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1 Advanced Data Structures

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2 Topics Data structures Algorithm design & analysis Queue Tree Graph

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3 What is a “data structure”?

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4 Why study data structures? Example problem Given: a set of N numbers Goal: search for number k Solution Store numbers in an array of size N Linearly scan array until k is found or array is exhausted Number of checks Best case: 1 Worst case: N Average case: N/

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5 Why study data structures? Solution #2 Store numbers in a binary search tree Search tree until find k Number of checks Best case: 1 Worst case: log 2 N Average case: (log 2 N)/

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6 Analysis Does it matter? N vs. log 2 N

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7 Analysis Does it matter? Assume N = 1,000,000,000 1 billion (Walmart transactions in 100 days) 1 GHz processor = 10 9 cycles per second 1 cycle per transaction O(N) algorithm 1 billion transactions = > 1 billion clock cycles O(lg N) algorithm 1 billion transactions => 30 clock cycles

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8 Example 2 Scheduling job in a printer Write a code to manage the printer queue Functions to support Insert, delete Special accommodations needed for: Priority Dynamic update Scheduling challenges

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9 Example 3 Exploring the Facebook connection network Write a code to tell who is connected to who (directly or indirectly) through your Facebook profile 6-degrees of separation

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10 Example 4 Pattern matching Write a code to do Google search on your web database

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11 Summary Keep the data organized Choice of data structures matters Appropriate data structures ease design & improve performance Challenge Design appropriate data structure & associated algorithms for a problem Analyze to show improved performance

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Queue ADT Like a stack, a queue is also a list. However, with a queue, insertion is done at one end, while deletion is performed at the other end. Accessing the elements of queues follows a First In, First Out (FIFO) order. Like customers standing in a check-out line in a store, the first customer in is the first customer served. 12

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: Basic operations enqueue: insert an element at the rear of the list dequeue: delete the element at the front of the list 13

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Implementation of Queue Just as stacks can be implemented as arrays or linked lists, so with queues. Dynamic queues have the same advantages over static queues as dynamic stacks have over static stacks 14

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Trees Linear access time of linked lists is prohibitive Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? Trees Basic concepts Tree traversal Binary tree Binary search tree and its operations 15

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Trees A tree T is a collection of nodes T can be empty (recursive definition) If not empty, a tree T consists of a (distinguished) node r (the root), and zero or more nonempty subtrees T 1, T 2,...., T k 16

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Tree Traversal Used to print out the data in a tree in a certain order Pre-order traversal (Root, left, right) Print the data at the root Recursively print out all data in the leftmost sub tree Recursively print out all data in the rightmost sub tree 18

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Continue…. Inorder traversal (left, root, right) Recursively print out all data in the leftmost sub tree Print the data at the root Recursively print out all data in the rightmost sub tree 20

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Continue.. Post-order traversal (left, right, root) Recursively print out all data in the leftmost sub tree Recursively print out all data in the rightmost sub tree Print the data at the root 22

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Binary Trees A tree in which no node can have more than two children 24

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Continue.. The depth of an “average” binary tree is considerably smaller than N, even though in the worst case, the depth can be as large as N – 1. 25

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Binary Search Trees (BST) A data structure for efficient searching, inser-tion and deletion Binary search tree property For every node X All the keys in its left subtree are smaller than the key value in X All the keys in its right subtree are larger than the key value in X 26

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Graph o A Simple graph G = (V, E) consists of a nonempty set V of vertices and a possibly empty set E of edges, each edge being a set of two vertices from V. The number of vertices and edges are denoted by |V| and |E|, respectively. 28

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Graph Representation There are variety of ways to represent a graph. Adjacency lists. An adjacency matrix of graph G = (V, E) is a binary |V| x |V| matrix such that each entry of this matrix An incident matrix of graph G = (V, E) is a binary |V| x |E| matrix such that each entry of this matrix 29

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Continue.. 31 Adjacency matrixIncident matrix

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Graph Traversal As in trees, traversing a graph consists of visiting each vertex only one time. The simple traversal algorithm used for trees can not be applied here because graph may include cycles (result in to infinite loop) or, isolated vertices (left some nodes). The most popular algorithms for traversing in graphs are: 1. Depth First Search 2. Breadth First search 32

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