Tries Data Structure

1. Tries Data Structure

2. 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”.

3. 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.

4. 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.

5. T S H I I E I N N $ N N S $ $ G $ $ $ $ Tries: Example

6. Tries: Example • 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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).

12. 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}

13. 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.

14. 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.

15. Abstract Data Types Based on Sets • INSERT(x,A): xis 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.

16. 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.

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