arithmetic encoding

Arithmetic coding is a technique for achieving near-optimal entropy encoding.

An arithmetic encoder takes a string of symbols as input and produces a rational number in the interval [0,1) as output. As each symbol is processed, the encoder will restrict the output to a smaller interval.

Let N be the number of distinct symbols in the input; let x1,x2xN represent the symbols, and let P1,P2PN represent the probability of each symbol appearing. At each step in the process, the output is restricted to the current interval [y,y+R). PartitionPlanetmathPlanetmath this interval into N disjoint subintervals:


Therefore the of Ii is PiR. If the next symbol is xi, then restrict the output to the new interval Ii.

Note that at each stage, all the possible intervals are pairwise disjoint. Therefore a specific sequenceMathworldPlanetmath of symbols produces exactly one unique output range, and the process can be reversed.

Since arithmetic encoders are typically implemented on binary computers, the actual output of the encoder is generally the shortest sequence of bits representing the fractional part of a rational number in the final interval.

Suppose our entire input string contains M symbols: then xi appears exactly PiM times in the input. Therefore, the size of the final interval will be


The number of bits necessary to write a binary fraction in this range is

-log2Rf = -log2i=1NPiPiM
= i=1N-log2PiPiM
= i=1N-PiMlog2Pi

By Shannon’s theoremMathworldPlanetmath, this is the total entropyMathworldPlanetmath of the original message. Therefore arithmetic encoding is near-optimal entropy encoding.

Title arithmetic encoding
Canonical name ArithmeticEncoding
Date of creation 2013-03-22 12:32:20
Last modified on 2013-03-22 12:32:20
Owner vampyr (22)
Last modified by vampyr (22)
Numerical id 4
Author vampyr (22)
Entry type Definition
Classification msc 68P30
Classification msc 94A24
Synonym arithmetic coding
Synonym arithmetic encoder
Synonym arithmetic coder
Related topic HuffmanCoding