1. Searching: Hash Tables
Chapter 12
6/9/15
Adapted from instructor resource slides
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

2. Info: Still grading exams. Will review answers Thursday
Review how to handle issues with project grading
Review Thursday’s material
Hashing (new)
break
Start sorting
Review project 2
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

3. Evolution of Reusability, Genericity
Major theme in development of programming languages
Reuse code
Avoid repeatedly reinventing the wheel
Trend contributing to this
Use of generic code
Can be used with different types of data
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

4. Function Genericity Overloading and Templates
Initially code was reusable by encapsulating it within functions
Example lines of code to swap values stored in two variables
Instead of rewriting those 3 lines
Place in a function void swap (int & first, int & second) { int temp = first; first = second; second = temp; }
Then call swap(x,y);
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

5. Template Mechanism Declare a type parameter
also called a type placeholder
Use it in the function instead of a specific type.
This requires a different kind of parameter list:
void Swap(______ & first, ______ & second)
{ ________ temp = first; first = second; second = temp; }
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

6. Instantiating Class Templates
Instantiate it by using declaration of form ClassName<Type> object;
Passes Type as an argument to the class template definition.
Examples:
Stack<int> intSt;
Stack<string> stringSt;
Compiler will generate two distinct definitions of Stack
two instances
one for ints and one for strings.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

7. STL (Standard Template Library)
A library of class and function templates
Components:
Containers:
Generic "off-the-shelf" class templates for storing collections of data
Algorithms:
Generic "off-the-shelf" function templates for operating on containers
Iterators:
Generalized "smart" pointers that allow algorithms to operate on almost any container
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

8. The vector Container A type-independent pattern for an array class
capacity can expand
self contained
Declaration
template <typename T>
class vector
{ } ;
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

9. vector Operations Information about a vector's contents
v.size()
v.empty()
v.capacity()
v.reserve()
Adding, removing, accessing elements
v.push_back()
v.pop_back()
v.front()
v.back()
What is difference between size and capacity?
What is reserve? Allows to set or increase, but not decrease
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

10. Increasing Capacity of a Vector
When vector v becomes full
capacity increased automatically when item added
Algorithm to increase capacity of vector<T>
Allocate new array to store vector's elements
use T copy constructor to copy existing elements to new array
Store item being added in new array
Destroy old array in vector<T>
Make new array the vector<T>'s storage array
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

11. Iterators Each STL container declares an iterator type
can be used to define iterator objects
Iterators are a generalization of pointers that allow a C++ program to work with different data structures (containers) in a uniform manner
To declare an iterator object
the identifier iterator must be preceded by
name of container
scope operator ::
Example: vector<int>::iterator vecIter = v.begin()
Would define vecIter as an iterator positioned at the first element of v
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

12. Iterators Contrast use of subscript vs. use of iterator
ostream & operator<<(ostream & out, const vector<double> & v) {
for (int i = 0; i < v.size(); i++)
out << v[i] << " "; return out; }
for (vector<double>::iterator it = v.begin(); it != v.end(); it++) out << *it << " ";
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

13. Iterator Functions Note Table 9-5
Note the capability of the last two groupings
Possible to insert, erase elements of a vector anywhere in the vector
Must use iterators to do this
Note also these operations are as inefficient as for arrays due to the shifting required
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

14. Contrast Vectors and Arrays
Capacity can increase
A self contained object
Is a class template (No specific type)
Has function members to do tasks
Fixed size, cannot be changed during execution
Cannot "operate" on itself
Bound to specific type
Must "re-invent the wheel" for most actions
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

15. STL's deque Class Template
Has the same operations as vector<T> except …
there is no capacity() and no reserve()
Has two new operations:
d.push_front(value); Push copy of value at front of d
d.pop_front(value); Remove value at the front of d
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

16. vector vs. deque vector deque Capacity of a vector must be increased
It must copy the objects from the old vector to the new vector
It must destroy each object in the old vector
A lot of overhead!
With deque this copying, creating, and destroying is avoided.
Once an object is constructed, it can stay in the same memory locations as long as it exists
If insertions and deletions take place at the ends of the deque.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

17. vector vs. deque
Unlike vectors, a deque isn't stored in a single varying-sized block of memory, but rather in a collection of fixed-size blocks (typically, 4K bytes).
One of its data members is essentially an array map whose elements point to the locations of these blocks.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

18. Linear Search Vector based search function template <typename t>
void LinearSearch (const vector<t> &v, const t &item, boolean &found, int &loc) {
found = false; loc = 0; while(loc < n && !found) {
if (found || loc == v.size()) return; if (item == x[loc])
found = true; else loc++;
} }
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

