Problem:
Score : 900 points

Problem Statement

In the city of Nevermore, there are 10^8 streets and 10^8 avenues, both numbered from 0 to 10^8-1.
All streets run straight from west to east, and all avenues run straight from south to north.
The distance between neighboring streets and between neighboring avenues is exactly 100 meters.

Every street intersects every avenue. Every intersection can be described by pair (x, y), where x is avenue ID and y is street ID.

There are N fountains in the city, situated at intersections (X_i, Y_i).
Unlike normal intersections, there's a circle with radius 10 meters centered at the intersection, and there are no road parts inside this circle.

The picture below shows an example of how a part of the city with roads and fountains may look like.

City governors don't like encountering more than one fountain while moving along the same road.
Therefore, every street contains at most one fountain on it, as well as every avenue.

Citizens can move along streets, avenues and fountain perimeters.
What is the shortest distance one needs to cover in order to get from intersection (x_1, y_1) to intersection (x_2, y_2)?

Constraints

0 \leq x_1, y_1, x_2, y_2 < 10^8

1 \leq N \leq 200,000

0 \leq X_i, Y_i < 10^8

X_i \neq X_j for i \neq j

Y_i \neq Y_j for i \neq j

Intersections (x_1, y_1) and (x_2, y_2) are different and don't contain fountains.

All input values are integers.

Input

Input is given from Standard Input in the following format:

x_1 y_1 x_2 y_2
N
X_1 Y_1
X_2 Y_2
:
X_N Y_N

Output

Print the shortest possible distance one needs to cover in order to get from intersection (x_1, y_1) to intersection (x_2, y_2), in meters.
Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{-11}.

Sample Input 1

1 1 6 5
3
3 2
5 3
2 4

Sample Output 1

891.415926535897938

One possible shortest path is shown on the picture below. The path starts at the blue point, finishes at the purple point and follows along the red line.

Sample Input 2

3 5 6 4
3
3 2
5 3
2 4

Sample Output 2

400.000000000000000

Sample Input 3

4 2 2 2
3
3 2
5 3
2 4

Sample Output 3

211.415926535897938

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)


import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;

public class Main {

  private static void solve() {
    int x1 = ni();
    int y1 = ni();
    int x2 = ni();
    int y2 = ni();
    
    int n = ni();
    int[] x = new int[n];
    int[] y = new int[n];
    int m = 0;
    int f1 = 1;
    int f2 = 1;
    if (x1 > x2) {
      f1 = -1;
      x1 = -x1;
      x2 = -x2;
    }
    if (y1 > y2) {
      f2 = -1;
      y1 = -y1;
      y2 = -y2;
    }
    for (int i = 0; i < n; i ++) {
      int nx = f1 * ni();
      int ny = f2 * ni();
      if (x1 <= nx && nx <= x2 && y1 <= ny && ny <= y2) {
        x[m] = nx;
        y[m] = ny;
        m ++;
      }
    }
    x = Arrays.copyOf(x, m);
    y = Arrays.copyOf(y, m);
    Compress cmp = new Compress(y);

    int[][] pos = new int[m + 1][2];
    for (int i = 0; i < m; i ++) {
      pos[i][0] = x[i];
      pos[i][1] = cmp.list[i];
    }
    pos[m][0] = x2;
    pos[m][1] = y2;
    
    Arrays.sort(pos, (o1, o2) -> o1[0] == o2[0] ? o1[1] - o2[1] : o1[0] - o2[0]);
    int ST = 201000;
    SegmentTreeRMQ st = new SegmentTreeRMQ(ST);
    for (int i = 0; i < ST; i ++) st.update(i, 0);
    st.update(pos[m][1], 0);
    for (int i = m - 1; i >= 0; i --) {
      int z = pos[i][1];
      int min = st.min(z, ST);
      st.update(z, min - 1);
    }
    
    int ball = -st.min(0,  201000);
    
