Radix tree

From Wikipedia, the free encyclopedia

A radix tree, Patricia trie/tree, or crit bit tree is a specialized set data structure based on the trie that is used to store a set of strings. These can be strings of characters, bit strings such as integers or IP addresses, or generally arbitrary sequences of objects in lexicographical order. Sometimes the names radix tree and crit bit tree are only applied to trees storing integers and Patricia trie is retained for more general inputs, but the structure works the same way in all cases.

1 Overview
2 Applications
3 History
4 Comparison to other data structures
5 External links

[edit] Overview

The radix tree is easiest to understand as a space-optimized tree where each node with only one child is merged with its child. The result is that every internal node has at least two children. Unlike in trees, edges can be labelled with sequences of characters as well as single characters. This makes them much more efficient for small sets (especially if the strings are long) and for sets of strings that share long prefixes.

It supports the following main operations, all of which are O(k), where k is the maximum length of all strings in the set:

Lookup: Determines if a string is in the set. This operation is identical to tries except that some edges consume multiple characters.
Insert: Add a string to the tree. We search the tree until we can make no further progress. At this point we either add a new outgoing edge labelled with all remaining characters in the input string, or if there is already an outgoing edge sharing a prefix with the remaining input string, we split it into two edges (the first labelled with the common prefix) and proceed. This splitting step ensures that no node has more children than there are possible string characters.
Delete: Delete a string from the tree. We delete the corresponding leaf, then if its parent only has one child remaining delete the parent and merge the two incident edges.
Find predecessor: Locates the largest string less than a given string, by lexicographic order.
Find successor: Locates the smallest string greater than a given string, by lexicographic order.

A common extension of radix trees uses two colors of nodes, 'black' and 'white'. To check if a given string is stored in the tree, the search starts from the top and follows the edges of the input string until no further progress can be made. If the search-string is consumed and the final node is a black node, the search has failed; if it is white, the search has succeeded. This enables us to add a large range of strings with a common prefix to the tree, using white nodes, then remove a small set of "exceptions" in a space-efficient manner by inserting them using black nodes.

[edit] Applications

As mentioned, radix trees are useful for constructing associative arrays with keys that can be expressed as strings. They find particular application in the area of IP routing, where the ability to contain large ranges of values with a few exceptions is particularly suited to the hierarchical organization of IP addresses. They are also used for inverted indexes of text documents in information retrieval.

[edit] History

Donald R. Morrison first described what he called "Patricia tries" in 1968; the name comes from the acronym PATRICIA, which stands for "Practical Algorithm to Retrieve Information Coded in Alphanumeric". Gernot Gwehenberger independently invented and described the data structure at about the same time.

[edit] Comparison to other data structures

(In the following comparisons, it is assumed that the keys are of length k and the data structure contains n elements.)

Unlike balanced trees, radix trees permit lookup, insertion, and deletion in O(k) time rather than O(log n). This doesn't seem like an advantage, since normally k ≥ log n, but in a balanced tree every comparison is a string comparison requiring O(k) worst-case time, many of which are slow in practice due to long common prefixes. In a trie, all comparisons require constant time, but it takes m comparisons to look up a string of length m. Radix trees can perform these operations with fewer comparisons and require many fewer nodes.

Radix trees also share the disadvantages of tries, however: as they can only be applied to strings of elements or elements with a reversible mapping (bijection) to strings, they lack the full generality of balanced search trees, which apply efficiently to any datatype with a total ordering (for example, floating-point numbers).

Hash tables have expected O(1) insertion and deletion times but are subject to attacks that degrade them to worst-case scenarios of O(n). Radix trees have worst-case O(k) insertion and deletion. The successor/predecessor operations of radix trees are also not implemented by hash tables.