# Yukicoder No.325 マンハッタン距離2

,

http://yukicoder.me/problems/no/325

## solution

やる。包除原理あたりを使って領域ごとに足し引きをする。原点や軸上に注意。

## implementation

• 原点
• 軸上
• それ以外

を求める関数をそれぞれ作って足し合わせた。

#!/usr/bin/env python3

# functions
def f(x, y, d): # (0, x) * (0, y)
assert 0 <= x
assert 0 <= y
if x == 0 or y == 0:
return 0
x -= 1
y -= 1
d = min(d, x + y)
ans = (d+1)*(d+2)//2
if x <= d:
dx = d-x
ans -= dx*(dx+1)//2
if y <= d:
dy = d-y
ans -= dy*(dy+1)//2
ans -= min(x,d)+1
ans -= min(y,d)+1
ans += 1
return ans
def g(x1, x2, y1, y2, d): # on the axes
if x1 <= 0 <= x2:
if 0 <= y1 <= y2:
if y1 <= d:
ans = min(d, y2) - min(d, y1) + 1
if y1 == 0:
ans -= 1
return ans
else:
return 0
elif y1 <= 0 <= y2:
return min(d, abs(y1)) + min(d, y2)
assert False
else:
return 0
def h(x1, x2, y1, y2, d): # on the origin
if x1 <= 0 <= x2 and y1 <= 0 <= y2:
return 1
else:
return 0

# input
x1, y1, x2, y2, d = map(int,input().split())
assert x1 < x2
assert y1 < y2
if x2 <= 0:
x1, x2 = - x2, - x1
if y2 <= 0:
y1, y2 = - y2, - y1
if x1 <= 0:
x1, y1 = y1, x1
x2, y2 = y2, x2

# calc
ans = 0
if x1 <= 0:
assert x1 <= 0 <= x2
assert y1 <= 0 <= y2
ans += f(abs(x1)+1, abs(y1)+1, d)
ans += f(abs(x1)+1, abs(y2)+1, d)
ans += f(abs(x2)+1, abs(y1)+1, d)
ans += f(abs(x2)+1, abs(y2)+1, d)
elif y1 <= 0:
assert 0 <= x1 < x2
assert y1 <= 0 <= y2
ans -= f(x1,   abs(y1)+1, d)
ans += f(x2+1, abs(y1)+1, d)
ans -= f(x1,   abs(y2)+1, d)
ans += f(x2+1, abs(y2)+1, d)
else:
assert 0 <= x1 < x2
assert 0 <= y1 < y2
ans += f(x1,   y1,   d)
ans -= f(x1,   y2+1, d)
ans -= f(x2+1, y1,   d)
ans += f(x2+1, y2+1, d)
ans += g(x1, x2, y1, y2, d)
ans += g(y1, y2, x1, x2, d)
ans += h(x1, x2, y1, y2, d)

# output
print(ans)