    double total = (Math.abs(x1 - x2) + Math.abs(y1 - y2)) * 100;
    double arg1 = 20 - Math.PI * 5;
    double arg2 = Math.PI * 10 - 20;
    
    int lenMin = Math.min(Math.abs(x2 - x2), Math.abs(y1 - y2));
    if (lenMin <= ball - 1) {
      out.println(total - arg1 * lenMin + arg2);
    } else {
      out.println(total - arg1 * ball);
    }
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = System.getProperty("debug");
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}

class Compress {
  public final Map<Integer, Integer> zip = new HashMap<>();;
  public final long[] unzip;
  public final int[] list;

  public Compress(int[] src) {
      int n = src.length;
      int[] tmp = Arrays.copyOf(src, n);
      Arrays.sort(tmp);

      unzip = new long[n];
      int index = 0;
      for (int i = 0; i < n; i ++) {
        int v = tmp[i];
          if (!zip.containsKey(v)) {
              zip.put(v, index ++);
              unzip[i] = v;
          }
      }

      list = new int[n];
      for (int i = 0; i < n; i ++) {
          list[i] = zip.get(src[i]);
      }
  }
}
class SegmentTreeRMQ {
  public int M, H, N;
  public int[] st;
  
  public SegmentTreeRMQ(int n)
  {
      N = n;
      M = Integer.highestOneBit(Math.max(N-1, 1))<<2;
      H = M>>>1;
      st = new int[M];
      Arrays.fill(st, 0, M, -1);
  }
  
  public SegmentTreeRMQ(int[] a)
  {
      N = a.length;
      M = Integer.highestOneBit(Math.max(N-1, 1))<<2;
      H = M>>>1;
      st = new int[M];
      for(int i = 0;i < N;i++){
          st[H+i] = a[i];
      }
      Arrays.fill(st, H+N, M, Integer.MAX_VALUE);
      for(int i = H-1;i >= 1;i--)propagate(i);
  }
  
  public void update(int pos, int x)
  {
      st[H+pos] = x;
      for(int i = (H+pos)>>>1;i >= 1;i >>>= 1)propagate(i);
  }
  
  private void propagate(int i)
  {
      st[i] = Math.min(st[2*i], st[2*i+1]);
  }
  
  public int minx(int l, int r){
      int min = Integer.MAX_VALUE;
      if(l >= r)return min;
      while(l != 0){
          int f = l&-l;
          if(l+f > r)break;
          int v = st[(H+l)/f];
          if(v < min)min = v;
          l += f;
      }
      
      while(l < r){
          int f = r&-r;
          int v = st[(H+r)/f-1];
          if(v < min)min = v;
          r -= f;
      }
      return min;
  }
  
  public int min(int l, int r){ return l >= r ? 0 : min(l, r, 0, H, 1);}
  
  private int min(int l, int r, int cl, int cr, int cur)
  {
      if(l <= cl && cr <= r){
          return st[cur];
      }else{
          int mid = cl+cr>>>1;
          int ret = Integer.MAX_VALUE;
          if(cl < r && l < mid){
              ret = Math.min(ret, min(l, r, cl, mid, 2*cur));
          }
          if(mid < r && l < cr){
              ret = Math.min(ret, min(l, r, mid, cr, 2*cur+1));
          }
          return ret;
      }
  }
  
  public int firstle(int l, int v) {
      int cur = H+l;
      while(true){
          if(st[cur] <= v){
              if(cur < H){
                  cur = 2*cur;
              }else{
                  return cur-H;
              }
          }else{
              cur++;
              if((cur&cur-1) == 0)return -1;
              if((cur&1)==0)cur>>>=1;
          }
      }
  }
  
  public int lastle(int l, int v) {
      int cur = H+l;
      while(true){
          if(st[cur] <= v){
              if(cur < H){
                  cur = 2*cur+1;
              }else{
                  return cur-H;
              }
          }else{
              if((cur&cur-1) == 0)return -1;
              cur--;
              if((cur&1)==1)cur>>>=1;
          }
      }
  }
}