Kraft's inequality

In coding theory, Kraft's inequality, named after Leon Kraft, gives a sufficient condition for the existence of a prefix code and necessary condition for the existence of a uniquely decodable code for a given set of codeword lengths. Its applications to prefix codes and trees often find use in computer science and information theory.

More specifically, Kraft's inequality limits the lengths of codewords in a prefix code: if one takes an exponential function of each length, the resulting values must look like a probability mass function. Kraft's inequality can be thought of in terms of a constrained budget to be spent on codewords, with shorter codewords being more expensive.

• If Kraft's inequality holds with strict inequality, the code has some redundancy.
• If Kraft's inequality holds with strict equality, the code in question is a complete code.
• If Kraft's inequality does not hold, the code is not uniquely decodable.

Kraft's inequality was published by Kraft (1949). However, Kraft's paper discusses only prefix codes, and attributes the analysis leading to the inequality to Raymond Redheffer. The inequality is sometimes also called the Kraft–McMillan theorem after the independent discovery of the result by McMillan (1956); McMillan proves the result for the general case of uniquely decodable codes, and attributes the version for prefix codes to a spoken observation in 1955 by Joseph Leo Doob.

Examples

Binary trees

9, 14, 19, 67 and 76 are leaf nodes at depths of 3, 3, 3, 3 and 2, respectively.

Any binary tree can be viewed as defining a prefix code for the leaves of the tree. Kraft's inequality states that

$\sum_{\ell \in \mathrm{leaves}} 2^{-\mathrm{depth}(\ell)} \leq 1.$

Here the sum is taken over the leaves of the tree, i.e. the nodes without any children. The depth is the distance to the root node. In the tree to the right, this sum is

$\frac{1}{4} + 4 \left( \frac{1}{8} \right) = \frac{3}{4} \leq 1.$

Chaitin's constant

In algorithmic information theory, Chaitin's constant is defined as

$\Omega = \sum_{p \in P} 2^{-|p|}.$

This is an infinite sum, which has one summand for every syntactically correct program that halts. |p| stands for the length of the bit string of p. The programs are required to be prefix-free in the sense that no summand has a prefix representing a syntactically valid program that halts. Hence the bit strings are prefix codes, and Kraft's inequality gives that $\Omega \leq 1$.

Formal statement

Let each source symbol from the alphabet

$S=\{\,s_1,s_2,\ldots,s_n\,\}\,$

be encoded into a uniquely decodable code over an alphabet of size $r$ with codeword lengths

$\ell_1,\ell_2,\ldots,\ell_n.\,$

Then

$\sum_{i=1}^{n} \left( \frac{1}{r} \right)^{\ell_i} \leq 1.$

Conversely, for a given set of natural numbers $\ell_1,\ell_2,\ldots,\ell_n\,$ satisfying the above inequality, there exists a uniquely decodable code over an alphabet of size $r$ with those codeword lengths.

A commonly occurring special case of a uniquely decodable code is a prefix code. Kraft's inequality therefore also holds for any prefix code.

Proof for prefix codes

Example for binary tree. Red nodes represent a prefix tree. The method for calculating the number of descendant leaf nodes in the full tree is shown.

Suppose that $\ell_1 \leq \ell_2 \leq ... \leq \ell_n$. Let $A$ be the full $r$-ary tree of depth $\ell_n$. Every word of length $\ell \leq \ell_n$ over an $r$-ary alphabet corresponds to a node in this tree at depth $\ell$. The $i$th word in the prefix code corresponds to a node $v_i$; let $A_i$ be the set of all leaf nodes in the subtree of $A$ rooted at $v_i$. Clearly

$|A_i| = r^{\ell_n-\ell_i}.$

Since the code is a prefix code,

$A_i \cap A_j = \varnothing,\quad i\neq j$.

Thus, given that the total number of nodes at depth $\ell_n$ is $r^{\ell_n}$,

$|\bigcup_{i=1}^n A_i| = \sum_{i=1}^n r^{\ell_n-\ell_i} \leq r^{\ell_n}$

from which the result follows.

Conversely, given any ordered sequence of $n$ natural numbers,

$\ell_1 \leq \ell_2 \leq \dots \leq \ell_n$

satisfying the Kraft inequality, one can construct a prefix code with codeword lengths equal to $\ell_i$ by pruning subtrees from a full $r$-ary tree of depth $\ell_n$. First choose any node from the full tree at depth $\ell_1$ and remove all of its descendants. This removes $r^{-\ell_1}$ fraction of the nodes from the full tree from being considered for the rest of the remaining codewords. The next iteration removes $r^{-\ell_2}$ fraction of the full tree for total of $r^{-\ell_1}+r^{-\ell_2}$. After $m$ iterations,

$\sum_{i=1}^m r^{-\ell_i}$

fraction of the full tree nodes are removed from consideration for any remaining codewords. But, by the assumption, this sum is less than 1 for all $m<n$. Thus prefix code with lengths $\ell_i$ can be constructed for all $n$ source symbols.

