Problem:
Score : 900 points

Problem Statement

There are N pinholes on the xy-plane. The i-th pinhole is located at (x_i,y_i).

We will denote the Manhattan distance between the i-th and j-th pinholes as d(i,j)(=|x_i-x_j|+|y_i-y_j|).

You have a peculiar pair of compasses, called Manhattan Compass.
This instrument always points at two of the pinholes.
The two legs of the compass are indistinguishable, thus we do not distinguish the following two states: the state where the compass points at the p-th and q-th pinholes, and the state where it points at the q-th and p-th pinholes.

When the compass points at the p-th and q-th pinholes and d(p,q)=d(p,r), one of the legs can be moved so that the compass will point at the p-th and r-th pinholes.

Initially, the compass points at the a-th and b-th pinholes.
Find the number of the pairs of pinholes that can be pointed by the compass.

Constraints

2≦N≦10^5

1≦x_i, y_i≦10^9

1≦a < b≦N

When i ≠ j, (x_i, y_i) ≠ (x_j, y_j)

x_i and y_i are integers.

Input

The input is given from Standard Input in the following format:

N a b
x_1 y_1
:
x_N y_N

Output

Print the number of the pairs of pinholes that can be pointed by the compass.

Sample Input 1

5 1 2
1 1
4 3
6 1
5 5
4 8

Sample Output 1

4

Initially, the compass points at the first and second pinholes.

Since d(1,2) = d(1,3), the compass can be moved so that it will point at the first and third pinholes.

Since d(1,3) = d(3,4), the compass can also point at the third and fourth pinholes.

Since d(1,2) = d(2,5), the compass can also point at the second and fifth pinholes.

No other pairs of pinholes can be pointed by the compass, thus the answer is 4.

Sample Input 2

6 2 3
1 3
5 3
3 5
8 4
4 7
2 5

Sample Output 2

4

Sample Input 3

8 1 2
1 5
4 3
8 2
4 7
8 8
3 3
6 6
4 8

Sample Output 3

7

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.util.ArrayList;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Scanner;
import java.util.TreeSet;
import java.util.function.Consumer;

public class Main {

    static class Point {
        @Override
        public String toString() {
            return "Point{" +
                   "x=" + x +
                   ", y=" + y +
                   '}';
        }

        public int getX() {
            return x;
        }

        public int getY() {
            return y;
        }

        int x;
        int y;

        Point(int x, int y) {
            this.x = x;
            this.y = y;
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) { return true; }
            if (o == null || getClass() != o.getClass()) { return false; }

            Point point = (Point) o;

            if (x != point.x) { return false; }
            return y == point.y;
        }

        @Override
        public int hashCode() {
            int result = x;
            result = 31 * result + y;
            return result;
        }
    }

    static Map<Integer, List<Point>> xSet = new HashMap<>();
    static Map<Integer, List<Point>> ySet = new HashMap<>();
    static Map<Integer, TreeSet<Point>> xTree = new HashMap<>();
    static Map<Integer, TreeSet<Point>> yTree = new HashMap<>();

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int n = scanner.nextInt();
        int a = scanner.nextInt() - 1;
        int b = scanner.nextInt() - 1;
        int[] xs = new int[n];
        int[] ys = new int[n];
        for (int i = 0; i < n; i++) {
            xs[i] = scanner.nextInt();
            ys[i] = scanner.nextInt();
        }
        int d = Math.abs(xs[a] - xs[b]) + Math.abs(ys[a] - ys[b]);
        for (int i = 0; i < n; i++) {
            Point point = new Point(xs[i] + ys[i], ys[i] - xs[i]);
            xSet.computeIfAbsent(point.x, integer -> {
                return new ArrayList<>();
            }).add(point);
            ySet.computeIfAbsent(point.y, integer -> {
                return new ArrayList<>();
            }).add(point);

            xTree.computeIfAbsent(point.x, integer -> {
                return new TreeSet<>(Comparator.comparing(Point::getY));
            }).add(point);
            yTree.computeIfAbsent(point.y, integer -> {
                return new TreeSet<>(Comparator.comparing(Point::getX));
            }).add(point);
        }

        xSet.values().forEach(new Consumer<List<Point>>() {
            @Override
            public void accept(List<Point> points) {
                points.sort(Comparator.comparingInt(Point::getY));
            }
        });
        ySet.values().forEach(new Consumer<List<Point>>() {
            @Override
            public void accept(List<Point> points) {
                points.sort(Comparator.comparingInt(Point::getX));
            }
        });

