Huffman Coding Example With Probabilities

Arithmetic coding is another statistical coding technique. At universities of all over the world many similar problems were solved, like at the one where David Huffman was studying. Consider system of n processes {p0, p1, … pn-1} Every process has critical…. Problem Summary: Read a text file. The first in the series is a paper by Gallager, "Variations on a Theme by Huffman", IEEE-IT, vol. Huffman codes are used for compressing data efficiently from 20% to 90%. Example implementation of Huffman coding in Python - huffman. Welcome to Compression Consulting's huffman coding hints. NET to Create DataSet. These are the types of questions asked in GATE based on Huffman Encoding. Also note that we are trying to code each quantized DCT 8x8 block of an image matrix. Huffman Algorithm. Luckily for us, American coinage works great for this approach. Custom Huffman code dictionary generator,encoder and decoder functions All functions support debug mode, which creates a log file of execution with several infos about each execution. • Second 1. Now, for example, we will give a coding using variable length strings that is based on the Huffman Tree for weighted data item as follows: - The Huffman Code for Ternary Tree. It will focus on practical issues you need to know for writing a fast and reasonable memory efficient huffman coder. There's Only 1 (Pretty Big) Problem. • Encode source symbol-by-symbol: symbol i → code i. Colors make it clearer, but they are not necessary to understand it (according to Wikipedia's guidelines): probability is shown in red, binary code is shown in blue inside a yellow frame. I wrote a small app that generated a sine wave via a C4353 DAC. Huffman (1951) Huffman coding uses frequencies of symbols in a string to build a variable rate prefix code Each symbol is mapped to a binary string More frequent symbols have shorter codes No code is a prefix of another 27/02/2011 Applied Algorithmics - week7 1 Example: A 0 B 100 C 101 D 11 D B C A 1 1 0 1 0 0 Variable. Submitted by Abhishek Kataria, on June 23, 2018 Huffman coding. The construction of a Huffman code is best illustrated by example. Example of Huffman Coding. A Huffman encoding can be computed by first creating a tree of nodes:. If 50% of the fish are bass and the rest are evenly divided among 15 other species, how many bits would be used to encode the species when a bass is tagged?. The expected output of a program for custom text with 100 000 words: 100 000 words compression (Huffman Coding algorithm) Algorithms Data structures Huffman. Reference Huffman coding. comp = huffmanenco(sig,dict) encodes the signal sig using the Huffman codes described by the code dictionary dict. I know it is a little messy, but it works (had to get it done quickly). A method for the construction of minimum-redundancy codes. The following Matlab project contains the source code and Matlab examples used for huffman code. Huffman coding - notes There are cases in which the Huffman coding does not uniquely determine codeword lengths, due to the arbitrary choice among equal minimum probabilities. 1 million symbols 16 MB of memory! Moreover traversing a tree from root to leaf involves follow a lot of pointers, with little locality of reference. Huffman Encoding is an important topic from GATE point of view and different types of questions are asked from this topic. Main Functions huffmandict_( alphabet, prob, debug ) Based on the probability of each symbol, generates the Huffman codebook. using the algorithm described by Li and Drew (algorithm 7. Huffman compression is a lossless compression algorithm that is ideal for compressing text or program files. Arithmetic coding is an alternative approach for efficient entropy encoding and it achieves compression efficiency very close to the entropy limit. The objective of this is to implement the Huffman coding algorithm using the binary tree data structure. the compression ratio is higher compared to huffman coding. Code for set of probabilities. If not, then it's not a Huffman code. 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. There are lots of papers that study exactly the problem you mention. The Huffman Coding Algorithm. The code can be used for study, and as a solid basis for modification and extension. Huffman Coding Huffman Encoding Algorithms use the probability distribution of the alphabet of the source to develop the code words for symbols. Colors make it clearer, but they are not necessary to understand it (according to Wikipedia's guidelines): probability is shown in red, binary code is shown in blue inside a yellow frame. Custom Huffman code dictionary generator,encoder and decoder functions All functions support debug mode, which creates a log file of execution with several infos about each execution. Suppose, for example, that we have six events with names and probabilities given in the table below. Notes on Huffman Code Frequencies computed for each input Must transmit the Huffman code or frequencies as well as the compressed input. ‘ Ibe ffrst result shows that a binary prefix condition code is a Huffmau code iff the. In the end, everything about Hollywood is basically absurd, so the same is true for the prosecution of two actresses, Felicity Huffman and Lori Loughlin, on charges they paid bribes to get their. Huffman coding (Section 3. Problem Summary: Read a text file. using the algorithm described by Li and Drew (algorithm 7. Submitted by Abhishek Kataria, on June 23, 2018 Huffman coding. 