Data Organization: Creating Order Out Of Chaos

Transcription

1 ata Organization: reating Order Out Of haos 3 Non-linear data structures 5-5 Principles of omputation, arnegie Mellon University - ORTIN Trees tree is a hierarchical (usually non-linear) data structure. very tree has a node called the root. ach node can have or more nodes as children. node that has no children is called a leaf. common tree in computer science algorithms is a binary tree. binary tree consists of nodes that have at most 2 children. complete binary tree has the maximum number of nodes on each of its levels. pplications: data compression, file storage, game trees 5-5 Principles of omputation, arnegie Mellon University - ORTIN 2

2 Trees 5-5 Principles of omputation, arnegie Mellon University - ORTIN 3 inary Tree The root has the data value 84. There are 4 leaves in this binary tree: 3, 3, 5, 98. This binary tree is not complete. 5-5 Principles of omputation, arnegie Mellon University - ORTIN 4 2

3 Implementation of Trees One common implementation of binary trees uses nodes like a linked list does. Instead of having a next pointer, each node has a left pointer and a right pointer. We could also use a vector. 45 Level Level 2 Level Level 2 Level Level Principles of omputation, arnegie Mellon University - ORTIN 5 xample inary Search Trees binary search tree (ST) is a binary tree such that ll nodes to the left of any node have data values less than that node ll nodes to the right of any node have data values greater than that node 5-5 Principles of omputation, arnegie Mellon University - ORTIN 6 3

4 xample inary Search Trees or each data value that you wish to insert into the binary search tree: Start at the root and compare the new data value with the root. If it is less, move down left. If it is greater, move down right. Repeat on the child of the root until you end up in a position that has no node. Insert a new node at this empty position. 5-5 Principles of omputation, arnegie Mellon University - ORTIN xample inary Search Trees Insert: 84, 4, 96, 24, 3, 5, 3, Principles of omputation, arnegie Mellon University - ORTIN 8 4

5 xample inary Search Trees Tree Sort 84 Start at the root node. or each node, display it after you display all the nodes to its left but before you display all the nodes to its right Principles of omputation, arnegie Mellon University - ORTIN 9 xample Heaps heap is a binary tree such that The largest data value is in the root or every node in the heap, its children contain smaller data. The heap is an almost-complete binary tree. n almost-complete binary tree is a binary tree such that every level of the tree has the maximum number of nodes possible except possibly the last level, where its nodes are attached as far left as possible. 5-5 Principles of omputation, arnegie Mellon University - ORTIN 5

6 These are not heaps! Why? Principles of omputation, arnegie Mellon University - ORTIN dding data to a heap Insert new data into next available tree position. xchange data with its parent(s) (if necessary) until the tree is restored to a heap insert here 5-5 Principles of omputation, arnegie Mellon University - ORTIN 2 6

7 Removing data from a heap Remove root (maximum value) and replace it with "last" node of the lowest level xchange data with its larger child (if necessary) until the tree is restored to a heap move last node to root 5-5 Principles of omputation, arnegie Mellon University - ORTIN 3 STs vs. Heaps Which tree is designed for easier searching? Which tree is designed for retrieving the maximum value? heap is guaranteed to be a complete or almost-complete tree. What about a ST? 5-5 Principles of omputation, arnegie Mellon University - ORTIN 4

8 raphs graph is a data structure that consists of a set of nodes and a set of edges connecting pairs of the nodes. graph doesn t have a root, per se. node can be connected to any number of other nodes using edges. n edge may be bidirectional or directed (one-way). n edge may have a weight on it that indicates a cost for traveling over that edge in the graph. pplications: computer networks, transportation systems, social relationships 5-5 Principles of omputation, arnegie Mellon University - ORTIN 5 raphs 5-5 Principles of omputation, arnegie Mellon University - ORTIN 6 8

9 9 5-5 Principles of omputation, arnegie Mellon University - ORTIN graph 5-5 Principles of omputation, arnegie Mellon University - ORTIN 8 Storing a graph djacency Matrix

10 Storing a graph djacency Lists null null null 5-5 Principles of omputation, arnegie Mellon University - ORTIN 9 directed graph 5-5 Principles of omputation, arnegie Mellon University - ORTIN 2

11 5-5 Principles of omputation, arnegie Mellon University - ORTIN 2 Storing a graph djacency Matrix 5-5 Principles of omputation, arnegie Mellon University - ORTIN 22 Storing a graph djacency Lists null null null

12 weighted graph an you think of examples in real life that can be modeled as weighted graphs? 5-5 Principles of omputation, arnegie Mellon University - ORTIN 23 Storing a graph djacency Matrix Principles of omputation, arnegie Mellon University - ORTIN 24 2

13 Storing a graph djacency Lists,,8, null,,2, null,2,6,,5 null Principles of omputation, arnegie Mellon University - ORTIN 25 raph xample Minimal Spanning Tree (MST) ind a tree T in the graph consisting of the minimum set of edges that connects all the nodes without cycles that has minimum total cost. lgorithm: hoose the lowest cost edge of graph for tree T While all nodes are not part of T do the following: hoose the lowest cost edge that connects with a node currently in the tree T If this edge does not form a cycle in T, add it to T Otherwise, remove this edge from consideration T is a Minimal Spanning Tree of graph 5-5 Principles of omputation, arnegie Mellon University - ORTIN 26 3

14 MST Trace MST shown in red (,) 3 (,) 4 (,) 5 (,) - (,) (,) - (,) 8 (,) Order of edges examined 5-5 Principles of omputation, arnegie Mellon University - ORTIN 2 MST Results TOTL MST OST = How did we detect if we were going to put a cycle in the tree? 5-5 Principles of omputation, arnegie Mellon University - ORTIN 28 4

15 atabases database is a large collection of data. schema describes the data values that are stored in the database, and the relationships that exist between these data values. The computer program used to manage and query a database is known as a database management system (MS). xamples: Oracle, Microsoft SQL Server 5-5 Principles of omputation, arnegie Mellon University - ORTIN 29 atabase Models Hierarchical links records together in a tree data structure such that each record type has only one owner xample: n order has only one customer. Network links records together in a graph data structure such that each record type can have more than owner xample: student can have more than one professor. 5-5 Principles of omputation, arnegie Mellon University - ORTIN 3 5

16 atabase Models (cont d) Relational links records together in a matrix based on relationships xample Relations(ields): ustomer(ustomer_i, Name, ddress, ity, State, Zip, Phone) Order(ustomer_I, Product_Number, Qty_Ordered) Invoice(Invoice_No, Product_Number, Qty_Shipped) Languages like SQL (Structured Query Language) can be used to write this query. SLT Name ROM ustomer ind all customer names from Pittsburgh who ordered product #55 but did not receive a full shipment yet. WHR ustomer.ity = "Pittsburgh" N ustomer.ustomer_i=order.ustomer_i N Order.Product_Number = 55 N Invoice.Product_Number = 55 N Order.Qty_Ordered > Invoice.Qty_Shipped 5-5 Principles of omputation, arnegie Mellon University - ORTIN 3 ealing with Huge ata Sets ata Mining procedure where computational techniques such as statistics and pattern matching are used to extract useful information from very large sets of data. ata Warehousing technique that is designed to provide efficient storage for large amounts of data that change infrequently or in localized ways. 5-5 Principles of omputation, arnegie Mellon University - ORTIN 32 6