        Point root = new Point(xs[a] + ys[a], ys[a] - xs[a]);
        xTree.get(root.x).remove(root);
        yTree.get(root.y).remove(root);
        System.out.println(dfs(root, d) / 2);
    }

    static long dfs(Point p, int d) {
        long ret = 0;

        List<Point> uSet = ySet.get(p.y + d);
        if (uSet != null) {
            int u2 = Collections.binarySearch(uSet, new Point(p.x + d, p.y + d),
                                              Comparator.comparingInt(Point::getX));
            int u1 = Collections.binarySearch(uSet, new Point(p.x - d, p.y + d),
                                              Comparator.comparingInt(Point::getX));
            ret += getRet(u2, u1);
            TreeSet<Point> points = yTree.get(p.y + d);
            while (true) {
                Point target = points.ceiling(new Point(p.x - d, p.y + d));
                if (target == null || target.x > p.x + d) {
                    break;
                }
                points.remove(target);
                xTree.get(target.getX()).remove(target);
                ret += dfs(target, d);
            }
        }

        List<Point> dSet = ySet.get(p.y - d);
        if (dSet != null) {
            int d2 = Collections.binarySearch(dSet, new Point(p.x + d, p.y - d),
                                              Comparator.comparingInt(Point::getX));
            int d1 = Collections.binarySearch(dSet, new Point(p.x - d, p.y - d),
                                              Comparator.comparingInt(Point::getX));
            ret += getRet(d2, d1);
            TreeSet<Point> points = yTree.get(p.y - d);
            while (true) {
                Point target = points.ceiling(new Point(p.x - d, p.y - d));
                if (target == null || target.x > p.x + d) {
                    break;
                }
                points.remove(target);
                xTree.get(target.getX()).remove(target);
                ret += dfs(target, d);
            }
        }

        List<Point> rSet = xSet.get(p.x + d);
        if (rSet != null) {
            int r2 = Collections.binarySearch(rSet, new Point(p.x + d, p.y + d - 1),
                                              Comparator.comparingInt(Point::getY));
            int r1 = Collections.binarySearch(rSet, new Point(p.x + d, p.y - d + 1),
                                              Comparator.comparingInt(Point::getY));
            ret += getRet(r2, r1);

            TreeSet<Point> points = xTree.get(p.x + d);
            while (true) {
                Point target = points.ceiling(new Point(p.x + d, p.y - d));
                if (target == null || target.y > p.y + d) {
                    break;
                }
                points.remove(target);
                yTree.get(target.getY()).remove(target);
                ret += dfs(target, d);
            }
        }

        List<Point> lSet = xSet.get(p.x - d);
        if (lSet != null) {
            int l2 = Collections.binarySearch(lSet, new Point(p.x - d, p.y + d - 1),
                                              Comparator.comparingInt(Point::getY));
            int l1 = Collections.binarySearch(lSet, new Point(p.x - d, p.y - d + 1),
                                              Comparator.comparingInt(Point::getY));
            ret += getRet(l2, l1);

            TreeSet<Point> points = xTree.get(p.x - d);
            while (true) {
                Point target = points.ceiling(new Point(p.x - d, p.y - d));
                if (target == null || target.y > p.y + d) {
                    break;
                }
                points.remove(target);
                yTree.get(target.getY()).remove(target);
                ret += dfs(target, d);
            }
        }
        return ret;
    }

    private static long getRet(long d2, long d1) {
        long ret = 0;
        if (d2 >= 0 & d1 >= 0) {
            ret += d2 - d1 + 1;
        }
        if (d2 >= 0 & d1 < 0) {
            ret += d2 + d1 + 1 + 1;
        }
        if (d2 < 0 & d1 >= 0) {
            ret += -(d2 + 1) - d1;
        }
        if (d2 < 0 & d1 < 0) {
            ret += -(d2 + 1) + d1 + 1;
        }
        return ret;
    }
}