0 Introduction. ; It uses variable length encoding where variable length codes are assigned to all the characters depending on how frequently they occur in the given text. This allows more efficient compression than fixed-length codes. net dictionary. The professor, Robert M. Building a Huffman Tree. What is the minimum number of bits to store the compressed database? ~2. Given An array of Alphabets and their frequency. Huffman codes are formulated to be an optimal code, i. In the "show steps" mode, this Demonstration illustrates the step-by-step procedure for finding the Huffman code for a set of characters with given probabilities. What is the minimum number of bits to store the compressed database? ~2. (iii) Huffman's greedy algorithm uses a table of the frequencies of occurrences of each character to build up an optimal way of representing each character as a binary string. implements Huffman coding over the remaining symbols, retaining the same probabilities proportionally; i. Greedy Algorithms – Huffman Coding • Huffman Coding Problem Example: Release 29. Huffman Coding For huffman coding one creates a binary tree of the source symbols, using the probabilities in P(x). Huffman Tree's C++ code Using Huffman Tree to code is an optimal solution to minimize the total length of coding. 15 Obtain Huffman coding. In our last article on compression we showed you how. In the alphabetic version, the alphabetic order of inputs and outputs must be identical. Huffman codes are of variable-length, and prefix-free (no code is prefix of any other). Prefix codes, and thus Huffman coding in particular, tend to have inefficiency on small alphabets, where probabilities often fall between these optimal (dyadic) points. A Dynamic Programming Approach To Length-Limited Huffman Coding Mordecai Golin, Member, IEEE, and Yan Zhang Abstract—The “state-of-the-art” in Length Limited Huffman Coding algorithms is the Θ(ND)-time, Θ(N)-space one of Hirschberg and Larmore, where D ≤ N is the length restriction on the code. NET with QODBC Sample Code of VB. Algorithm FGK transmits 47 bits for this ensemble while the static Huffman code requires 53. Huffman codes are formulated to be an optimal code, i. I mean without using the recursion in huffmandict? The explanation is in question itself. Tags application of huffman coding in data compression c program to implement huffman code huffman code generator huffman coding algorithm in c huffman coding c program huffman coding c++ huffman coding example huffman coding example pdf huffman coding example with probabilities huffman coding in c huffman coding java huffman coding program in. comp = huffmanenco(sig,dict) encodes the signal sig using the Huffman codes described by the code dictionary dict. I'm using it because. Arithmetic coding is another statistical coding technique. Now, we know what is Huffman code and how it works. " The sum is then positioned. For Huffman to work well there must be values in the data that occur more frequently than others. In the field of data compression, Shannon-Fano coding, named after Claude Shannon and Robert Fano, is a technique for constructing a prefix code based on a set of symbols and their probabilities (estimated or measured). (points: 1) D. 6400 = 0 and 0. The technique for finding this code is sometimes called Huffman-Shannon-Fano coding, since it is optimal like Huffman coding, but alphabetic in weight probability, like Shannon-Fano coding. What does Huffman coding mean? Information and translations of Huffman coding in the most comprehensive dictionary definitions resource on the web. 4 Variable-Length Coding Assuming that some of the symbols are more likely than others (and assuming we know the respective probabilities of occurrence), the key idea to obtaining a more efficient coding is to use variable-length coding. Code for set of probabilities. That is, we assign shorter. Create a binary tree out of these two symbols, labeling one branch with a "1" and the other with a "0". When doing extended Huffman coding, I understand that you do for example a1a1,a1a2,a1a3 etc and you do their probabilities times, however, how do you get the codeword?. This page assumes that you are familiar with huffman coding. For Example. Education Secretary Betsy DeVos wipes her brow during an October 2017 appearance in Bellevue, Washington. Thus, for example, could not be assigned code , but instead should be assigned either or. The process for deriving an optimal prefix code for a given alphabet (the essence of Huffman coding) is straightforward. 