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.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.InputMismatchException;
import java.util.NoSuchElementException;

public class Main {

	public static void main(String[] args) {
		IO io = new IO();
		int n = io.nextInt();
		int s1 = io.nextInt() - 1;
		int s2 = io.nextInt() - 1;
		long[] x = new long[n];
		long[] y = new long[n];
		for(int i=0;i<n;i++) {
			int xraw = io.nextInt();
			int yraw = io.nextInt();
			x[i] = xraw + yraw;
			y[i] = xraw - yraw;
		}
		long d = Math.max(Math.abs(x[s1] - x[s2]), Math.abs(y[s1] - y[s2]));
		UnionFind uf = new UnionFind(n);
		int[] degree = new int[n];
		solveHalf(n, x, y, d, d, degree, uf);
		solveHalf(n, y, x, d, d-1, degree, uf);
		long sum = 0;
		for(int i=0;i<n;i++) {
			if (uf.isConnected(s1, i)) {
				sum += degree[i];
			}
		}
		System.out.println(sum / 2);
	}
	

	@SuppressWarnings("unchecked")
	public static void solveHalf(int n,long[] x,long[] y,long dx, long dy,int[] degree,UnionFind uf) {
		long[] unzipx = new long[n];
		HashMap<Long, Integer> zipx = new HashMap<>();
		
		ArrayList<Pair> al = new ArrayList<>();
		for(int i=0;i<n;i++) {
			al.add(new Pair(i,x[i], y[i]));
		}
		al.sort((p1,p2)->{
			if (p1.x != p2.x) return Long.compare(p1.x, p2.x);
			return Long.compare(p1.y, p2.y);
		});
		int nx = 0;
		for(int i=0;i<n;i++) {
			Pair p = al.get(i);
			if (nx == 0 || unzipx[nx-1] != p.x) {
				unzipx[nx] = p.x;
				zipx.put(p.x, nx);
				nx++;
			}
		}
		
		ArrayList<Pair>[] points = new ArrayList[nx];
		for(int i=0;i<nx;i++) {
			points[i] = new ArrayList<>();
		}
		for(int i=0;i<n;i++) {
			Pair p = al.get(i);
			points[zipx.get(p.x)].add(p);
		}
		
		int[][] imos = new int[nx][];
		for(int i=0;i<nx;i++) {
			imos[i] = new int[points[i].size()+1];
		}
		
		for(Pair p: al) {
			union(dy, uf, points, imos, p, degree, zipx.get(p.x - dx));
			union(dy, uf, points, imos, p, degree, zipx.get(p.x + dx));
		}
		
		for(int i=0;i<nx;i++) {
			int m = points[i].size();
			for(int j=0;j<m;j++) {
				if (j > 0) {
					imos[i][j] += imos[i][j-1];
				}
				if (imos[i][j] > 0) {
					uf.union(points[i].get(j).i, points[i].get(j+1).i);
				}
			}
		}
	}


	private static void union(long dy, UnionFind uf, ArrayList<Pair>[] points, int[][] imos, Pair p, int[] degree, Integer x2) {
		if (x2 == null) {
			return;
		}
		int i1 = Bound.lowerBound(points[x2], p.y - dy);
		int i2 = Bound.upperBound(points[x2], p.y + dy);
		if (i1 > i2) {
			return;
		}
		degree[p.i] += i2 - i1 + 1;
		imos[x2][i1]++;
		imos[x2][i2]--;
		uf.union(p.i, points[x2].get(i1).i);
	}
}
class Bound {
	static int lowerBound(ArrayList<Pair> a,int fromIndex,int toIndex,long key) {
		if (a.get(toIndex-1).y < key) {
			return toIndex;
		}else if(a.get(fromIndex).y >= key) {
			return fromIndex;
		}
		int l = fromIndex;
		int r = toIndex - 1;
		while(l + 1 < r) {
			int c = (l + r) >>> 1;
			if (a.get(c).y >= key) {
				r = c;
			}else{
				l = c;
			}
		}
		return r;
	}
	static int lowerBound(ArrayList<Pair> a,long key) {
		return lowerBound(a, 0, a.size(), key);
	}
	static int upperBound(ArrayList<Pair> a,int fromIndex,int toIndex,long key) {
		if (a.get(toIndex-1).y <= key) {
			return toIndex - 1;
		}else if(a.get(fromIndex).y > key) {
			return fromIndex - 1;
		}
		int l = fromIndex;
		int r = toIndex - 1;
		while(l + 1 < r) {
			int c = (l + r) >>> 1;
			if (a.get(c).y > key) {
				r = c;
			}else{
				l = c;
			}
		}
		return l;
	}
	static int upperBound(ArrayList<Pair> a,long key) {
		return upperBound(a, 0, a.size(), key);
	}
}

