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Topic: Huffman tree

	Huffman coding - Wikipedia, the free encyclopedia
		Huffman was able to design the most efficient compression method of this type: no other mapping of individual source symbols to unique strings of bits will produce a smaller average output size when the actual symbol frequencies agree with those used to create the code.
		Huffman, unable to prove any codes were the most efficient, was about to give up and start studying for the final when he hit upon the idea of using a frequency-sorted binary tree, and quickly proved this method the most efficient.
		Huffman coding with unequal letter costs is the generalization in which this assumption is no longer assumed true: the letters of the encoding alphabet may have non-uniform lengths, due to characteristics of the transmission medium.
	en.wikipedia.org /wiki/Huffman_encoding (1817 words)

	PlanetMath: Huffman coding
		Huffman coding is a method of lossless data compression, and a form of entropy encoding.
		The basic idea is to map an alphabet to a representation for that alphabet, composed of strings of variable size, so that symbols that have a higher probability of occurring have a smaller representation than those that occur less often.
		The mapping is obtained by the path from the root of the Huffman tree to the leaf associated with a symbol's weight.
	planetmath.org /encyclopedia/HuffmanCoding.html (660 words)

	Huffman Code Discussion and Implementation
		Huffman coding is a statistical technique which attempts to reduce the amount of bits required to represent a string of symbols.
		The Huffman code for an alphabet (set of symbols) may be generated by constructing a binary tree with nodes containing the symbols to be encoded and their probabilities of occurrence.
		By using the symbol counts and the same tree generation algorithm that the encoding algorithm use, a tree that matching the encoding tree may be constructed.
	michael.dipperstein.com /huffman/huffman.html (2581 words)

	Huffman Codes
		A Huffman tree is a binary tree in which the leaves hold symbols from the alphabet.
		The string that a symbol is mapped to is determined by tracing a path from the root of the tree to the symbol.
		The frequency of the new tree is the sum of the frequencies of the two trees used to construct the new tree.
	www.cs.unm.edu /~maccabe/classes/259/huf.html (676 words)

	oop2.huffman
		This class represents compound Huffman trees, where a bit must be used to encode which of two Huffman trees the encoded symbols is contained in.
		Note that the frequency of a tree is the sum of the frequencies of the trees it is formed from.
		This tree is the Huffman tree for the entire set of messages, and gives the minimal average bit length for encoding messages occurring with the specified frequencies.
	www.cs.umd.edu /~ykim/oop2/huffman/package-summary.html (1034 words)

	Huffman Codes
		A Huffman tree is an extended binary tree in which the leaf nodes contain the characters of the alphabet.
		The tree is structured such that the path to a leaf node is determined by the bit code, where 0 is interpreted as left and 1 is interpreted as right.
		Huffman codes are optimal in that sense that they produce minimal length encodings given the frequency with which characters occur.
	www.cs.uiowa.edu /~kearney/22c30Fall01/program4-F01.html (708 words)

	Huffman Coding: A CS2 Assignment
		All trees are one node, with the weight of the tree equal to the weight of the character in the node.
		Note that this tree is different from the tree we used to illustrate Huffman coding above, and the bit patterns for each character are different, but the total number of bits used to encode "go go gophers" is the same.
		The tree that actually results from a programmed implementation of Huffman's algorithm will be the same each time the program is run for the same weights (assuming no randomness is used in creating the tree).
	www.cs.duke.edu /csed/poop/huff/info (3345 words)

	Huffman Compression
		The purpose of the tree is to associate each byte value with a bit string of variable length.
		To compress the file, the Huffman algorythm reads the file a second time, converting each byte value into the bit string assigned to it by the Huffman Tree and then writing the bit string to a new file.
		Beginning at the root node in the Huffman Tree and depending on the value of the bit, you take the right or left branch of the tree and then return to read another bit.
	www.geocities.com /SiliconValley/2151/huffman.html (1073 words)

	Algorithms / Huffman Compression Codes - Games++
		Huffman Compression, also known as Huffman Encoding, is one of many compression techniques in use today.
		To compress the file, the Huffman agorithm reads the file a second time, converting each byte value into the bit string assigned to it by the Huffman Tree and then writing the bit string to a new file.
		Compression then involves traversing the tree beginning at the leaf node for the character to be compressed and navigating to the root.
	www.gamespp.com /algorithms/huffmanCompressionCodes.html (955 words)

	Huffman Trees (Site not responding. Last check: 2007-10-22)
		This tree describes a varying-length binary encoding for each character in the input string such that the length of the encoded string is minimized.
		The Huffman tree is displayed, along with a table giving, for each character in the string, its original encoding in hexadecimal (this helps to identify non-printing characters), its binary Huffman code, and its number of occurrences in the string.
		The Huffman code actually describes the path from the root of the tree to the node containing the encoded character: a 0 represents an edge to a left child, and a 1 represents an edge to a right child.
	www.cis.ksu.edu /~rhowell/viewer/huffman.html (623 words)