5 Lempel-Ziv Coding. The professor, Robert M. Most frequent characters have the smallest codes and longer codes for least frequent. adaptive Huffman coding, Huffman decoding, prefix codes, binary search 1. Sort the symbols according to their probabilities. Arithmetic coding is another statistical coding technique. Code for set of probabilities. Create a binary tree out of these two symbols, labeling one branch with a "1" and the other with a "0". Algorithm of Huffman Code with daa tutorial, introduction, Algorithm, Asymptotic Analysis, Control Structure, Recurrence, Master Method, Recursion Tree Method. Now, for example, we will give a coding using variable length strings that is based on the Huffman Tree for weighted data item as follows: - The Huffman Code for Ternary Tree. Suppose that all of the symbol probabilities are negative powers of 2. The MATLAB program output for the example is given below: Enter the probabilities: [0. Chapter 1 Huffman Coding Steven Pigeon Universit´e de Montr´eal [email protected] A Huffman code is an example of a prefix code—no character has a code word that is a prefix of another character's code word. For example, a static Huffman coding process may be used in which a Huffman dictionary is first created for the dataset and then used to encode the data. 5 As a generalization of Problem A. For example, does the sequence 010 correspond to AC or BA? One way of preventing this problem is to require that the binary code of each character cannot be a prefix of any other. Huffman Coding - probabilities. Here’s the basic idea: each ASCII character is usually represented with 8 bits, but if we had a text filed composed of only the lowercase a-z letters we could represent each character with only 5 bits (i. The Huffman code is a way of compressing data streams by encoding the more frequent items with shorter words. decoding a given code word to find the corresponding encoded characters against the given Huffman Tree. If we consider groups of codes at a time, symbol-by-symbol Huffman coding is only optimal if the probabilities of the symbols are independent and are some power of a half, i. 1 Introduction Codes may be characterized by how general they are with respect to the distribution of symbols they are meant to code. 5, making the upper limit of inefficiency unbounded. Example 1 – Huffman code •Find Huffman code for Gray Level Probability 0 0. Save it in a file The original (uncompressed) image representation uses 8 bits/pixel. Huffman Encoding is an important topic from GATE point of view and different types of questions are asked from this topic. In particular, it is a prefix-free code (no codeword is the prefix of any other codeword) and hence uniquely decodable. An article on fast Huffman coding technique. 3) and arithmetic coding (Section 3. This is an. Working Code Example. Huffman-coding English words Codebook sample for Huffman coding: a problem is that if the codeword is a single letter, we can only encode lowercase and title. This post talks about fixed length and variable length encoding, uniquely decodable codes, prefix rules and construction of Huffman Tree. This is an. " The younger man's excitement and apprehension for White Fang was growing. (points: 1) D. Makes use of statistical coding - more frequently utilized symbols have shorter code words. 2 For Huffman must be kept small but for AC it can be VERY large! 13 1. Then the Huffman coding assigns to each symbol one bit, therefore each symbols is encoded exactly with one bits. 3 (B) and 0. Entropy coding • Entropy is a lower bound on the average number of bits needed to represent the symbols (the data compression limit). If you look closely, you may see I'm using TypeScript in the code. 3 An Application of Binary Trees: Huffman Code Construction REF. No libraries. Measure the coding efficiency of an encoding scheme. Huffman Algorithm was developed by David Huffman in 1951. Huffman and his MIT information theory classmates were given the choice of a term paper or a final exam. This is an implementation of the algorithm in C. The higher the probability, the shorter the code-sequence for this letter will be. Huffman coding is divided in to two categories:- 1. Huffman, unable to prove any codes were. It has many interesting properties. 1 Almost every lossy compression system contains a lossless compression system We discuss the basics of lossless compression first, then move on to lossy compression Lossless compression in lossy compression systems Transform Quantizer Lossless Encoder Lossless. The full source code is available at GitHub, written using C++11. The program first generates the dictionary of messages. A little information about huffman coing--- In computer science and information theory. INFORMATION, ENTROPY, AND CODING 8. In the PCM format, each of these values would be converted to a three-bit binary number. The Process: We begin with a table of symbols and their probabilities. You can learn more about TypeScript coding with VS Code here. Cancer-Linked Contaminants Found In Tap Water - Toms River, NJ - A new study found drinking water is often less safe than what the federal government may deem legal. IntroductionAn effective and widely used Application ofBinary Trees and Priority QueuesDeveloped by David. 1 of 15-Feb-2005 of TrEMBL Protein Database contains 1,614,107 sequence entries, comprising 505,947,503 amino acids. 