Tries Data Structure

Tries  Trie is a special structure to represent sets of character strings.  Can also be used to represent data types that are objects of any type e.g. strings of integers.  The word “trie” is derived from the middle letters of the word “retrieval”.

Tries: Example One way to implement a spelling checker is  Read a text file.  Break it into words( character strings separated by blanks and new lines).  Find those words not in a standard dictionary of words.  Words in the text but not in the dictionary are printed out as possible misspellings.

Tries: Example It can be implemented by a set having operations of :  INSERT  DELETE  MAKENULL  PRINT A Trie structure supports these set operations when the element of the set are words.

Tries: Example T H E $ $$$ $ $ $ N I I N G S SNN I

 Tries are appropriate when many words begin with the same sequence of letters.  i.e; when the number of distinct prefixes among all words in the set is much less than the total length of all the words.  Each path from the root to the leaf corresponds to one word in the represented set.  Nodes of the trie correspond to the prefixes of words in the set.

Tries: Example  The symbol $ is added at the end of each word so that no prefix of a word can be a word itself.  The Trie corresponds to the set {THE,THEN THIN, TIN, SIN, SING}  Each node has at most 27 children, one for each letter and $  Most nodes will have many fewer than 27 children.  A leaf reached by an edge labeled $ cannot have any children.

Tries nodes as ADT  A node in a trie can be viewed as:  Mapping whose domain is {A,B,…Z, $}  And whose value set is the type “Pointer to trie node”.  A trie can be identified with its root.  => ADT’s TRIE and TRIENODE have the same data type.  However, there operations are different.

Operations on Tries nodes  ASSIGN(node,c,p): Assign value p (a pointer to a node) to character c in node node.  VALUEOF(node, c): Produce the value associated with character c in node.  GETNEW(node, c): Make the value of node for character c be a pointer to a new node.  MAKENULL(node): Makes node to be null mapping.

Sets  A Set is a collection of members (or elements).  Each member of a set is either itself a set or is a primitive element called an atom.  All elements of a set are different.

Sets  Set can be integers, characters or strings.  All elements can be of the same type.  Atoms in a set can be linearly ordered.  A linear order (denoted by <) on a set S (“less than” or precedes”) satisfies two properties:  For any a and b in S, exactly one of a < b, a = b, or b < a is true.  For all a, b and c in S, if a < b and b < c, then a < c (transitivity).

Set Notation  A set of atoms is generally exhibited by putting curly brackets around its members.  Example: {1,4}, denotes the set whose members are 1 and 4.  Set is not a list, since order of elements in a set is not important.  {4,1} is the same set as {1,4}

Operations on Set  UNION: If A and B are sets then A  B is the set of elements that are members of A or B or both.  INTERSECTION: A  B is the set of elements, that are present both in A and B.  DIFFERENCE: A – B is the set of elements that are members of A but are not members of B.

Abstract Data Types Based on Sets The Set ADT can incorporate some other operations as well.  MERGE(A,B,C): Assigns to the set variable C the value A  B, the operator is not defined if A  B  Ø  MEMBER(x,A): Returns true if x  A and returns false if x  A.  MAKENULL(A): makes the Null set be the value for set variable A.

Abstract Data Types Based on Sets  INSERT(x,A): x is an element of the type of A’s members. Makes x a member of A. A = A  {x}  DELETE(x,A): removes x from A. A = A – { x }  ASSIGN(A,B): sets the value of set variable A to be equal to the value of set variable B.

Abstract Data Types Based on Sets  MIN(A): Returns the least element in set A.This operator is applicable only when the member of A are linearly ordered.  MAX(A): Returns the largest element in set A.This operator is applicable only when the member of A are linearly ordered.  EQUAL(A,B): Returns true if and only if sets A and B consists of the same elements.  FIND(x): Works for collection of disjoint sets. Returns the name of the unique set of which x is a member.

Reference “Data Structures and Algorithms” by A. V. Aho, J. E. Hopcroft, J. D. Ullman.