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Some of the programs can be simplified by using more advanced language features (some not available in Pascal), but this is true less often than one might think. However, we’ll see that the study of methods with good worstcase performance will uncover some fundamental properties of point sets that should be understood even if simpler methods turn out to be more suitable.

A goal of this book is to present the algorithms in as simple and direct form as possible. The general approach that we’ll be examining provides yet another example of the use of doubly recursive procedures to intertwine processing along the two coordinate directions.

When interesting, the relationship of the practical algorithms being discussed to purely theoretical results is described. Once we’ve found an intersection, we can’t simply press on and hope to find others, because the two lines that intersect should swap places in the ordering directly after the point of intersection.

More information of the orientation and coverage of the material in the book may be found in the Introduction which follows. One way to handle this problem would be to use a priority queue instead of a binary tree for the “ y sort”: initially put lines on the priority queue according to the y coordinates of their endpoints, then work the scan line up by successively taking the smallest y coordinate from the priority queue and doing a binary tree insert or delete as above.

It is appropriate for use as a textbook for a second, third or fourth course in computer science: after students have acquired some programming skills and familiarity with computer systems, but before they have specialized courses in advanced areas of computer science or computer applications. Except for the use of this function whenever a comparison is needed, the standard binary search tree procedures (even balanced trees, if desired) can be used.

Reprinted Copyright 0 with corrections, August 1984 1983 by Addison-Wesley Publishing Company, Inc. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission of the publisher. ISBN o-201-06672-6 FGHIJ-HA-8987654 Preface This book is intended to survey the most important algorithms in use on computers today and to teach fundamental techniques to the growing number of people who are interested in becoming serious computer users. This generalized “line comparison” operation is a simple extension of the same procedure of Chapter 24.

Some of the chapters involve mtroductory treatment of advanced material. It is essential to note this, because the binary tree deletion procedure assumes that the comparison operation is transitive: when B is deleted from the last tree in the above sequence, the tree 358 CHAPTER 27 is formed without F and D ever having been explicitly compared.

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.

An elementary course on “data structures and algorithms” might omit some of the mathematical algorithms and some of the advanced graph algorithms and other advanced topics, then emphasize the ways in which various data structures are used in the implementation. Again, if this comparison procedure is used in the above method, we have an algorithm for testing for intersection among a set of circles. In the horizontal-vertical line intersection algorithm, how many pairs of lines are tested for intersection in a set of lines with no intersections in the worst case? 7 Describe how you would solve the general line intersection problem given the additional fact that the minimum separation between two lines is greater than the maximum length of the lines. Give the binary tree structure that exists when the line intersection algorithm detects an intersection in the following set of lines: 9.

An intermediate course on “design and analysis of algorithms” might omit some of the more practically-oriented sections, then emphasize the identification and study of the ways in which good algorithms achieve good asymptotic performance. The problem of returning all intersections in such cases is much more complicated, though the brute-force method mentioned in the previous paragraph will always work if few intersections are expected. Are the comparison procedures for circles and Manhattan rectangles that are described in the text transitive? Write a program to find the number of intersecting pairs among a set of N random lines, each line generated with random integer coordinates between . Closest Point Problems Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances between the points.