Richard Johnsonbaugh

A graph generation software package

We discuss a software package that generates graphs of specified sizes and properties. Among the types of graphs are

The four-peg Tower of Hanoi puzzle

We discuss a version of the Tower of Hanoi puzzle in which there are four pegs rather than three. The fourpeg puzzle provides a rich source of exercises (samples of which are included) for students after the familiar three-peg version has been presented. We give an algorithm that solves the four-peg puzzle in the claimed minimum number of moves (see [2, 4]). Our algorithm solves the four-peg puzzle in O-(4√n) moves whereas the best algorithm for the three-peg puzzle requires 2n - 1 moves. As far as we know, the minimum number of moves required to solve the four-peg puzzle is an open question.

A real-time people counter

This paper describes an implementation method for the people counting system which detects and tracks moving people using a fixed single camera. This system counts the number of moving objects (people) entering the security door. Moreover, the detected objects are tracked by the proposed tracking algorithm before entering the door. The proposed system with Intel Pentium IV operates at an average rate of 10 frames a second on real world scenes where up to 6 persons come into the view of a vertically mounted camera.

Tiling and recursion

A tiling problem is presented that demonstrates the power of recursion in the design of algorithms. When implemented as a program, the solution can be shown using a computer graphics display.

Converses of pumping lemmas

Pumping lemmas appear in courses that study formal languages such as automata theory and the theory of computation. Converses of pumping lemmas, which a.re generally false, are ignored by most of the books tha,t treat formal languages. This is unfortunate since converses of pumping lemmas arise in a natural way and students typically ask whether converses of particular pumping lemmas are true. We give counterexamples to the converse of a pumping lemma for regular langua.ges and to th e converse of Ogden’s Lemma, a pumping lemma for context-free languages. We also show tha.t converses to these lemmas are true for languages over a single symbol. We conclude by discussing the counterexa.mple to the converse of Ogden’s Lemma with reference to Par&h’s necessary condition for a language to be context-free.

A Discrete Intermediate Value Theorem

Many theories in mathematics (for instance, difference and differential equations) come in discrete and continuous versions. Indeed, an entire book, Excursions in Calculus: An Interplay ofthe Continuous and the Discrete by Robert M. Young (MAA, Washington, DC, 1992), has been devoted to this topic. In this note, I give a discrete intermediate value theorem and apply it in an appealing induction proof.