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



Sample input:
5 1 2
1 1
4 3
6 1
5 5
4 8


Sample output:
4

Submission status: Accepted

Solution code:
import bisect
import sys
from collections import defaultdict


class RandomizedBinarySearchTree:
    def __init__(self):
        self.children = [[-1], [-1]]
        self.values = [0]
        self.counts = [0]
        self.root = 0

    def merge(self, left_root, right_root):
        children = self.children
        counts = self.counts
        li = left_root
        ri = right_root
        stack = []
        switch = 0
        while li != 0 and ri != 0:
            if switch:
                stack.append((li, 1))
                li = children[1][li]
            else:
                stack.append((ri, 0))
                ri = children[0][ri]
            switch ^= 1

        i = li if li != 0 else ri
        while stack:
            pi, is_right = stack.pop()
            children[is_right][pi] = i
            counts[pi] = counts[children[0][pi]] + counts[children[1][pi]] + 1
            i = pi

        return i

    def split(self, root, x):
        i = root
        lefts, rights = self.children
        values = self.values
        counts = self.counts
        l_stack = []
        r_stack = []
        while i != 0:
            if x < values[i]:
                r_stack.append(i)
                i = lefts[i]
            else:
                l_stack.append(i)
                i = rights[i]

        li, ri = 0, 0
        while l_stack:
            pi = l_stack.pop()
            rights[pi] = li
            counts[pi] = counts[lefts[pi]] + counts[li] + 1
            li = pi
        while r_stack:
            pi = r_stack.pop()
            lefts[pi] = ri
            counts[pi] = counts[ri] + counts[rights[pi]] + 1
            ri = pi

        return li, ri

    def insert(self, x):
        ni = len(self.values)
        self.children[0].append(0)
        self.children[1].append(0)
        self.values.append(x)
        self.counts.append(1)

        li, ri = self.split(self.root, x)
        self.root = self.merge(self.merge(li, ni), ri)

    def delete(self, x):
        li, mri = self.split(self.root, x - 1)
        mi, ri = self.split(mri, x)

        if mi == 0:
            self.root = self.merge(li, ri)
            return

        self.root = self.merge(li, ri)
        return

    def upper_bound(self, x, default=-1):
        i = self.root
        lefts, rights = self.children
        values = self.values
        counts = self.counts
        y = default
        c = counts[i]
        j = 0
        while i != 0:
            if x < values[i]:
                y = values[i]
                c = j + counts[lefts[i]]
                i = lefts[i]
            else:
                j += counts[lefts[i]] + 1
                i = rights[i]
        return y, c

    def lower_bound(self, x, default=-1):
        i = self.root
        lefts, rights = self.children
        values = self.values
        counts = self.counts
        y = default
        c = counts[i]
        j = 0
        while i != 0:
            if x <= values[i]:
                y = values[i]
                c = j + counts[lefts[i]]
                i = lefts[i]
            else:
                j += counts[lefts[i]] + 1
                i = rights[i]
        return y, c

    def get_k_th(self, k, default=-1):
        i = self.root
        children = self.children
        values = self.values
        counts = self.counts
        if counts[i] <= k:
            return default
        j = k
        while i != 0:
            left_count = counts[children[0][i]]
            if left_count == j:
                return values[i]
            elif left_count > j:
                i = children[0][i]
            else:
                j -= left_count + 1
                i = children[1][i]
        return default

    def debug_print(self):
        print('Lefts ', self.children[0])
        print('Rights', self.children[1])
        print('Values', self.values)
        print('Counts', self.counts)
        self._debug_print(self.root, 0)

    def _debug_print(self, i, depth):
        if i != -1:
            self._debug_print(self.children[0][i], depth + 1)
            print('      ' * depth, self.values[i], self.counts[i])
            self._debug_print(self.children[1][i], depth + 1)


def x_check(x_dict, nx, y, d, stack, stacked):
    counter = 0
    if nx in x_dict:
        rbst, lst = x_dict[nx]
        ny, _ = rbst.lower_bound(y - d, y + d + 1)
        while ny <= y + d:
            z = (nx << 32) | ny
            if z not in stacked:
                stack.append(z)
                stacked.add(z)
            rbst.delete(ny)
            ny, _ = rbst.lower_bound(y - d, y + d + 1)
        i = bisect.bisect_left(lst, y - d)
        j = bisect.bisect(lst, y + d)
        counter = j - i
    return counter


