挑战程序竞赛系列（94）：3.6凸包（5）

挑战程序竞赛系列（94）：3.6凸包（5）

传送门：POJ 2079: Triangle

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题意：

求三个点构成的最大三角形面积。

思路：
可以证明，三点构成的最大三角形面积一定在凸包上，传统算法for循环三次，时间复杂度为O(n3)，实际上可以把复杂度降为O(n2)。对于给定的两点(P, Q)，求得第三点R，使得PQR的三角形面积最大。

性质１：

因为凸包上满足性质：循环遍历R时，三角形面积先增后减，找到最大即可停止搜索。

性质2：

利用(P, Q)求得的点R，在搜索(Q, Q_next)的最大三角形面积时，可以从点R开始继续向后搜索，而不需要从头开始。

综上，时间复杂度降为O(n2)。

代码如下：

import java.io.BufferedReader;import java.io.File;import java.io.FileInputStream;import java.io.IOException;import java.io.InputStream;import java.io.InputStreamReader;import java.io.PrintWriter;import java.util.Arrays;import java.util.Map;import java.util.StringTokenizer;public class Main{    String INPUT = "./data/judge/201710/P2079.txt";    public static void main(String[] args) throws IOException {        new Main().run();    }    class P implements Comparable<P>{        int x;        int y;        P(int x, int y){            this.x = x;            this.y = y;        }        P sub(P a) {            return new P(x - a.x, y - a.y);        }        int det(P a) {            return x * a.y - y * a.x;        }        @Override        public int compareTo(P o) {            return x - o.x != 0 ? x - o.x : y - o.y;        }    }    int N;    P[] p;    double area(P a, P b, P c) {        double ans = (c.x - a.x) * (b.y - a.y) - (c.y - a.y) * (b.x - a.x);        return 0.5 * Math.abs(ans);    }    P[] convexHull() {        Arrays.sort(p);        P[] qs = new P[2 * N];        int k = 0;        for (int i = 0; i < N; ++i) {            while (k > 1 && qs[k - 1].sub(qs[k - 2]).det(p[i].sub(qs[k - 2])) < 0) k --;            qs[k++] = p[i];        }        for (int i = N - 2, t = k; i >= 0; --i) {            while (k > t && qs[k - 1].sub(qs[k - 2]).det(p[i].sub(qs[k - 2])) < 0) k--;            qs[k++] = p[i];        }        P[] res = new P[k - 1];        System.arraycopy(qs, 0, res, 0, k - 1);        return res;    }    void solve() {        P[] qs = convexHull();        int n = qs.length;        double ans = 0;        for (int offset = 1; offset < (n + 1) / 2; ++offset) {            int fir = 0;            int pos = (fir + offset + 1) % n;            do {                int sec = (fir + offset) % n;                while ((pos + 1) % n != fir && area(qs[fir % n], qs[sec % n], qs[(pos + 1) % n])                            >= area(qs[fir % n], qs[sec % n], qs[pos % n])) {                    pos = (pos + 1) % n;                }                ans = Math.max(ans, area(qs[fir % n], qs[sec % n], qs[pos % n]));                fir = (fir + 1) % n;            }            while (fir != 0);        }        out.printf("%.2f\n", ans);    }    void read() {        while (true) {            N = ni();            if (N == -1) break;            p = new P[N];            for (int i = 0; i < N; ++i) {                p[i] = new P(ni(), ni());            }            solve();        }    }    FastScanner in;    PrintWriter out;    void run() throws IOException {        boolean oj;        try {            oj = ! System.getProperty("user.dir").equals("F:\\oxygen_workspace\\Algorithm");        } catch (Exception e) {            oj = System.getProperty("ONLINE_JUDGE") != null;        }        InputStream is = oj ? System.in : new FileInputStream(new File(INPUT));        in = new FastScanner(is);        out = new PrintWriter(System.out);        long s = System.currentTimeMillis();        read();        out.flush();        if (!oj){            System.out.println("[" + (System.currentTimeMillis() - s) + "ms]");        }    }    public boolean more(){        return in.hasNext();    }    public int ni(){        return in.nextInt();    }    public long nl(){        return in.nextLong();    }    public double nd(){        return in.nextDouble();    }    public String ns(){        return in.nextString();    }    public char nc(){        return in.nextChar();    }    class FastScanner {        BufferedReader br;        StringTokenizer st;        boolean hasNext;        public FastScanner(InputStream is) throws IOException {            br = new BufferedReader(new InputStreamReader(is));            hasNext = true;        }        public String nextToken() {            while (st == null || !st.hasMoreTokens()) {                try {                    st = new StringTokenizer(br.readLine());                } catch (Exception e) {                    hasNext = false;                    return "##";                }            }            return st.nextToken();        }        String next = null;        public boolean hasNext(){            next = nextToken();            return hasNext;        }        public int nextInt() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Integer.parseInt(more);        }        public long nextLong() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Long.parseLong(more);        }        public double nextDouble() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Double.parseDouble(more);        }        public String nextString(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more;        }        public char nextChar(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more.charAt(0);        }    }}