Hash Tables Maps and Sets Chapter 15 Wherein

Hash Tables, Maps and Sets Chapter 15 Wherein we throw all the data into random array slots and somehow obtain O(1) retrieval time Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 1

Hash Tables • Recall order of magnitude of searches – Linear search O(n) – Binary search O(log 2 n) – Balanced binary tree search O(log 2 n) – Unbalanced binary tree can degrade to O(n) Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 2

Hash Tables • In some situations faster search is needed – Solution is to use a hash function – Value of key field given to hash function – Location in a hash table is calculated Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 3

Hash Functions • Simple function could be to mod the value of the key by some arbitrary integer int h(int i) { return i % some. Int; } • Note the max number of locations in the table will be same as some. Int • Note that we have traded speed for wasted space – Table must be considerably larger than number of items anticipated Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 4

Hash Functions • Observe the problem with same value returned by h(i) for different values of i – Called collisions • A simple solution is linear probing – Linear search begins at collision location – Continues until empty slot found for insertion Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 5

Hash Functions • When retrieving a value linear probe until found – If empty slot encountered then value is not in table • If deletions permitted – Slot can be marked so it will not be empty and cause an invalid linear probe Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 6

Hash Functions • Strategies for improved performance – Increase table capacity (less collisions) – Use different collision resolution technique – Devise different hash function • Hash table capacity – Size of table must be 1. 5 to 2 times the size of the number of items to be stored – Otherwise probability of collisions is too high Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 7

Collision Strategies • Linear probing can result in primary clustering • Consider quadratic probing – Probe sequence from location i is i + 1, i – 1, i + 4, i – 4, i + 9, i – 9, … – Secondary clusters can still form • Double hashing – Use a second hash function to determine probe sequence Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 8

Collision Strategies • Chaining – Table is a list or vector of head nodes to linked lists – When item hashes to location, it is added to that linked list Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 9

Improving the Hash Function • Ideal hash function – Simple to evaluate – Scatters items uniformly throughout table • Modulo arithmetic not so good for strings – Possible to manipulate numeric (ASCII) value of first and last characters of a name Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 10

Implementations Interfaces Hash Table Resizable Balanced Linked Hash Table + Linked Array Tree List Set Hash. Set Tree. Set Linked. Hash. Set List Array. List Linked. List Map Hash. Map Tree. Map Linked. Hash. Map Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 11

• Hash. Map -- an associative data structure • Hash. Set -- a container for holding values • Tree. Map – just like Hash. Map only items are ordered by key (internal red-black tree) Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 12

the Map Interface • Map ADT methods: – get(k): if the map M has an entry with key k, return its associated value; else, return null – put(k, v): insert entry (k, v) into the map M; if key k is not already in M, then return null; else, return old value associated with k – remove(k): if the map M has an entry with key k, remove it from M and return its associated value; else, return null – size(), is. Empty() – keys(): return an iterator of the keys in M – values(): return an iterator of the values in M Hash Tables 13

the Set Interface • Set ADT methods: – add(o): Adds the specified element to this set if it is not already present. Returns true if the set does not already contain element (new item added) – contains(o): returnts true if the set contains the specified element – remove(o): Removes the specified element from this set if it is present. Returns true if the set contained the specified element – size(), is. Empty() Hash Tables 14

Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson

Using “Chaining” to resolve collisions • A linked list in every cell of array • No rehashing required • Unlimited collisions can be handled • Lousy hash codes will still reduce performance Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0 -13 -140909 -3 23