310C Lecture Notes Spring 2010 Shannon’s noiseless coding theorem tells us how compactly we can compress messages in which all letters are drawn independently from an alphabet Aand we are given the probability p a of each letter a2Aappearing in the message. Arithmetic coding is another statistical coding technique. The average length, however, will be the same for all of these codes. The Core of Huffman Coding. • Compute probabilities of each symbol by histogramming source. Greedy Algorithms Huffman Coding Huffman Coding Problem Example Release 291 of from COT 6405 at Florida International University. , they achieve the shortest average code length (minimum average codeword length), which may still be greater than or equal to the entropy of source. This post talks about fixed length and variable length encoding, uniquely decodable codes, prefix rules and construction of Huffman Tree. efficiency is greater comparatively. Now construct a Huffman code eg. Describe the Huffman code. We have a text like "aaabc", with the probabilities you can see below. Even an asymptotically optimal universal code cannot compare with static Huffman coding on a source for which the probabilities of the messages are known. So let's look at a couple of examples where we'll apply this procedure and see that indeed it's quite straightforward to produce a Huffman code. My uncle, David A. Observation. The worst case for Huffman coding can happen when the probability of the most likely symbol far exceeds 2 −1 = 0. (ii) It is a widely used and beneficial technique for compressing data. Add the root node fx i;x. Let's first look at the binary tree given below. Custom Huffman code dictionary generator,encoder and decoder functions - lostpfg/Huffman-Matlab. Arithmetic coding is discussed in this segment which addresses some of the shortcomings of Huffman coding. ' Ibe ffrst result shows that a binary prefix condition code is a Huffmau code iff the. Huffman Coding is such a widespread method for creating prefix-free codes that the term "Huffman Code" is widely used as synonym for "Prefix Free Code". The first in the series is a paper by Gallager, "Variations on a Theme by Huffman", IEEE-IT, vol. The worst case for Huffman coding can happen when the probability of the most likely symbol far exceeds 2 −1 = 0. Homework Submission: 1. I am new to Huffman coding and I find myself facing a lot of confusion as to how to determine if a code is Huffman or not without having the probabilities associated to each codeword. 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. I try to imagine a Huffman tree for it, but it does not appear to make much sense, as trees for Huffman encoding aren't always balanced (so I can't use tree's height to calculate this I guess). Select two symbols with the lowest probabilities (if multiple symbols have the same probability, select two arbitrarily). There are lots of papers that study exactly the problem you mention. Huffman coding with unequal letter costs. I know one way is look at whether or not the code is prefix. One of the authors of that algorithm, Robert Shannon proposed the problem about searching for optimal variable-length code to his student David Huffman who at last came upon brilliant idea - to build the code-tree in "reverse" order - i. Arithmetic coding is another statistical coding technique. net dictionary. A Huffman tree represents Huffman codes for the character that might appear in a text file. The given probabilities are just suggestive. Take a list of symbols and their probabilities. GitHub Gist: instantly share code, notes, and snippets. It is an algorithm which works with integer length codes. A little information about huffman coing--- In computer science and information theory. Currently, there is a Java version there. Huffman Algorithm was developed by David Huffman in 1951. E-mail: [email protected] We have discussed the Modified Huffman coding scheme to enhance its efficiency. Combine n least likely probabilities. Are drugs a problem in your local fishery? Talk to us about it. What is the expected code length of the Grinch's code tree, in terms of L, the expected code length of the original code tree, as well as p1? Explain. Extended Huffman Code (1/2) If a symbol a has probability 0. You can extend this range by changing in the source code. While getting his masters degree, a professor gave his students the option of solving a difficult problem instead of taking the final exam. The Huffman-Shannon-Fano code corresponding to the example is {000,001,01,10,11} , which, having the same codeword lengths as the original solution, is also. Huffman coding. Huffman Coding (also known as Huffman Encoding) is a algorithm for doing data compression and it forms the basic idea behind file compression. Main Functions huffmandict_( alphabet, prob, debug ) Based on the probability of each symbol, generates the Huffman codebook. The average length, however, will be the same for all of these codes. What is the minimum number of bits to store the compressed database? ~2. A '1' when is added to the code when we move right in the binary tree. And that is all for Huffman