def y_check(y_dict, ny, x, d, stack, stacked):
    counter = 0
    if ny in y_dict:
        rbst, lst = y_dict[ny]
        nx, _ = rbst.lower_bound(x - d, x + d + 1)
        while nx <= x + d:
            z = (nx << 32) | ny
            if z not in stacked:
                stack.append(z)
                stacked.add(z)
            rbst.delete(nx)
            nx, _ = rbst.lower_bound(x - d, x + d + 1)
        i = bisect.bisect(lst, x - d)
        j = bisect.bisect_left(lst, x + d)
        counter = j - i
    return counter


n, a, b, *xy = map(int, sys.stdin.buffer.read().split())
xxx = xy[0::2]
yyy = xy[1::2]

x_dict = defaultdict(lambda: [RandomizedBinarySearchTree(), []])
y_dict = defaultdict(lambda: [RandomizedBinarySearchTree(), []])
OFFSET = 10 ** 9
MASK = (1 << 32) - 1
for i in range(n):
    x = xxx[i] - yyy[i] + OFFSET
    y = xxx[i] + yyy[i]
    x_dict[x][0].insert(y)
    y_dict[y][0].insert(x)
    x_dict[x][1].append(y)
    y_dict[y][1].append(x)

for _, lst in x_dict.values():
    lst.sort()
for _, lst in y_dict.values():
    lst.sort()

a -= 1
b -= 1
ax_, ay_ = xxx[a], yyy[a]
bx_, by_ = xxx[b], yyy[b]
ax, ay = ax_ - ay_ + OFFSET, ax_ + ay_
bx, by = bx_ - by_ + OFFSET, bx_ + by_
az = (ax << 32) | ay
bz = (bx << 32) | by
d = max(abs(ax - bx), abs(ay - by))
x_dict[ax][0].delete(ay)
x_dict[bx][0].delete(by)
y_dict[ay][0].delete(ax)
y_dict[by][0].delete(bx)

stack = [az, bz]
stacked = {az, bz}
ans = 0
while stack:
    z = stack.pop()
    y = z & MASK
    x = z >> 32
    ans += x_check(x_dict, x - d, y, d, stack, stacked)
    ans += x_check(x_dict, x + d, y, d, stack, stacked)
    ans += y_check(y_dict, y - d, x, d, stack, stacked)
    ans += y_check(y_dict, y + d, x, d, stack, stacked)

print(ans // 2)

Submission status: Accepted

Solution code:
import sys
input = sys.stdin.readline
from collections import defaultdict
import bisect

INF = 10 ** 12

N,a,b = map(int,input().split())
X_to_Y = defaultdict(lambda: [-INF,INF])
Y_to_X = defaultdict(lambda: [-INF,INF])
for i in range(1,N+1):
    x,y = map(int,input().split())
    x,y = x+y,x-y
    X_to_Y[x].append(y)
    Y_to_X[y].append(x)
    if i == a:
        sx,sy = x,y
    elif i == b:
        tx,ty = x,y

R = max(abs(sx-tx), abs(sy-ty))

for key, arr in X_to_Y.items():
    arr.sort()
    L = len(arr)
    move_right = list(range(1,L)) + [None]
    X_to_Y[key] = [arr,move_right]
    
for key, arr in Y_to_X.items():
    arr.sort()
    L = len(arr)
    move_right = list(range(1,L)) + [None]
    Y_to_X[key] = [arr,move_right]

equiv_class = set([(sx,sy), (tx,ty)])

answer = 0
task = [(sx,sy), (tx,ty)]

