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A course on “software tools” might omit the mathematical and advanced algorithmic material, then emphasize means by which the implementations given here can be integrated for use into large programs or systems. Another approach that will suffice for many applications is simply to consider complicated objects as sets of lines and to use the line intersection procedure. How would you determine whether two triangles intersect? For example, a very natural problem which arises in many applications is the nearestneighbor problem: find the point among a set of given points closest to a given new point.

Some supplementary material might be required for each of these examples to reflect their particular orientation (on elementary data structures for “data structures and algorithms,” on mathematical analysis for “design and analysis of algorithms,” and on software engineering techniques for “software tools”); in this book, the emphasis is on the algorithms themselves. This seems to involve checking the distance from the given point to each point in the set, but we’ll see that much better solutions are possible.

To a large extent, the individual chapters in the book can each be read independently of the others. For example, if we implement a procedure which GEOMETRIC INTERSECTION 359 compares two rectangles whose edges are horizontal and vertical according to the trivial rule that rectangle 5 is to the left of rectangle y if the right edge of x is to the left of the left edge of y, then we can use the above method to test for intersection among a set of such rectangles.

The material can be adapted for use for various courses by selecting perhaps thirty of the forty chapters. For circles, we can use the x coordinates of the centers for the ordering, but explicitly test for intersection (for example, compare the distance between the centers to the sum of the radii). What happens if the horizontal-vertical line intersection procedure is used on a set of lines with arbitrary slope? Write a program to find the number of intersecting pairs among a set of N random horizontal and vertical lines, each line generated with two random integer coordinates between and a random bit to distinguish horizontal from vertical. Give a method for testing whether or not a given polygon is simple (doesn’t intersect itself). Give a method for testing whether one polygon is totally contained within another.

The emphasis is on t,eaching students the tools of their trade to the point that they can confidently implement, run and debug useful algorithms. Of course, this comparison could result in the detection of an intersection, as it does in our example.

Full implementations of the methods discussed (in an actual programming language) are included in the text, along with descriptions of the operations of these programs on a consistent set of examples. In summary, to test for an intersection among a set of N lines, we use the program above, but with the call to range removed, and with the binary tree routines extended to use the generalized comparison as described above.

The treatment of sorting, searching and string processing (which may not be covered in other courses) is somewhat more complete than the treatment of mathematical algorithms (which may be covered in more depth in applied mathematics or engineering courses), or geometric and graph algorithms (which may be covered in more depth in advanced computer science courses). In the example above, F is to the left of B (because B is to the right of F) and B is to the left of D, but F is not to the left of D.

There is a great deal of flexibility in the way that the material in the book can be taught. An interesting feature of the above procedure is that it can be adapted to solve the problem for testing for the existence of an intersecting pair among a set of more general geometric shapes just by changing the generalized comparison procedure.

MY ADDISON-WESLEY PUBLISHING COM P ANY Reading, Massachusetts l Menlo Park, California London l Amsterdam l Don Mills, Ontario l Sydney To Adam, Brett, Robbie and especially Linda This book is in the Addison-Wesley Series in Computer Science Consulting Editor Michael A. De Wolfe Library of Congress Cataloging in Publication Data Sedgewick, Robert, 1946Algorithms. The book consists of forty chapters which are grouped into seven major parts: mathematical algorithms, sorting, searching, string processing, geometric algorithms, graph algorithms and advanced topics. B 9 B H G % H F G D Each “comparison” performed during the tree manipulation procedures is actually a line intersection test: if the binary search tree procedure can’t decide to go right or left, then the two lines in question must intersect, and we’re finished.

Additionally, the book may be useful as a reference for those who already have some familiarity with the material, since it contains a number of computer implementations of useful algorithms. For example, the following sequence of diagrams shows the manipulation of the tree for our example between the time that line C is encountered and the time that line D is encountered.

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Though not emphasized, connections to theoretical computer science and the analysis of algorithms are not ignored. If there is no intersection, we’ll start with a null tree and end with a null tree without finding any incomparable lines.

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