introduction to trees,graphs,hashing

WHAT IS TREE?
• A tree is a finite nonempty set of elements.
• It is an abstract model of a hierarchical structure.
• consists of nodes with a parent-child relation.
Types of trees:
1.Binary tree
2.Binary search tree
3.AVL trees
4.B trees
5.B+trees
6.Huffman tree
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BINARY TREE:
 Binary Tree is a rooted tree in which root can have maximum two
children such that each of them is again a binary tree. That means,
there can be 0,1, or 2 children of any node.

 Why do we use trees/need:
1. store information that naturally forms a hierarchy.
2. we can search for a given key in moderate time
3. Manipulate sorted lists of data.
 Applications:
1. Organizing charts
2. File systems
3. Programming environment
 Advantages:
1. Allows easier processing of data
2. It stored on disk very efficiently.
 Disadvantage:
1. Requires more memory space
2. Many rules and restrictions for making connections

GRAPHS:
 A Non linear data structure that consists of a set of nodes (vertices)
and a set of edges that relate the nodes to each other.
 The set of edges describes relationships among the vertices.
Types of graphs:
1.directed graph
2.undirected graph
Graph traversals:
1.Depth-first-search
2.Breadth-first-search
Vertices
Edges=
=
Directed graph Undirected graph

 Why do we use graph:
1. They represent relationship between two or more objects.
2. We can represent the info in network model.
 Applications:
1. facebook.
2. Google maps.
 Advantages:
1. Adapts easily to different kinds of graphs.
2. solve problems from checking whether the nodes are
connected to finding the shortest paths.
 Disadvantage:
In fact the worst case time could be proportional to the number of
vertices.

HASHING:
 In computing, a hash table (hash map) is a data structure which implements
an associative array abstract data type, a structure that can map keys to values. A
hash table uses a hash function to compute an index into an array
of buckets or slots, from which the desired value can be found.
 Hash Table Operations
 Search: compute f(k) and see if a pair exists
 Insert: compute f(k) and place it in that position
 Delete: compute f(k) and delete the pair in that position
 In ideal situation, hash table search, insert or delete takes O(1)
o The size of the array is TableSize.
o Each key is mapped into some number in the range 0 to TableSize – 1.

introduction to trees,graphs,hashing

SOME METHODS
 Truncation:
 e.g. 123456789 map to a table of 1000 addresses by picking 3 digits of the key.
 Folding:
 e.g. 123|456|789: add them and take mod.
 Key mod N:
 N is the size of the table, better if it is prime.
 Squaring:
 Square the key and then truncate
 When an element is inserted, if it hashes to the same value as an already inserted element, then we have a collision.
Collision resolving techniques:
 Separate chaining
Each cell of hash table point to a linked list of records that have same hash function value.
o Open addressing
• In Open Addressing, all elements are stored in the hash table itself.
• Size of table must be greater than or equal to total number of keys
• Linear probing
In linear probing, we linearly probe for next slot.
H(x)=(hash(x)+i)%tablesize
 Quadratic probing
We probe to the i2‘th slot in i’th iteration.
H(x)=(hash(x)+i2)%tablesize
 Double hashing
we perform i*hash2(x) slot in i’th rotation.
COLLISION:

 Why do we need hash tables:
 Internal routers
 We could get O(1) access without a lot of space
 Applications:
 File management- working out where to store records
 Comparing complex values
 Dictionaries
 Security systems
 Advantages:
 It takes O(1) for insertion,searching,deletion
 Hash tables turn out to be more efficient than search trees or any other
table lookup structure.
 They are widely used for associative arrays, database indexing, caches
and sets.
 Disadvantage:
 Hash collisions are practically unavoidable. when hashing a random
subset of a large set of possible keys.
 Hash tables become quite inefficient when there are many collisions.
 Hash table does not allow null values, like hash map.

introduction to trees,graphs,hashing

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