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5 Recursive Algorithms

EBook Name5 Recursive Algorithms
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EBook Description: Recursion is a form of definition and of algorithms that is very important in computer science theory as well as in practice. Recursive algorithms can be inefficient or efficient.1 A recursive definition or a recursive algorithm is characterized by self-reference.

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Electronic Transactions on Numerical Analysis

its book of mathematics on numerical analysis.

6 Induction

The principle of induction is a one of the most fundamental laws of mathematics. Students come across it, as here, as a method of proof. However, it is a basic property of the natural numbers (i.e. the positive integers) and can be considered as part of the definition of the natural numbers. Since we do not define the natural numbers here, the principle will be simply given as an axiom.1

7 Graph Searching

Algorithms for searching graphs go back at least to the nineteenth century when they were used for mazes. Until the advent of computers they were an extremely obscure topic (as was graph theory). We are going to look at two algorithms which represent the two main approaches to searching graphs.

8 Horner's Algorithm

Horner's algorithm is an excellent example of an algorithm. It is a simple and useful algorithm, but it does not have much to do with discrete mathematics.

9 Shortest Paths

A fundamental problem in graphs is finding the shortest path from vertex A to vertex B. Fortunately there are several simple (and efficient algorithms for doing this). We will look at the three best of these: Dijkstra’s algorithm, Floyd’s algorithm, and the Bellman-Ford algorithm. First we need to discuss different types of shortest path problems and various conventions that we will use in solving them.

10 Euclidean Algorithm

Number theory is the mathematics of integer arithmetic. In this chapter we will restrict ourselves to integers,

11 Division Mod n

Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we get zero. If x is !1 and y is 1 we get 2. In fact a little experimentation will convince you that you can get all the even integers but only even integers.

12 Chinese Remainder Theorem

Many classroom exercises involve dealing cards. In this chapter we will focus on a simple problem: Write an algorithm to randomly select one card out of an ordinary 52-card deck. My students frequently derive an efficient algorithm to solve this problem. The algorithm goes as follows we use a random number generator to select a number between 1 and 52 (or between 0 and 51; either way works fine).

13 Euler's Theorem and Fermat's Little Theorem

The formulas of this section are the most sophisticated number theory results in this book. The reason I am presenting them is that by use of graph theory we can understand them easily. Fermat was a great mathematician of the 17th century and Euler was a great mathematician of the 18th century.

14 Wilson's Theorem

Wilson’s Theorem is elegant. It is not very useful, but like a lot of other people, I like it. So that is why it is here. Consider an integer n > 1. If the integer n-1! + 1 is divided by any number from 2 to n-1, it yields a remainder of 1.

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