A game field is a strip of 1 × n square cells. In some cells there are Packmen, in some cells — asterisks, other cells are empty.

Packman can move to neighboring cell in 1 time unit. If there is an asterisk in the target cell then Packman eats it. Packman doesn't spend any time to eat an asterisk.

In the initial moment of time all Packmen begin to move. Each Packman can change direction of its move unlimited number of times, but it is not allowed to go beyond the boundaries of the game field. Packmen do not interfere with the movement of other packmen; in one cell there can be any number of packmen moving in any directions.

Your task is to determine minimum possible time after which Packmen can eat all the asterisks.


-----Input-----

The first line contains a single integer n (2 ≤ n ≤ 10^5) — the length of the game field.

The second line contains the description of the game field consisting of n symbols. If there is symbol '.' in position i — the cell i is empty. If there is symbol '*' in position i — in the cell i contains an asterisk. If there is symbol 'P' in position i — Packman is in the cell i.

It is guaranteed that on the game field there is at least one Packman and at least one asterisk.


-----Output-----

Print minimum possible time after which Packmen can eat all asterisks.


-----Examples-----
Input
7
*..P*P*

Output
3

Input
10
.**PP.*P.*

Output
2



-----Note-----

In the first example Packman in position 4 will move to the left and will eat asterisk in position 1. He will spend 3 time units on it. During the same 3 time units Packman in position 6 will eat both of neighboring with it asterisks. For example, it can move to the left and eat asterisk in position 5 (in 1 time unit) and then move from the position 5 to the right and eat asterisk in the position 7 (in 2 time units). So in 3 time units Packmen will eat all asterisks on the game field.

In the second example Packman in the position 4 will move to the left and after 2 time units will eat asterisks in positions 3 and 2. Packmen in positions 5 and 8 will move to the right and in 2 time units will eat asterisks in positions 7 and 10, respectively. So 2 time units is enough for Packmen to eat all asterisks on the game field.

Difficulty: MEDIUM_HARD
Tags: Dynamic programming

Sorting
Skill types: Dynamic programming

Sorting

Starter code:

Solutions:
from bisect import bisect_right, bisect_left

def check(t):
	pt = 0
	contain_t = 0
	(pt1, pt2) = (0, 0)
	(contain_t1, contain_t2) = (0, 0)
	for i in range(l_p):
		(contain_t1, contain_t2) = (t, t)
		if pt >= l_s:
			return 1
		if ash[pt] < pokemon[i] and t < pokemon[i] - ash[pt]:
			return 0
		if ash[pt] < pokemon[i]:
			contain_t1 = (contain_t1 - (pokemon[i] - ash[pt])) // 2
			contain_t2 = max(0, contain_t2 - (pokemon[i] - ash[pt]) * 2)
		pt = bisect_right(ash, pokemon[i] + max(contain_t1, contain_t2))
	if pt >= l_s:
		return 1
	return 0
n = int(input())
st = input()
a = []
for i in range(n):
	a.append(st[i])
(ans1, ans2) = (0, 0)
ash = []
pokemon = []
for i in range(n):
	if a[i] == '*':
		ash.append(i)
	elif a[i] == 'P':
		pokemon.append(i)
l_p = len(pokemon)
l_s = len(ash)
i = 0
j = 10 ** 10
while i < j:
	m = (i + j) // 2
	if check(m):
		j = m - 1
	else:
		i = m + 1
if check(i):
	print(i)
else:
	print(i + 1)


import bisect
n = int(input())
s = input()
packmans = []
stars = []
for i in range(n):
	if s[i] == '*':
		stars.append(i)
	elif s[i] == 'P':
		packmans.append(i)
if len(stars) == 0:
	print(0)
	return

def check(t):
	first_to_eat = 0
	for i in range(len(packmans)):
		x = stars[first_to_eat]
		if packmans[i] > x:
			if packmans[i] - x > t:
				return False
			d1 = t - 2 * (packmans[i] - x)
			d2 = (t - (packmans[i] - x)) // 2
			first_to_eat = bisect.bisect_right(stars, packmans[i] + max(d1, d2))
			if first_to_eat < len(stars) and stars[first_to_eat] == packmans[i] + max(d1, d2):
				first_to_eat += 1
		else:
			j = bisect.bisect_right(stars, packmans[i] + t)
			if first_to_eat < len(stars) and stars[first_to_eat] == packmans[i] + t:
				first_to_eat += 1
			first_to_eat = max(j, first_to_eat)
		if first_to_eat >= len(stars):
			return True
	return first_to_eat >= len(stars)
l = 0
r = 2 * n + 1
while r - l > 1:
	m = (l + r) // 2
	if check(m):
		r = m
	else:
		l = m
print(r)


