Computer Science for Engineers

Transcription

1 Computer Science for Engineers Lecture 8 Data structures part 2 Prof. Dr. Dr.-Ing. Jivka Ovtcharova Dipl. Wi.-Ing. Dan Gutu 18 th of December 2009

2 Outline Lecture Content 1. Preface 2. Basics 3. Object orientation 4.1. Introduction 4.2. Graphs 4.3. Use of Graphs 4.4. Trees 4.5. Use of Trees 4.6. Linked Lists 4.7. Queues and Stacks 5. Algorithms Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 2

3 Modeling and Simulation of Dynamic Systems General dynamic system: mechanical, electrical, hydraulic, thermal, chemical systems, among others Example of a dynamic system: traveling vehicle, micro-electronic switch, satellite positioning system Classic modeling approach - Creation of an ideal model of a real system - Transformation of the ideal model into mathematical 4.3 Use of Graphs equations, esp. block diagram. Disadvantages to the classic modeling approach - Elaborate process - With new changes to the ideal model, new equations must then be derived - Discipline specific approach (mechanic, electric, hydraulic ) Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 3

4 The Bond Graph Approach Founded by H. Paynter, MIT, MA, 1959 Recursive top-down disassembly into sub systems that exchange energy with one another Interdisciplinary representation of the dynamic behavior of physical systems System models are described with a uniform notation for all types of physical systems A powerful tool for modeling technical systems, esp. when more than one discipline is involved. 4.3 Use of Graphs Contains the information about the physical structure: the structure of a bond graph results from the topologic structure of the schematic representation of the system. With changes to the ideal model, only the concerned model parts must be updated: very good for model driven design and What-If? situations The Bond graph-approach is an object oriented approach Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 4

5 Bond Graphs: Discipline Independent Representation of Physical Systems Hydraulic Mechanic Electric Thermodynamic Magnetism and chemical 4.3 Use of Graphs Where can bond graphs be used? Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 5

6 Usage Areas of Bond Graphs Mechanic Rotation Mechanic Translation Hydraulic/Pneumatic 4.3 Use of Graphs Electric Thermodynamic Chemical engineering Magnetism Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 6

7 Outline Lecture Content 1. Preface 2. Basics 3. Object orientation 4.1. Introduction 4.2. Graphs 4.3. Use of Graphs 4.4. Trees 4.5. Use of Trees 4.6. Linked Lists 4.7. Queues and Stacks 5. Algorithms Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 7

8 From Graphs to Trees A tree represents a special kind of graph. a tree is a graph in which additional conditions must be given (similar to: every rectangle is a polygon). Through this, the structure and operations are simplified. Graph tree What are these conditions? (next slide) Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 8

9 General Tree Definition 3.7: The Graph G = (V,E) is a Tree if and only if 1. G is loop free 2. G contains no simple edged circle 3. G is associating Counter examples loop circle Non-associating Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 9

10 Analogy leaves (End of the branching) Branching (vertices) twigs (edges) Analogy root Tree as a graph Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 10

11 General Tree Important: According to later implementation in Java, as a result we must consider the edges of a tree as directed edges. Distinguishing the root vertex is not necessary any more. root tree Tree with directed edges Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 11

12 Terms (1) Let T = (V,E) be a tree. The successor of a vertex v are also called children, or sons of v. siblings root The predecessor of vertex v is also called father or parent of v. A vertex is the root of the tree when it does not have a father. A vertex that does not have any children is called a leaf. Vertices are called siblings when they have the same father. leaves Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 12

13 Terms (2) Let T = (V,E) be a tree with the root v 0. Depth of a vertex v n = length of the path π = (v 0,,v n ) = π = (v 0,,v n ) = displacement from the root Height of a tree = maximal depth root v 0 v 2 v 3 T has the order d = the branching factor d(t) = Each vertex of T has a maximum of d children Nomenclature: d-nary tree i.e.: d = 2: binary tree d = 3: ternary tree v 4 v 5 v 6 Depth of v 4 = 2 Height = 3 v 7 Order = 3 Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 13

14 Trees as dynamic data structures (1) In order to implement data structures based on graphs, one can use this object oriented approach. A vertex is an administration object with a controlled amount of child-vertices and a data object. An edge is created through reference to the respective child-vertices. :Node v 0 n 1, n k data v 1 v k Tree representation :Node n 1, n k data Implementation :Node n 1, n k data Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 14

15 Trees as dynamic data structures (2) The actual information is contained in the data object of the vertex The sum of all the data objects yields the amount of data that will be managed through the tree structure. :Node :Object n 1, n k data :Object :Node n 1, n k data Tree structure :Node n 1, n k data :Object Amount of data Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 15

16 From a general tree to a binary tree A binary tree represents a special case of a general tree. Binary = 2, in a pair, composed of 2 basic entities The structure of a binary tree and the operations associated with the tree are realized relatively easy on a digital computer through the tree characteristics. General tree structure Binary tree structure Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 16

17 Binary tree Definition 3.8: The tree T = (V,E) is a binary tree only if d(t) = 2 That means, each vertex from T contains a maximum of 2 child-vertices. examples Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 17

18 Binary trees as dynamic data structures (1) Binary trees are created in Java with the help of the BNode class. The data attribute references a data element of type object. (Similar to the data structure of a general tree) The linkage to the managing element results in the tree structure: left references the previous, right the next successor. :BNode v W left, right data v L v R Tree representation :BNode left, right data Implementation :BNode left, right data Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 18

19 Binary trees: class diagrams BTree 2 recursive associations BNode BNode BNode Object Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 19

20 Operations of a binary tree Entire readout Traversing of all vertices in a certain order. Tree organization Splitting of a tree into tree parts: Tree[] split() Assembly of multiple trees to one new tree: Merge(Tree t) Data access insert: add(object o) = insert delete: remove(object o) = delete Search/ ask: boolean contains(object o) Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 20

21 Traversing Binary trees Traversing process is the process that runs through each vertex of a tree-forming graph exactly once. In conjunction with binary trees, one could also be talking about a linearization. Prevalent traversing strategies: Preorder: 1 WLR root, left part, right part 2 3 Postorder: 3 LRW left part, right part, root 1 2 Inorder: LWR left part, root, right part Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 21

22 Traversing Binary trees - example 1 L B 2 10 P A 1 D 4 7 N 9 R C E O 8 Z 11 Preorder WLR yields L, B, A, D, C, E, P, N, O, R, Z Postorder LRW yields A, C, E, D, B, O, N, Z, R, P, L Inorder LWR yields A, B, C, D, E, L, N, O, P, R, Z Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 22

23 Binary Search Trees - Example < A < B < C < D < E < L < N < O < P < R < Z L B P A D N R C E O Z The value of the user data of all vertexes of the left part of the tree are smaller than the roots. The value of the user data of all vertexes of the right part of the tree are bigger than the roots Prof. Dr. Dr.-Ing. Jivka Ovtcharova CSE-Lecture Ch. 4 - WS 09/10 - Slide 23