CS 225 Data Structures and Software Principles

CS 225 Data Structures and Software Principles Section 12 Tries

Agenda • Tries • Basics • Jason’s Code • Patricia Trees • De La Briandais Trees • Hybrid Trees

Tries • Data structure optimized for lookups on a key that can be decomposed into characters • Take into account overlapping strings • Regular expression search • Pattern searches in images • Non-compact tries • De La Briandais Trees, Patricia Trees

Trie Representation • Tries are represented using a tree of arrays • For a character set of size k, the corresponding Trie structure is a (k+1)-ary tree

Tries • The i-th character (starting at 0) in the data corresponds to the node at depth i • Need a mapping of character to an index in the array • The extra one cell in the array is used to hold a “null character” represented by  • Points to a leaf • Ideally, no need to store key in a leaf, since it is completely determined by path followed • Info stored at the leaf • Spend only constant time at each level

a b c … r s … z 0 a t 1 1 f a i 2 2 t r r 3 3 3 t 4 4 4 star stir raft 5 start Trie Example Words in Trie raft star start stir

Tries • Running time of Find operation: O(L) where L is the length of the string we are looking for • Unique trie for any set of search keys • Advantage: NOT dependent on the number of strings we have in the Trie structure • Disadvantage: memory waste • 27 cell array, one per character needed for Strings • Space: (k+1) * #nodes * sizeof(pointer)

Jason’s Code:TrieNode Data TrieNode { int nodeLevel; // level of the node bool isLeaf; Array<TrieNode*> subtries; // array of ptrs to nodes String key; // string key in leaf nodes Etype storedInfo; }

Patricia Trees • Acronym: Practical Algorithm To Retrieve Information Coded In Alphanumeric • Trick: only allocate arrays that make a “decision” • Do not store nodes with only one non-NULL cell • Store in each node the index of the character position on which it discriminates • Tradeoff: Less space required, but more work for Insert and Remove • Key no longer uniquely determined by path • Now we must store keys in the leaf

One (Patricia) Tree Application • Communication Networks (CS 438) • Task of efficiently finding the longest match between an IP address and variable-length prefixes in a forwarding table (due to CIDR) • Given packet for 128.174.5.130, where would it go? 1000 0000 1011 1100

De La Briandais Trees • Trick: convert arrays in Trie to sparse arrays • Allocate space only for used cells in the arrays • At each level we now have a linked list • Array cells are now nodes that not only point down, but to the next used character on that level • Advantage: can save much space; good when the linked lists are not long • Disadvantage: search is now dependent on k (alphabet size)

head r s a t f a i r r t t star raft stir start de la Briandais Tree Example Words in Trie raft star start stir

Hybrid Structures • Patricia/de la Briandais • Uses both optimizations • We eliminate all one node linked lists in the de la Briandais tree • Trie/Patricia/de la Briandais • Highly optimized data structure • Some levels have arrays and others have linked lists

CS 225 Data Structures and Software Principles