that such cost is a constant amount paid per reallocation. That is, we aim to minimize also the number of reallocations. We present three online reallocation algorithms for Windows Scheduling. We evaluate experimentally multiple variants of these protocols showing that, in practice, all three achieve constant amortized reallocations with close to optimal channel usage. Our simulations also expose interesting tradeoffs between reallocations and channel usage. We introduce a new objective function for WS with reallocations that can be also applied to models where reallocations are not possible. We analyze this metric for one of the algorithms that, to the best of our knowledge, is the first online WS protocol with theoretical guarantees that applies to scenarios where clients may leave and the analysis is against current load rather than peak load. Using previous results, we also observe bounds on channel usage for one of the algorithms.

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Sequential Adaptive Compressed Sampling via Huffman Codes

Sampling Theory, Signal Processing, and Data Analysis ◽

10.1007/bf03549543 ◽

2011 ◽

Vol 10 (3) ◽

pp. 231-254

Author(s):

Akram Aldroubi ◽

Haichao Wang ◽

Kourosh Zarringhalam

Keyword(s):

Huffman Codes ◽

Compressed Sampling

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A Huffman Codes Based Approach to Achieve Minimum Redundancy

2010 Second International Conference on Computer Modeling and Simulation ◽

10.1109/iccms.2010.177 ◽

2010 ◽

Author(s):

J. S. Bhullar ◽

Parvinder S. Sandhu ◽

Manish Gupta

Keyword(s):

Huffman Codes

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Monochromatic solutions to equations with unit fractions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700029221 ◽

1991 ◽

Vol 43 (3) ◽

pp. 387-392 ◽

Cited By ~ 7

Author(s):

Tom C. Brown ◽

Voijtech Rödl

Keyword(s):

Solutions To Equations ◽

Unit Fractions ◽

Image Position ◽

Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

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Modified HuffBit Compress Algorithm – An Application of R

Journal of Integrative Bioinformatics ◽

10.1515/jib-2017-0057 ◽

2018 ◽

Vol 15 (3) ◽

Cited By ~ 5

Author(s):

Nahida Habib ◽

Kawsar Ahmed ◽

Iffat Jabin ◽

Mohammad Motiur Rahman

Keyword(s):

Dna Sequences ◽

Binary Tree ◽

Deoxyribonucleic Acid ◽

Repetitive Sequences ◽

Huffman Codes ◽

R Language ◽

Living Organisms ◽

Encoding Method ◽

Dna Sequence Compression ◽

Sequence Compression

Abstract The databases of genomic sequences are growing at an explicative rate because of the increasing growth of living organisms. Compressing deoxyribonucleic acid (DNA) sequences is a momentous task as the databases are getting closest to its threshold. Various compression algorithms are developed for DNA sequence compression. An efficient DNA compression algorithm that works on both repetitive and non-repetitive sequences known as “HuffBit Compress” is based on the concept of Extended Binary Tree. In this paper, here is proposed and developed a modified version of “HuffBit Compress” algorithm to compress and decompress DNA sequences using the R language which will always give the Best Case of the compression ratio but it uses extra 6 bits to compress than best case of “HuffBit Compress” algorithm and can be named as the “Modified HuffBit Compress Algorithm”. The algorithm makes an extended binary tree based on the Huffman Codes and the maximum occurring bases (A, C, G, T). Experimenting with 6 sequences the proposed algorithm gives approximately 16.18 % improvement in compression ration over the “HuffBit Compress” algorithm and 11.12 % improvement in compression ration over the “2-Bits Encoding Method”.

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