Coding implementation, with compression and decompression. I mean without using the recursion in huffmandict? The explanation is in question itself. Say, for example, a file starts out with a series of a character that are not repeated again in the file. An example of how to implement huffman in python. A Dynamic Programming Approach To Length-Limited Huffman Coding Mordecai Golin, Member, IEEE, and Yan Zhang Abstract—The "state-of-the-art" in Length Limited Huffman Coding algorithms is the Θ(ND)-time, Θ(N)-space one of Hirschberg and Larmore, where D ≤ N is the length restriction on the code. Huffman Code - College of More Examples. For example, if you use letters as symbols and have details of the frequency of occurrence of those letters in typical strings, then you could just en. Example: Suppose a frame of 10 values has the sequence 2,3,4,3,2,1,0,1,2,2. Huffman coding is a lossless data encoding algorithm. Even an asymptotically optimal universal code cannot compare with static Huffman coding on a source for which the probabilities of the messages are known. Huffman Coding Problem: Given a set of symbols and their frequency of usage, find a binary code for each symbol, such that: a. There's Only 1 (Pretty Big) Problem. This article contains basic concept of Huffman coding with their algorithm, example of Huffman coding and time complexity of a Huffman coding is also prescribed in this article. 5 G bits or 300MB. 6400 = 0 and 0. The description is mainly taken from Professor Vijay Raghunathan. Supposing you already read the story about Shannon-Fano Coding (and even probably solved the exercise) let us now learn the sequel of it. The Huffman Coding is a lossless data compression algorithm, developed by David Huffman in the early of 50s while he was a PhD student at MIT. There is an optimal prefix code with tree T* where. I know that at least two least-probable symbols will have longest codings, but how to relate element's probability with its Huffman code length?. The question is whether Huffman's algorithm can be generalize to handle ternary codewords, that is codewords using the symbols 0,1 and 2. Assign Huffman codes to a set of symbols of known probabilities. Arithmetic coding is another statistical coding technique. Argue that for an optimal Huffman-tree, anysubtree is optimal (w. But Microsoft says it will ship with the new Surface Pro X. The huffmandict, huffmanenco, and huffmandeco functions support Huffman coding and decoding. Huffman codes are of variable-length, and prefix-free (no code is prefix of any other). The average length, however, will be the same for all of these codes. We have discussed the Modified Huffman coding scheme to enhance its efficiency. Problem Create Huffman codewords for the characters. Lemma: Let be a full binary tree representing an optimal prefix code over an alphabet , where fre-quency is defined for each character. You can extend this range by changing in the source code. How to implement Huffman Encoding without using the inbuilt function 'Huffmandict' in matlab. Huffman coding (Section 3. Let's first look at the binary tree given below. Huffman coding example. Add the root node fx i;x. Example implementation of Huffman coding in Python - huffman. Frozen raspberries sold under the Aldi and Raley’s supermarket private label brands were recalled Thursday because of possible contamination with the hepatitis A virus, the Food and Drug. Any researcher can access the paper and the code, test out. These messages are nothing but codes or bitstreams from 00 to 1001 in this example. It compresses data very effectively saving from 20% to 90% memory, depending on the characteristics of the data being compressed. No code is a prefix of another. Count the number of occurrences of each symbol (character) in the file. ECE264: Huffman Coding. Now construct a Huffman code eg. 5, making the upper limit of inefficiency unbounded. In some cases, a sufficiently accurate source model is difficult to obtain, especially when several types of data (such as text, graphics, and natural pictures. In fact, the code for C is the same as the code for N repeated twice. We consider the data to be a sequence of characters. So the first tree you drew is incorrect, in that it is not the result of applying Huffman's algorithm. We have a text like "aaabc", with the probabilities you can see below. Opting for what he thought was the easy way out, my uncle tried to find a solution to the “smallest code” problem. Count the number of occurrences of each symbol (character) in the file. • The resulting code is called a Huffman code. In this assignment, you will utilize your knowledge about priority queues, stacks, and trees to design a file compression program and file decompression program (similar to zip and unzip). Static Huffman Coding example Example: Information to be transmitted over the internet contains the following characters with their associated frequencies: Use Huffman technique to answer the following questions: Build the Huffman code tree for the message. The code can be used for study, and as a solid basis for modification and extension. Prefix codes, and thus Huffman coding in particular, tend to have inefficiency on small alphabets, where probabilities often fall between these optimal (dyadic) points. The letters of Table 7. This is first assuming that the coding alphabet is binary, as it is within the computer, a more general case will be shown after. 