Proof for binary trees

Let $T$ be any binary tree and let $T'$ be the binary tree obtained by attaching a leaf to any node of $T$ that has exactly one child so that $T'$ is full; every node of $T'$ has either two or zero children. Every leaf of $T$ is present in $T'$ so we immediately have

$\sum_{\ell \in \mathrm{leaves}(T)} 2^{-\mathrm{depth}(\ell)} \leq \sum_{\ell \in \mathrm{leaves}(T')} 2^{-\mathrm{depth}(\ell)} \;.$

Consider the random walk that starts at the root of $T'$ and repeatedly moves to the left or right child of the current node, with equal probability, until it reaches a leaf. The probability that this walk reaches a particular leaf, $\ell$, is exactly $2^{-\mathrm{depth}(\ell)}$. Therefore, $\langle 2^{-\mathrm{depth}(\ell)} : \ell \in \mathrm{leaves}(T') \rangle$ is a probability distribution so

$\sum_{\ell \in \mathrm{leaves}(T')} 2^{-\mathrm{depth}(\ell)} = 1 \;.$

The proof of the converse half of the result is given above.

Proof of the general case

Consider the generating function in inverse of x for the code S

$F(x) = \sum_{i=1}^n x^{-|s_i|} = \sum_{\ell=\min}^\max p_\ell \, x^{-\ell}$

in which $p_\ell$—the coefficient in front of $x^{-\ell}$—is the number of distinct codewords of length $\ell$. Here min is the length of the shortest codeword in S, and max is the length of the longest codeword in S.

For any positive integer m consider the m-fold product S^m, which consists of all the words of the form $s_{i_1}s_{i_2}\dots s_{i_m}$, where $i_1, i_2, \dots, i_m$ are indices between 1 and n. Note that, since S was assumed to uniquely decodable, if $s_{i_1}s_{i_2}\dots s_{i_m}=s_{j_1}s_{j_2}\dots s_{j_m}$, then $i_1=j_1, i_2=j_2, \dots, i_m=j_m$. In other words, every word in $S^m$ comes from a unique sequence of codewords in $S$. Because of this property, one can compute the generating function $G(x)$ for $S^m$ from the generating function $F(x)$ as

$G(x) = \left( F(x) \right)^m = \left( \sum_{i=1}^n x^{-|s_i|} \right)^m = \sum_{i_1=1}^n \sum_{i_2=1}^n \cdots \sum_{i_m=1}^n x^{-\left(|s_{i_1}| + |s_{i_2}| + \cdots + |s_{i_m}|\right)} = \sum_{i_1=1}^n \sum_{i_2=1}^n \cdots \sum_{i_m=1}^n x^{-|s_{i_1} s_{i_2}\cdots s_{i_m}|} = \sum_{\ell=m \cdot \min}^{m \cdot \max} q_\ell \, x^{-\ell} \; .$

Here, similarly as before, $q_\ell$—the coefficient in front of $x^{-\ell}$ in $G(x)$—is the number of words of length $\ell$ in $S^m$. Clearly, $q_\ell$ cannot exceed $r^\ell$. Hence for any positive x

$\left( F(x) \right)^m \le \sum_{\ell=m \cdot \min}^{m \cdot \max} r^\ell \, x^{-\ell} \; .$

Substituting the value x = r we have

$\left( F(r) \right)^m \le m \cdot (\max-\min)+1$

for any positive integer $m$. The left side of the inequality grows exponentially in $m$ and the right side only linearly. The only possibility for the inequality to be valid for all $m$ is that $F(r) \le 1$. Looking back on the definition of $F(x)$ we finally get the inequality.

$\sum_{i=1}^n r^{-\ell_i} = \sum_{i=1}^n r^{-|s_i|} = F(r) \le 1 \; .$

References

• Kraft, Leon G. (1949), A device for quantizing, grouping, and coding amplitude modulated pulses, Cambridge, MA: MS Thesis, Electrical Engineering Department, Massachusetts Institute of Technology .

• McMillan, Brockway (1956), "Two inequalities implied by unique decipherability", IEEE Trans. Information Theory 2 (4): 115–116, doi:10.1109/TIT.1956.1056818 .

External links

• Kraft's inequality (NIST)