while task:
    x,y = task.pop()
    # 既出の元も含めて辺の個数を数える
    # 未出の元を同値に追加
    # x,y両方満たすものは、x側にだけ入れる
    for x1 in [x-R,x+R]:
        if not (x1 in X_to_Y):
            continue
        arr, move_right = X_to_Y[x1]
        left = bisect.bisect_left(arr, y-R)
        right = bisect.bisect_right(arr, y+R)
        # [left, right) が辺で結べるターゲット
        answer += right - left # 辺の個数
        i = left
        while i < right:
            y1 = arr[i]
            if (x1,y1) not in equiv_class:
                equiv_class.add((x1,y1))
                task.append((x1,y1))
            # なるべく、既に居ない元は見ないで済ませるように
            next_i = move_right[i]
            if next_i >= right:
                break
            move_right[i] = right
            i = next_i
    for y1 in [y-R,y+R]:
        if not y1 in Y_to_X:
            continue
        arr, move_right = Y_to_X[y1]
        left = bisect.bisect_left(arr, x-R+1)
        right = bisect.bisect_right(arr, x+R-1)
        # [left, right) が辺で結べるターゲット
        answer += right - left # 辺の個数
        i = left
        while i < right:
            x1 = arr[i]
            if (x1,y1) not in equiv_class:
                equiv_class.add((x1,y1))
                task.append((x1,y1))
            # なるべく、既に居ない元は見ないで済ませるように
            next_i = move_right[i]
            if next_i >= right:
                break
            move_right[i] = right
            i = next_i

answer //= 2 # 両方向から辺を見た
print(answer)

Submission status: Accepted

Solution code:
N, a, b = map(int, input().split()); a -= 1; b -= 1
P = []
Q = []
for i in range(N):
    x, y = map(int, input().split())
    P.append((x-y, x+y, i))
    Q.append((x+y, x-y, i))

d = max(abs(P[a][0] - P[b][0]), abs(P[a][1] - P[b][1]))

*parent, = range(N)
def root(x):
    if x == parent[x]:
        return x
    y = parent[x] = root(parent[x])
    return y
def unite(x, y):
    px = root(x); py = root(y)
    if px < py:
        parent[py] = px
    else:
        parent[px] = py
C = [0]*N
D = [0]*N

def check(P0, i0, j0):
    return abs(P0[i0][0] - P0[j0][0]) == abs(P0[i0][1] - P0[j0][1])

def solve(P0):
    P = P0[:]
    P.sort()

    s = t = 0; prev = -1
    for i in range(N):
        x, y, i0 = P[i]
        while t < N and P[t][0] < x-d or (P[t][0] == x-d and P[t][1] <= y+d): t += 1
        while s < N and (P[s][0] < x-d or (P[s][0] == x-d and P[s][1] < y-d)): s += 1
        if s < t:
            j0 = P[s][2]
            unite(i0, j0)
            if check(P0, i0, j0):
                D[i0] += 1
            else:
                C[i0] += 1
            if s < t-1:
                j0 = P[t-1][2]
                if check(P0, i0, j0):
                    D[i0] += 1
                    C[i0] += t-s-2
                else:
                    C[i0] += t-s-1
            for j in range(max(prev, s), t-1):
                unite(P[j][2], P[j+1][2])
            prev = t-1
solve(P)
solve(Q)

S = T = 0
r = root(a)
for i in range(N):
    if root(i) == r:
        S += C[i]; T += D[i]
print(S + T//2)

Submission status: Runtime Error

Solution code:
import numpy as np

n, a, b = map(int, input().split())

holes = np.fromiter((complex(*map(int, input().split())) for _ in range(n)), dtype=complex)
ab = holes[a - 1] - holes[b - 1]
d = abs(ab.real) + abs(ab.imag)
links = [set() for _ in range(n)]

for i in range(0, n, 10000):
    xy1, xy2 = np.meshgrid(holes, holes[i:i + 10000])
    diff = xy1 - xy2
    dists = abs(diff.real) + abs(diff.imag)

    for x, y in zip(*np.where(dists == d)):
        links[x].add(y)

visited = set()
queue = [a - 1]
while queue:
    p = queue.pop()
    if p in visited:
        continue
    visited.add(p)
    queue.extend(q for q in links[p] if q not in visited)

print(sum(p < q for p in visited for q in links[p]))

Submission status: Runtime Error

Solution code:
import numpy as np

n, a, b = map(int, input().split())

holes = np.fromiter((complex(*map(int, input().split())) for _ in range(n)), dtype=complex)
xy1, xy2 = np.meshgrid(holes, holes)
diff = xy1 - xy2
dists = abs(diff.real) + abs(diff.imag)

d = dists[a - 1][b - 1]
w = np.where(dists == d)

links = [set() for _ in range(n)]
for x, y in zip(*w):
    links[x].add(y)

visited = set()
queue = [a - 1]
while queue:
    p = queue.pop()
    if p in visited:
        continue
    visited.add(p)
    queue.extend(q for q in links[p] if q not in visited)

print(sum(p < q for p in visited for q in links[p]))