19. Binary Search found = false; int first = 0;
Binary search function for vector
template <typename t>
void LinearSearch (const vector<t> &v, const t &item, boolean &found, int &loc) {
found = false; int first = 0;
int last = v.size() - 1; while(first <= last && !found)
if (found || first > last)
return;
loc = (first + last) / 2; if (item < v[loc]) last = loc + 1;
else if (item > v[loc]) first = loc + 1; else /* item == v[loc] */
found = true; } }
NEED TO HAVE AN ORDERED, OR SORTED LIST!!!
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

20. Binary Search Usually outperforms a linear search Disadvantage:
Requires a sequential storage
Not appropriate for linked lists (Why?)
It is possible to use a linked structure which can be searched in a binary-like manner
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

21. Trees Tree terminology Root node Children of the parent (3)
Siblings to each other
Leaf nodes
Root, children, parent, ancestor, descendant, etc
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

22. Binary Trees Each node has at most two children
Useful in modeling processes where
a comparison or experiment has exactly two possible outcomes
the test is performed repeatedly
Example
multiple coin tosses
encoding/decoding messages in dots and dashes such as Morse code
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

23. Binary Trees Each node has at most two children
Useful in modeling processes where
a comparison or experiment has exactly two possible outcomes
the test is performed repeatedly
Example
multiple coin tosses
encoding/decoding messages in dots and dashes such as Morse code
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

24. Array Representation of Binary Trees
Works OK for complete trees, not for sparse trees
Balanced trees are binary trees with the property that for each node, the height of its left subtree and the height of its right subtree differ by at most one, where a tree’s height is the number of levels in the tree.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

25. Linked Representation of Binary Trees
Uses space more efficiently
Provides additional flexibility
Each node has two links
one to the left child of the node
one to the right child of the node
if no child node exists for a node, the link is set to NULL
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

26. Binary Trees as Recursive Data Structures
A binary tree is either empty … or
Consists of
a node called the root
root has pointers to two disjoint binary (sub)trees called …
right (sub)tree
left (sub)tree
Anchor
Inductive step
Which is either empty … or …
Which is either empty … or …
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

27. ADT Binary Search Tree (BST)
Collection of Data Elements
binary tree
each node x,
value in left child of x value in x in right child of x
Basic operations
Construct an empty BST
Determine if BST is empty
Search BST for given item
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

28. ADT Binary Search Tree (BST)
Basic operations (ctd)
Insert a new item in the BST
Maintain the BST property
Delete an item from the BST
Traverse the BST
Visit each node exactly once
The inorder traversal must visit the values in the nodes in ascending order
View BST class template, Fig. 12-1
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

29. BST Traversals Note that recursive calls must be made
To left subtree
To right subtree
Must use two functions
Public method to send message to BST object
Private auxiliary method that can access BinNodes and pointers within these nodes
Similar solution to graphic output
Public graphic method
Private graphAux method
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

30. BST Searches Search begins at root
If that is desired item, done
If item is less, move down left subtree
If item searched for is greater, move down right subtree
If item is not found, we will run into an empty subtree
View search()
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

31. Inserting into a BST Insert function View insert() function
Uses modified version of search to locate insertion location or already existing item
Pointer parent trails search pointer locptr, keeps track of parent node
Thus new node can be attached to BST in proper place
View insert() function R
Looking to insert ‘R’
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

32. Recursive Deletion
Three possible cases to delete a node, x, from a BST
1. The node, x, is a leaf
Make parent pointer a null pointer, In deleting D, make right pointer in parent C a null.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

33. Recursive Deletion 2. The node, x has one child
In deleting E, set right pointer of parent A to point at C
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

34. Recursive Deletion x has two children
Delete node pointed to by xSucc as described for cases 1 and 2
Replace contents of x with inorder successor
Want to delete J. To replace with inorder successor, go right and then descent left as far as possible: K, then delete xSucc(original K) K
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

35. Problem of Lopsidedness
Trees can be totally lopsided
Suppose each node has a right child only
Degenerates into a linked list
Processing time affected by "shape" of tree
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

36. Hash Tables Recall order of magnitude of searches Linear search O(n)
Binary search O(log2n)
Balanced binary tree search O(log2n)
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

37. 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

38. Hash Functions
Simple function could be to mod the value of the key by the size of the table
H(x) = x % tableSize
Note that we have traded speed for wasted space
Table must be considerably larger than number of items anticipated
Suggested to be 1.5-2x larger
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

39. Hash Functions
Observe the problem with same value returned by h(x) for different values of x
Called collisions
A simple solution is linear probing
Empty slots marked with -1
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

40. 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
Ex. -1 for unused slots, -2 for slots which used to contain data
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

41. Collision Reduction Strategies
Strategies for improved performance
Increase table capacity (less collisions)
Use different collision resolution technique
Devise different hash function
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved

42. Collision Reduction Strategies
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

43. Collision Reduction 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

44. Collision Reduction 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

45. 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