Data Structure II So Pak Yeung

Outline Review  Array  Sorted Array  Linked List Binary Search Tree Heap Hash Table

Review Operation  Find an element  Find the min/max element  Insert an element  Remove an element Time complexity?  O(1)?  O(lg N)?  O(N)?

Array Find an element  O(N) Find the smallest element  O(N) Insert an element  O(1) Remove an element  O(1)

Sorted Array Find an element  O(lg N) Find the smallest element  O(1) Insert an element  O(N) Remove an element  O(N)

Linked List Find an element  O(N) Find the smallest element  O(N) Insert an element  O(1) Remove an element  O(1)

Binary Search Tree Binary Tree  At most 2 children for each node For each Node i,  Node j <= Node i, for all Node j in left subtree of Node i  Node j > Node i, for all Node j in right subtree of Node i

Binary Search Tree

Binary Search Tree Find 13

Binary Search Tree Find 3 ???

Binary Search Tree Insert 1 1

Binary Search Tree Insert 3 1 3

Binary Search Tree Find an element  Seems O(lg N)? Find min/max  Seems O(lg N)? Insert an element  Seems O(lg N)?

Binary Search Tree Worst Case: O(N)!!!

Binary Search Tree Remove a leaf node  Just Do it! Remove a node with single child  Replace the node by its child Remove a node with 2 children  Replace the node by the max node of left subtree / min node of right subtree Lazy Deletion  Mark the node as deleted instead of deleting it

Binary Search Tree Again, seems O(lg N), worst Case: O(N)!!!

Heap Priority Queue Binary Tree For each node, it is no greater/less than all nodes of its subtree Operation  Extract min/max  Insert

Heap

Heap Extract min/max  Get the value from the root (O(1) to find min)  Replace the root by the last node  Shift down Time complexity  O(lg N)

Heap Get 1

Heap

Heap

Heap

Heap Insert an element  Add the node at the end  Shift up Time complexity  O(lg N)

Heap Add 1 1

Heap

Heap

Heap Build heap  Insert each node  O(N lg N)  There exists a faster way!!  Only need to perform shift down process from the bottom  O(N)

Heap Find an element  Not support  O(N), if implement by array Remove an element  Consider that subtree  O(lg N)

Hash Table Using limited memory, storing data of unlimited range Convert data to integers that are in a limited range by a hash function

Hash Table Mark 5 number Each number is between [1, ] Not a good idea to use an array of size of A[n%5]=n

Hash Table Insert an element  O(1) Find an element  Using the same Hash function!  O(1) Delete an element  Lazy Deletion  O(1)

Hash Table Collision?  E.g. 56 and 111  Open Hashing  Close Hashing

Hash Table Opening Hashing

Hash Table Close hashing Insert 1 If the cell is occupied, find the next empty cell