17 hours ago · Sampo Group’s profit before taxes for January - September 2019 amounted to EUR 1,073 million (1,643). The equivalent fixed-length code would require about five bits. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. Now, we know what is Huffman code and how it works. /** * Huffman encoding obeys the huffman algorithm. a L R L R b c Figure 1: Tree. Example: Suppose a frame of 10 values has the sequence 2,3,4,3,2,1,0,1,2,2. A detailed explaination of Huffman coding along with the examples is solved here. The question is whether Huffman's algorithm can be generalize to handle ternary codewords, that is codewords using the symbols 0,1 and 2. Generate a Huffman code for compressed image. The argument sig can have the form of a numeric vector, numeric cell array, or alphanumeric cell array. No code is a prefix of another. The Huffman code is a way of compressing data streams by encoding the more frequent items with shorter words. • Entropy coding methods: • Aspire to achieve the entropy for a given alphabet, BPS Entropy • A code achieving the entropy limit is optimal BPS : bits per symbol original message encoded message BPS 8. The story of David Huffman and his coding The year 1951 was written. Assign Huffman codes to a set of symbols of known probabilities. net dictionary. The path from the top or root of this tree to a particular event will determine the code group we associate with that event. 1 are atypical in that there are too many common letters compared to the number of rare letters. Alternatively, an adaptive Huffman coding process may be used in which the Huffman tree is updated as new symbols are encountered without first creating the Huffman dictionary. It's called greedy because the two smallest nodes are chosen at each step, and this local decision results in a globally optimal encoding tree. When working under. We have a text like "aaabc", with the probabilities you can see below. For example, the code for a is L since we go down the left subtree from the root to reach a. Example of Huffman Coding. DeVos’ formula for success: Trash public schools and push privatization U. 1 Introduction Codes may be characterized by how general they are with respect to the distribution of symbols they are meant to code. A Huffman tree represents Huffman codes for the character that might appear in a text file. using the algorithm described by Li and Drew (algorithm 7. For this example Entropy < Avg. SUMMARY The average code length of ordinary Huffman coding seems to be better than the Dynamic version,in this exercise. Copyright © 2000–2017, Robert Sedgewick and Kevin Wayne. Alternatively, an adaptive Huffman coding process may be used in which the Huffman tree is updated as new symbols are encountered without first creating the Huffman dictionary. Let's first look at the binary tree given below. Assign Huffman codes to a set of symbols of known probabilities. An example will help illustrate the advantages of Huffman coding. When doing extended Huffman coding, I understand that you do for example a1a1,a1a2,a1a3 etc and you do their probabilities times, however, how do you get the codeword? For example from the image below how do you get that 0. This is an implementation of the algorithm in C. Works well with regard to text as well as fax transmissions. Do comment for any doubts. Huffman Coding Huffman Encoding Algorithms use the probability distribution of the alphabet of the source to develop the code words for symbols. The Core of Huffman Coding. Huffman Coding Huffman (1951) Uses frequencies of symbols in a string to build a variable rate prefix code. Huffman coding for all 26 letters would yield an expected cost of 4. The problem with static coding is that the tree has to be constructed in the transmitter and sent to the receiver. Just given a sequence of dots and dashes, it is not possible to uniquely break the sequence into letters. 29 bits per letter. 3 Bits Per Symbol? B. This post talks about fixed length and variable length encoding, uniquely decodable codes, prefix rules and construction of Huffman Tree. At the second step those are 0. Adaptive Huffman coding. Redundancy is much reduced. Huffman coding makes a stab at this by, in effect, approximating all the probabilities pi by integer powers of 1/2, so that all the Li’s are integral. Huffman-coding English words Codebook sample for Huffman coding: a problem is that if the codeword is a single letter, we can only encode lowercase and title. Let x i and x j, with probabilities p i and p j, respectively, be the two least probable symbols Remove them from the list and connect them in a binary tree. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under  Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to $585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over $1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: