Cracking the Coding Interview, 4 Edition - 150 Programming Interview Questions and Solutions

Solutions to Chapter 10 | Mathematical
10.6 Given a two dimensional graph with points on it, find a line which passes the most
number of points.
SOLUTION
pg 68
If we draw a line between every two points, we can check to see which line is the most common.
A brute force approach would be to simply iterate through each line segment (formed
by pairs of points) and count how many points fall on it. This would take O(N^3) time.
Before we discuss if we can do better, let’s figure out how we can represent a line. A line can
be represented in (at least) two different ways: (1) as a pairing of points or (2) as a slope and
a y-intercept.
Because our line is infinite, the slope and y-intercept approach seems more appropriate.
The slope and y-intercept approach has an additional advantage: every line segment on the
same greater line will have identical slopes and y-intercepts.
Let’s re-think our solution. We have a bunch of line segments, represented as a slope and
y-intercept, and we want to find the most common slope and y-intercept. How can we find
the most common one?
This is really no different than the old “find the most common number in a list of numbers”
problem. We just iterate through the lines segments and use a hash table to count the number
of times we’ve seen each line.
1 public static Line findBestLine(GraphPoint[] points) {
2 Line bestLine = null;
3 HashMap line_count = new HashMap();
4 for (int i = 0; i < points.length; i++) {
5 for (int j = i + 1; j < points.length; j++) {
6 Line line = new Line(points[i], points[j]);
7 if (!line_count.containsKey(line)) {
8 line_count.put(line, 0);
9 }
10 line_count.put(line, line_count.get(line) + 1);
11 if (bestLine == null ||
12 line_count.get(line) > line_count.get(bestLine)) {
13 bestLine = line;
14 }
15 }
16 }
17 return bestLine;
18 }
19
20 public class Line {
21 private static double epsilon = .0001;
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Cracking the Coding Interview | Concepts and Algorithms