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146 149 151 152 155 157 158 160 164

196 199 202 204

224 228 231 233 235





389 392 393 395 The Huffman Encoding Algorithm .................................................................................................................. 398 Justification ..................................................................................................................................................... 401

Huffman Codes.............................................................................................................................. 396

Searching........................................................................................................................................................ Lower Bounds on Searching .......................................................................................................................... Binary Tree Search......................................................................................................................................... Sorting.............................................................................................................................................................

Decision Trees............................................................................................................................... 389

5.4 Decision Trees and Huffman Codes..........................................................................388

Directed Graphs and Binary Relations ........................................................................................... 375 Reachability.................................................................................................................................... 377 Warshall’s Algorithm....................................................................................................................... 382

5.3 Directed Graphs and Binary Relations and Warshall’s Algorithm.........................374

Adjacency Matrix............................................................................................................................ 363 Adjacency List................................................................................................................................ 364 Binary Tree Representation ........................................................................................................... 367

5.2 Computer Representations of Graphs......................................................................362

Trees.............................................................................................................................................. 346 Directed Graphs............................................................................................................................. 347 Applications.................................................................................................................................... 349

Planarity.......................................................................................................................................................... 335 Colorability...................................................................................................................................................... 341

Graphs........................................................................................................................................... 326 Isomorphic Graphs......................................................................................................................... 330 Other Properties of Graphs ............................................................................................................ 334

5.1 Graph Terminology and Applications.......................................................................326

Graphs and Trees............................................................................................................. 325

4.4 Matrices........................................................................................................................ 307

4.3 Functions...................................................................................................................... 270

Databases...................................................................................................................................... 251 Topological Sorting (PERT−chart)................................................................................................. 261

4.2 Relations and Databases and Topological Sorting.................................................251

Binary Relations............................................................................................................................. Properties of Relations ................................................................................................................... Closures of Relations ..................................................................................................................... Partial Orderings............................................................................................................................ Equivalence Relations [x] ..............................................................................................................

4.1 Relations....................................................................................................................... 224

Relations, Functions and Matrices................................................................................ 223

3.5 The Binominal Theorem..............................................................................................211

Permutations.................................................................................................................................. Combinations................................................................................................................................. Eliminating Dublicates.................................................................................................................... Permutations and Combinations with Repetitions ..........................................................................




Principle of Inclusion and Exclusion ............................................................................................... 189 The Pigeonhole Principle ............................................................................................................... 193

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3.4 Permutations and Combinations...............................................................................196

mathematical structures for computer science − TOC




3.3 Principle of Inclusion and Exclusion and the Pigeonhole Principle......................189

The Multiplication Principle ............................................................................................................. 179 The Addition Principle .................................................................................................................... 181 Decision Trees............................................................................................................................... 184

3.2 Counting....................................................................................................................... 178

Notation.......................................................................................................................................... Relationships between Sets ........................................................................................................... Sets of Sets (powerset)................................................................................................................. Binary and Unary Operations ......................................................................................................... Operations on Sets ........................................................................................................................ Cartesian Product A x B................................................................................................................ Set Identities .................................................................................................................................. Sets in Programming...................................................................................................................... Countable and Uncountable Sets ..................................................................................................

3.1 Sets............................................................................................................................... 145

Sets and Combinatorics.................................................................................................. 145

Recursive Algorithms...................................................................................................................... 106 Solving Recurrence Relations ........................................................................................................ 112

Recursive Sequences .................................................................................................................................... 100 Recursive Sets ............................................................................................................................................... 103 Recursive Operations ..................................................................................................................................... 105

Recursive Definitions ...................................................................................................................... 100

2.3 Recursion and Recurrence Relations.......................................................................100

The Method...................................................................................................................................... 85 Inductive Proofs ............................................................................................................................... 86 Complete Induction .......................................................................................................................... 92

2.2 Induction......................................................................................................................... 85

Direct Proof....................................................................................................................................................... 78 Contraposition................................................................................................................................................... 79 Contradiction..................................................................................................................................................... 81

Method’s of Attack ............................................................................................................................ 77

2.1 Proof Techniques.......................................................................................................... 75

Proofs, Recursion and Analysis of Algorithms............................................................. 75

Quantifiers and Predicates (∀x) (...), (∃x) (...) ................................................................................. 18 Validity.............................................................................................................................................. 24

1.2 Quantifiers, Predicates and Validity............................................................................18

Connectives and Truth Values ∧, ∨, →, ↔........................................................................................ 2 Tautologies......................................................................................................................................... 7 Tautological Equivalences ................................................................................................................ 10

1.1 Statements, Symbolic Representation and Tautologies.............................................1

Formal Logic.......................................................................................................................... 1





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Modeling Arithmetic, Computation and Languages................................................... 535

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09/13/1999 Michael Rauch I1C [email protected]

Important Note: This table of contents for the book ’Mathematical Structures For Computer Science’ is NOT complete. It’s main purpose is to offer faster access to the topics treated in the course ’Computer Basics’ at the BSE. Chapters which were not treated in the course (or perhaps I only missed these lessons... ) are not listed in this TOC. Hope this will be useful for the ’Vordiploma’. :−)

8.4 Formal Languages....................................................................................................... 621

8.3 Turing Machines.......................................................................................................... 597

Unreachable States........................................................................................................................................ 578 A Minimization Procedure .............................................................................................................................. 579

Definition........................................................................................................................................ Examples of Finite−State Machines ............................................................................................... Recognition .................................................................................................................................... Regular Sets and Kleene’s Theorem ............................................................................................. Machine Minimization.....................................................................................................................

8.2 Finite−State Machines................................................................................................. 568

8.1 Algebraic Structures...................................................................................................536

Boolean Algebra and Computer Logic.......................................................................... 467

The Problem Statement ................................................................................................................. 454 The Idea behind the Algorithm ....................................................................................................... 456 The Algorithm Itself ........................................................................................................................ 458

6.4 Articulation Points and Computer Networks...........................................................454

Tree Traversal................................................................................................................................ 445

Depth−First Search......................................................................................................................................... 436 Breadth−First Search..................................................................................................................................... 438 Analysis .......................................................................................................................................................... 441 Applications .................................................................................................................................................... 442

Depth−First Search and Breadth−First Search ............................................................................... 435

6.3 Traversal Algorithms................................................................................................... 435

The Shortest−Path Problem ........................................................................................................... 421 The Minimal Spanning Tree Problem ............................................................................................. 428

6.2 Shortest Path and Minimal Spanning Tree...............................................................421

The Euler Path Problem ................................................................................................................. 412 The Hamiltonian Circuit Problem .................................................................................................... 417

6.1 Euler Path and Hamiltonian Circuit...........................................................................412

Graph Algorithms............................................................................................................. 411




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mathematical structures for computer science − TOC

by m. rauch