7.6 Given a two-dimensional graph with points on it, find a line which passes the most number of points.

這道題給了我們許多點，讓我們求經過最多點的一條直線。給之前那道7.5 A Line Cut Two Squares in Half 平均分割兩個正方形的直線一樣，都需要自己寫出點類和直線類。在直線類中，我用我們用斜率和截距來表示直線，為了應對斜率不存在情況，我們還需用一個flag來標記是否為垂直的線。在直線類中，我們要有判斷兩條直線是否相等的函式。判斷相等的方法和之前那道7.3 Line Intersection 直線相交

class Point {
public:
    double _x, _y;
    Point(double x, double y): _x(x), _y(y) {};
};

class Line {
public:
    static constexpr double
 
 _epsilon = 0.0001;
    double _slope, _intercept;
    bool _infi_slope = false;
    Line(Point p, Point q) {
        if (fabs(p._x - q._x) > _epsilon) {
            _slope = (p._y - q._y) / (p._x - q._x);
            _intercept = p._y - _slope * p._x;
        } else {
            _infi_slope 
 
= true;
            _intercept = p._x;
        }
    }
    static double floorToNearestEpsilon(double d) {
        int r = (int)(d / _epsilon);
        return ((double)r) * _epsilon;
    }
    bool isEquivalent(double a, double b) {
        return (fabs(a - b) < _epsilon);
    }
    bool isEquivalent(Line other) {
        if (isEquivalent(_slope, other._slope) && isEquivalent(_intercept, other._intercept) && (_infi_slope == other._infi_slope)) {
            return true;
        }
        return false;
    }
};

class Solution {
public:
    Line findBestLine(vector<Point> &points) {
        Line res(points[0], points[1]);
        int bestCnt = 0;
        unordered_map<double, vector<Line> > m;
        for (int i = 0; i < (int)points.size(); ++i) {
            for (int j = i + 1; j < (int)points.size(); ++j) {
                Line line(points[i], points[j]);
                insertLine(m, line);
                int cnt = countEquivalentLines(m, line);
                if (cnt > bestCnt) {
                    res = line;
                    bestCnt = cnt;
                }
            }
        }
        return res;
    }
    void insertLine(unordered_map<double, vector<Line> > &m, Line &line) {
        vector<Line> lines;
        double key = Line::floorToNearestEpsilon(line._slope);
        if (m.find(key) != m.end()) {
            lines = m[key];
        } else {
            m[key] = lines;
        }
        lines.push_back(line);
    }
    int countEquivalentLines(unordered_map<double, vector<Line> > &m, Line &line) {
        double key = Line::floorToNearestEpsilon(line._slope);
        double eps = Line::_epsilon;
        return countEquivalentLines(m[key], line) + countEquivalentLines(m[key - eps], line) + countEquivalentLines(m[key + eps], line);
    }
    int countEquivalentLines(vector<Line> &lines, Line &line) {
        if (lines.empty()) return 0;
        int res = 0;
        for (auto &a : lines) {
            if (a.isEquivalent(line)) ++res;
        }
        return res;
    }
};