Splay Trees

Splay tree is a balanced binary search tree.

Splay tree is not perferctly balanced. Worst case time can be O(n) for one single operation. But average running time is O(n lg n)

Tree Rotation

Operations

Find

Find min key

Find Max key

Insert

Remove

Graphs

A graph G is a set of vertices and a set of edges that connects the vertices together.

It is written as

G = (V, E)

 

Undirected graphs

Directed graphs

Weighted graph

Graph Traversal

[incomplete]

Disjoint Sets

Disjoint set is a mathematical term which represents sets among which no two set has any item common. So intersection od any two sets results to empty set.

Operations

Union

Merges two sets into one set.

 

Find

Given an item it returns the set identifier in which the item belongs.

List based disjoint set

 

Tree based disjoint set

Path compression

Union by size

Implementing quick union using Array

[incomplete]

Tree

A tree is a set of nodes and edges that connects the nodes. Between there could be exactly one path.

Rooted Tree

When a node is distingushed as root the thre is a rooted tree. In such a tree every node has a parent except the root node. From any node to the path of root parent is the first node in that path.

Definations

Leaf node
Siblings
Ancestors
Descendents
Length of path
Depth of node
Height of node
Height of tree
Subtree

Binary tree

Tree traversal

Preorder traversal

Postorder traversal

Inorder traversal for binary tree

Level order traversal

 

Expression tree

Prefix expression

Postfix expression

 

Infix expression

[incomplete] Complete binary tree can not have a right child without a left child.

Index

Data Structures

Array
Linked List
Stack
Queue
Double ended queue - Deque
Hash Tables
Tree  
Binary Search Tree
Trie
Priority Queue / Binary Heap
Disjoint Sets
Splay Trees
B-Tree
Graphs

Algorithms

Greedy algorithms

Divide and Conquer

String operations

Knuth-Morris-Pratt algorithm for string search

Search

Linear search
Binary search

Sort

Insertion Sort
Selection Sort
Mergesort
Heapsort
Quicksort and Quick Select
Bucket sort
Counting sort
Radix sort
Sort stability

Graph algorithms

Minimum spanning tree -
    Kruskals algorithm
Sortest path-
    Dijkstra's algorithm
    Bellman Ford algorithm
Topological sorting
Gift wrapping algorithm (convex hull)

Computer Graphics

Bresenham's line drawing algorithm
Flood fill algorithm
Ray traching algorithm

Encoding

Huffman coding

Cryptography

RSA algorithm for public key encryption

Game Tree Search

Priority Queue and Binary Heap

Priority Queue

Data structure with nodes and keys and data  are associated with each node. Total order is defined for the keys.

Its is simple to identify or remove smallest key. But no other key can be identified or removes as easily.

Insertion is possible at any time.

Priority Queue can be represented using Binary Heap.

Binary Heap

A complete binary tree- A tree in which every level is filled except bottom level which is filled in the left to right order.

No child of any node may have key less than its parents key.

Level order traversal gives a sorted list of keys.

Binary heap using Array

A binary heap can be maintained using an array. For any node X with an index i its children are at i*2 and i*2+1. Like for a node with index 3, its children nodes are at 6 nad 7.

Binary heap time complexity

	Average case	Worst case
Search	O(n)	O(n)
Insert	O(lg n)	O(lg n)
Delete	O(lg n)	O(lg n)

Operations

Get min

Just return the node's key at root.

Insert

Place the new key at the boottom of the tree- ie the after the last entry- set this as current node.
Now the entry needs to bubble up untill the heaporder property is satisfied-
    Repeat untill no swap possible
        If current node's key is less than its parents' key swap current node with parent.

Remove min

Remove the root entry and replace root with last item in the tree (array). Then while possible bubble down the entry by comparing and swaping with child that has smaller key than current nodes key untill no swap is possible.

Bottom up heap construction

Insert all the items in the tree any order. That is we can keep the array of numbers as it is.
Take the last internal node and bubble down to tree like removeMin() operation in the heap.

Binary Search Tree

Binary search tree is a ordered dictionary in which keys for each node in the tree has total order. It is also called ordered or sorted binary tree.

We can find, insert, remove entries relatively fast. Quickly find node with minumum or maximum keys or keas near another key.

We can find words with close spelling without knowing exact spelling or can find something like smallest number that is greater than 100.

BST is used as data structures for sets, multisets, associated arrays etc.

Binary search tree time complexity

	Average	Worst case
Search	O(lg n)	O(n)
Insert	O(lg n)	O(n)
Delete	O(lg n)	O(n)

Binary Search Tree Invariant

For any node X, the tree the left subtree of X contains nodes with keys less than X's key and right subtree contains nodes with greater than X's key nad both subtree must be binary search trees themselves.

Any node may have data attached to it.

We can get all the keys in sorted order by doing inorder traversal of the tree.

Operations

Find a node with key K

Start from the root as current node
Untill the key is found or no more node to visit
    If the current node's key is equal to K return the current node
    If current node's key is smaller than K select right node as current node
    If current node's key is greater than K select left node as current node
Return NULL

Find smallest key >=K and largest key <=K

While searching for a key K we encounter both of these keys if K is not in the tree. We need to keep track of the keys. When we go right node we have key smaller than K. When we go left we have key greater than K. Last of these two is the smallest key >=K and largest key <=K.

Find a node with smallest key

Starting form root continously select left child as long as possible. Last node will be the node with smallest key of the tree.

Find a node with largest key

Starting form root continously select right child as long as possible. Last node will be the node with largest key of the tree.

Insert a node

Let K be the key we want to insert.
Search the tree for key K untill it is found or we hit NULL node
    If NULL replace the NULL node with  K.
    If K is found insert new node as left or right node with key K

Delete a node

Let K be the key of the node we want to delete.
Search for the node X with Key K.
    (a)If not found return.
    (b)If found and node has no child detach the key from its parent.
    (c)If found and node has one child replace the node with its child node.
    (d)If found and node has two children-
            Find the node Y with smallest key in the RIGHT subtree of the node to delete
                Y do not have any left subtree so it can be removed easily using (a)  or (b)
                Replace X with Y