	Class notes CS251B -- Winter 1997
		A Huffman tree is a binary tree which minimizes the sum of f(i)D(i) over all leaves i, where f(i) is the frequency or weight of leaf i and D(i) is the length of the path from the root to leaf i.
		A Huffman code it is obtained by using the codewords obtained by constructing the Huffman tree for the probabilities.If the probabilities are exact, then the Huffman code gives optimal average number of bits per input symbol among all prefix codes, but not necessarily optimal among all compression methods (see also Lempel-Ziv Compression)!
		One of the fundamental properties of Huffman trees is the smallest frequencies are the furthest away from the root and siblings merge into their parent.
	www.cs.mcgill.ca /~cs251/OldCourses/1997/topic22 (2256 words)

	Huffman 10/95
		Huffman trees are binary trees which are based on the probability distribution of a symbol set and the principle that symbols occurring more frequently will be represented by shorter codes than other, less probable ones.
		Huffman's scheme is based on statistical coding which means that the probability of a symbol has a direct bearing on the length of its representation.
		Huffmanís variable length storage scheme elegantly solves that problem by providing means to detect "spaces" between symbols during the decoding process.
	www.sbg.ac.at /geo/agit/papers96/skupin.htm (2882 words)

	Adaptive Huffman Coding
		Huffman coding suffers from the fact that the uncompresser need have some knowledge of the probabilities of the characters in the compressed files.
		Because the Huffman tree is constructed from the character counts of the file as a whole, it works effectively at a universal level.
		This algorithm is called adaptive huffman coding because the tree is adaptive- it is created simultaneously with either the compressed or uncompressed file as it reads in the other.
	www.cs.duke.edu /csed/curious/compression/adaptivehuff.html (1951 words)

	HUFFMAN VS DYNAMIC HUFFMAN -- Winter 1999 (Site not responding. Last check: 2007-10-22)
		Generaly, in Huffman coding, a binary tree is used to construct minimal length encodings for messages, when the frequency of letters or symbols used in the message is known.
		The dynamic Huffman coding, allows the encoder and the decoder to build the Huffman tree dynamically as the characters are transmitted, and hence construct the codeword table on the fly.
		However since the dynamic Huffman tree, follows the same pattern as the ordinary Huffman tree (lower frequencies at the bottom, higher frequencies at the top) there might be a violat ion of this rule if we increase the frequencies, in a random manner..
	www.cs.mcgill.ca /~yfaras/project.html (1063 words)

	DataCompression.info - Huffman Coding (Site not responding. Last check: 2007-10-22)
		The first pass generates a huffman tree and the second pass encodes the file.
		The decoder is one pass and uses a huffman code table at the beginning of the compressed file to decode the file.
		This type of Huffman coding follows some specific rules regarding the structure of the Huffman tree that simplify the process of transmitting the tree.
	datacompression.info /Huffman.shtml (3319 words)

	Compression and the Huffman Code
		The bit sequence obtained from passing from the tree's root to the symbol is its Huffman code.
		The binary tree created by the algorithm extends to the right, with the root node (the one at which the tree begins) defining the codewords.
		The Huffman coding tree for the second set of probabilities is identical to that for the first (figure 1).
	cnx.org /content/m0092/latest (769 words)

	CS216: Problem Set 4: Huffman Coding (Site not responding. Last check: 2007-10-22)
		There are many implementations of Huffman encoding and decoding available on the web, and it would certainly defeat the purpose of this assignment if you copied one of them instead of thinking on your own.
		Huffman worked on the problem for months, developing a number of approaches, but none that he could prove to be the most efficient.
		Huffman encoding was developed in a term paper David Huffman wrote instead of taking the final exam in an information theory course, and is now used in many applications including MPEG and MP3.
	www.cs.virginia.edu /cs216/ps/ps4 (1710 words)

	Huffman algorithm, making codes from probabilities
		To achieve optimality huffman joins the two symbols with lowest probability and replaces them with a new fictive node whose probability is the sum of the other nodes' probabilities (a fictive node is a node that doesn't hold a symbol).
		Now we can make the code table by recursion: a function is called with a pointer to the root, it then calls itself with the pointer of the left child, and after that with a pointer to the root node.
		Because the codes are restricted to be of an integer length, huffman can't achieve the ideal code length, it's however close.
	www.arturocampos.com /cp_ch3-1.html (996 words)

	[No title]
		To read the codes from a Huffman tree, start from the root and add a '0' every time you go left to a child, and add a '1' every time you go right.
		Notice that since all the characters are at the leafs of the tree (the ends), there is never a chance that one code will be the prefix of another one (eg.
		The decoder then can use the Huffman tree to decode the string by following the paths according to the string and adding a character every time it comes to one.
	www.cs.sfu.ca /cs/CC/365/li/squeeze/Huffman.html (502 words)