class Pair {
	int i;
	long x,y;
	public Pair(int i, long x, long y) {
		super();
		this.i = i;
		this.x = x;
		this.y = y;
	}
}
class UnionFind {
	private int[] parent;
	public UnionFind(int size) {
		parent = new int[size];
		Arrays.fill(parent, -1);
	}
	public boolean union(int x,int y) {
		x = root(x);
		y = root(y);
		if (x!=y) {
			if (parent[y] < parent[x]) {
				int tmp = y;
				y = x;
				x = tmp;
			}
			parent[x] += parent[y];
			parent[y] = x;
			return true;
		}
		return false;
	}
	public boolean isConnected(int x,int y) {
		return root(x)==root(y);
	}
	private int root(int x) {
		return parent[x] < 0 ? x : (parent[x] = root(parent[x]));
	}
	public int size(int x) {
		return -parent[root(x)];
	}
	public ArrayList<ArrayList<Integer>> groups() {
		int n = parent.length;
		ArrayList<ArrayList<Integer>> g = new ArrayList<ArrayList<Integer>>();
		HashMap<Integer,Integer> hm = new HashMap<Integer, Integer>();
		for(int i=0;i<n;i++) {
			int r = root(i);
			if (!hm.containsKey(r)) {
				hm.put(r, g.size());
				g.add(new ArrayList<Integer>());
			}
			g.get(hm.get(r)).add(i);
		}
		return g;
	}
	public String toString() {
		return Arrays.toString(parent);
	}
}

class IO extends PrintWriter {
	private final InputStream in;
	private final byte[] buffer = new byte[1024];
	private int ptr = 0;
	private int buflen = 0;
	
	public IO() { this(System.in);}
	public IO(InputStream source) { super(System.out); this.in = source;}
	private boolean hasNextByte() {
		if (ptr < buflen) {
			return true;
		}else{
			ptr = 0;
			try {
				buflen = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			if (buflen <= 0) {
				return false;
			}
		}
		return true;
	}
	private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
	private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
	private static boolean isNewLine(int c) { return c == '\n' || c == '\r';}
	public boolean hasNext() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++; return hasNextByte();}
	public boolean hasNextLine() { while(hasNextByte() && isNewLine(buffer[ptr])) ptr++; return hasNextByte();}
	public String next() {
		if (!hasNext()) {
			throw new NoSuchElementException();
		}
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while(isPrintableChar(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	public char[] nextCharArray(int len) {
		if (!hasNext()) {
			throw new NoSuchElementException();
		}
		char[] s = new char[len];
		int i = 0;
		int b = readByte();
		while(isPrintableChar(b)) {
			if (i == len) {
				throw new InputMismatchException();
			}
			s[i++] = (char) b;
			b = readByte();
		}
		return s;
	}
	public String nextLine() {
		if (!hasNextLine()) {
			throw new NoSuchElementException();
		}
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while(!isNewLine(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	public long nextLong() {
		if (!hasNext()) {
			throw new NoSuchElementException();
		}
		long n = 0;
		boolean minus = false;
		int b = readByte();
		if (b == '-') {
			minus = true;
			b = readByte();
		}
		if (b < '0' || '9' < b) {
			throw new NumberFormatException();
		}
		while(true){
			if ('0' <= b && b <= '9') {
				n *= 10;
				n += b - '0';
			}else if(b == -1 || !isPrintableChar(b)){
				return minus ? -n : n;
			}else{
				throw new NumberFormatException();
			}
			b = readByte();
		}
	}
	public int nextInt() {
		long nl = nextLong();
		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) {
			throw new NumberFormatException();
		}
		return (int) nl;
	}
	public char nextChar() {
		if (!hasNext()) {
			throw new NoSuchElementException();
		}
		return (char) readByte();
	}
	public double nextDouble() { return Double.parseDouble(next());}
	public int[] nextIntArray(int n) { int[] a = new int[n]; for(int i=0;i<n;i++) a[i] = nextInt(); return a;}
	public long[] nextLongArray(int n) { long[] a = new long[n]; for(int i=0;i<n;i++) a[i] = nextLong(); return a;}
	public double[] nextDoubleArray(int n) { double[] a = new double[n]; for(int i=0;i<n;i++) a[i] = nextDouble(); return a;}
	public void nextIntArrays(int[]... a) { for(int i=0;i<a[0].length;i++) for(int j=0;j<a.length;j++) a[j][i] = nextInt();}
	public int[][] nextIntMatrix(int n,int m) { int[][] a = new int[n][]; for(int i=0;i<n;i++) a[i] = nextIntArray(m); return a;}
	public char[][] nextCharMap(int n,int m) { char[][] a = new char[n][]; for(int i=0;i<n;i++) a[i] = nextCharArray(m); return a;}
	public void close() { super.close(); try {in.close();} catch (IOException e) {}}
}