	Static Huffman by Arturo Campos
		Huffman's method is efficient, while Shannon-Fano isn't always efficient, probably this is because huffman sorts the probabilities after combining them, and Shannon-Fano doesnn't.
		The '1' is the root of the tree.
		The pointer to the binary tree points to the start of the binary tree, and when the tree is being build to the next position.
	www.arturocampos.com /ac_static_huffman.html (2728 words)

	Huffman Codes (Site not responding. Last check: 2007-10-22)
		Binary trees can be used in an interesting way to construct minimal length encodings for messages when the frequency of letters used in the messages is known.
		A special kind of binary tree, called a Huffman coding tree is used to accomplish this.
		To build the Huffman tree, we sort the frequencies into increasing order (4, 5, 7, 8, 12, 29).
	www.cs.uidaho.edu /~karenv/cs213/cs213.useful.pages/huffman.html (337 words)

	Adaptive Huffman Coding
		The key is to have both encoder and decoder to use exactly the same initialization and update_model routines.
		During the updates, the Huffman tree will be maintained its sibling property, i.e.
		The Huffman tree could look very different after node swapping (Fig 7.2), e.g., in the third tree, node A is again swapped and becomes the #5 node.
	www.cs.cf.ac.uk /Dave/Multimedia/node212.html (279 words)

	Data Compression -- Section 4
		The basis for algorithm FGK is the Sibling Property, defined by Gallager [Gallager 1978]: A binary code tree has the sibling property if each node (except the root) has a sibling and if the nodes can be listed in order of nonincreasing weight with each node adjacent to its sibling.
		First, the number of interchanges in which a node is moved upward in the tree during a recomputation is limited to one.
		When an exchange of nodes is required to maintain the sibling property, algorithm V requires that the node being promoted be moved to the position currently occupied by the highest-numbered node in the target block.
	www.ics.uci.edu /~dan/pubs/DC-Sec4.html (1768 words)

	Encoding - Huffman Coding (Site not responding. Last check: 2007-10-22)
		The idea behind Huffman coding is simply to use shorter bit patterns for more common characters, and longer bit patterns for less common characters.
		Label all the left branches of the tree with a 0 and all the right branches with a 1.
		The code for each of the letters is the sequence of 0's and 1's that lead to it on the tree, starting from the symbol with a probability of 1.
	educ.queensu.ca /~compsci/units/encoding/huffman.html (493 words)

	Self-Similar Huffman Trees
		Any binary tree can be thought of as a prefix code, whose codewords are taken to be the bit-strings corresponding to paths from the root to the leaves of the tree.
		It turns out [1] that the entropy of a Huffman tree is a measure of the optimal compression of a sequence of random objects drawn according to probabilities p
		This document identifies Huffman trees in which the chance of beating the carny not only falls below even odds, but in fact goes to zero in the limiting case.
	www.ima.umn.edu /~pliam/doc/huff (590 words)

	Building a Huffman tree
		Each step involves taking off the two lowest weight trees and replacing them with a new tree having their combined weight.
		The program shows the steps in the construction and displays the Huffman tree.We also need a revised priority tree header file and the heap file heap.cpp.
		We have taken off nodes with frequencies 2 and 4 We have taken off nodes with frequencies 5 and 6 We have taken off nodes with frequencies 8 and 10 We have taken off nodes with frequencies 11 and 16 We have taken off nodes with frequencies 18 and 27 This is the Huffman tree.
	www.csm.astate.edu /~rossa/cs3363/hufftree.html (786 words)

	Math Forum Discussions - Huffman codes and Fibonacci numbers
		The binary tree size is number of its terminal nodes.
		----------------- on a left-sided binary tree L(n) --------------
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?forumID=13&threadID=49181 (1432 words)

	Huffman Coding of ACIS Pixel Data
		The integers {i} become leaves of a tree, associated with which are bit strings whose values may be determined by traversing the tree, starting at the top, and concatenating 1 or 0 depending on which sub-branch was taken.
		The Huffman tree is not particularly useful when compressing the original integers.
		After creating the Huffman tree and table, the length of the bit string corresponding to this out-of-range value is inspected.
	acis.mit.edu /huff (3934 words)

	Huffman coding
		The Huffman algorithm converts characters into bit strings using a binary tree containing all possible characters.
		We are not concerned with the construction of the tree initially.
		The Huffman code for a character may be obtained by traversing the tree, where a left branch is choosen we have the bit 0, if a right branch is taken the bit is 1.
	www.dcs.napier.ac.uk /~andrew/comp/huffman (615 words)

	Practical Huffman Coding (Site not responding. Last check: 2007-10-22)
		Example: ((a,b),c) denotes a binary tree where the left child of the root is a node with a as left child, b as right child.
		Textbooks usually will not tell you, but typically there is more than one huffman tree for a distribution, so there is more than one huffman code.
		The first code was derived directly from the tree; code2, code3 and the code labeled canonical are some other prefix codes with the same length.
	www.compressconsult.com /huffman (3148 words)

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