from sys import stdin, stdout
n = int(stdin.readline())
s = stdin.readline().strip()
mins = []
packs = []
for i in range(len(s)):
	if s[i] == '*':
		mins.append(i)
	elif s[i] == 'P':
		packs.append(i)
(l, r) = (-1, 2 * len(s) + 1)
while r - l > 1:
	m = l + r >> 1
	test1 = mins[:]
	test2 = packs[:]
	while test2 and test1:
		cnt = m
		pos = test2.pop()
		if pos > test1[-1]:
			while test1 and abs(pos - test1[-1]) <= cnt:
				cnt -= abs(pos - test1[-1])
				pos = test1[-1]
				test1.pop()
		else:
			(cntl, cntr) = (0, 0)
			if abs(test1[-1] - pos) > m:
				break
			lpos = (m + pos - test1[-1]) // 2
			rpos = m - 2 * abs(test1[-1] - pos)
			(lb, rb) = (-1, len(test1))
			while rb - lb > 1:
				mb = lb + rb >> 1
				if pos - test1[mb] <= lpos:
					rb = mb
				else:
					lb = mb
			cntl = len(test1) - rb
			(lb, rb) = (-1, len(test1))
			while rb - lb > 1:
				mb = lb + rb >> 1
				if pos - test1[mb] <= rpos:
					rb = mb
				else:
					lb = mb
			cntr = len(test1) - rb
			cnt = max(cntl, cntr)
			while test1 and cnt:
				test1.pop()
				cnt -= 1
	if not test1:
		r = m
	else:
		l = m
stdout.write(str(r))


def check(mid, p, f):
	u = 0
	for i in range(len(p)):
		left = p[i] - f[u]
		left = max(0, left)
		if mid < left:
			return 0
		right = max(mid - 2 * left, 0)
		right = max(right, (mid - left) // 2)
		while u < len(f) and f[u] <= p[i] + right:
			u += 1
		if u == len(f):
			return 1
	return 0

def f(s, n):
	p = []
	f = []
	for i in range(n):
		if s[i] == '*':
			f.append(i)
		if s[i] == 'P':
			p.append(i)
	lo = 0
	hi = 10 ** 9
	while lo <= hi:
		mid = (lo + hi) // 2
		if check(mid, p, f):
			hi = mid - 1
		else:
			lo = mid + 1
	return lo
n = int(input())
s = list(input().strip())
print(f(s, n))


def check(mid, p, f):
	ptr = 0
	for x in p:
		From = x
		To = x
		while ptr < len(f):
			From = min(From, f[ptr])
			To = max(To, f[ptr])
			need = To - From + min(To - x, x - From)
			if need > mid:
				break
			ptr += 1
		if ptr == len(f):
			return 1
	return 0

def f(s, n):
	p = []
	f = []
	for i in range(n):
		if s[i] == '*':
			f.append(i)
		if s[i] == 'P':
			p.append(i)
	lo = 0
	hi = 10 ** 9
	while lo <= hi:
		mid = (lo + hi) // 2
		if check(mid, p, f):
			hi = mid - 1
		else:
			lo = mid + 1
	return lo
n = int(input())
s = list(input().strip())
print(f(s, n))


import math
import random
import heapq, bisect
import sys
from collections import deque, defaultdict
from fractions import Fraction
import sys
import threading
from collections import defaultdict
threading.stack_size(10 ** 8)
mod = 10 ** 9 + 7
mod1 = 998244353
import os
import sys
from io import BytesIO, IOBase
sys.setrecursionlimit(300000)
BUFSIZE = 8192

class FastIO(IOBase):
	newlines = 0

	def __init__(self, file):
		self._fd = file.fileno()
		self.buffer = BytesIO()
		self.writable = 'x' in file.mode or 'r' not in file.mode
		self.write = self.buffer.write if self.writable else None

	def read(self):
		while True:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			if not b:
				break
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines = 0
		return self.buffer.read()

	def readline(self):
		while self.newlines == 0:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			self.newlines = b.count(b'\n') + (not b)
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines -= 1
		return self.buffer.readline()

	def flush(self):
		if self.writable:
			os.write(self._fd, self.buffer.getvalue())
			(self.buffer.truncate(0), self.buffer.seek(0))

class IOWrapper(IOBase):

	def __init__(self, file):
		self.buffer = FastIO(file)
		self.flush = self.buffer.flush
		self.writable = self.buffer.writable
		self.write = lambda s: self.buffer.write(s.encode('ascii'))
		self.read = lambda : self.buffer.read().decode('ascii')
		self.readline = lambda : self.buffer.readline().decode('ascii')
(sys.stdin, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline().rstrip('\r\n')

class TreeNode:

	def __init__(self, k, v):
		self.key = k
		self.value = v
		self.left = None
		self.right = None
		self.parent = None
		self.height = 1
		self.num_left = 1
		self.num_total = 1

class AvlTree:

	def __init__(self):
		self._tree = None

	def add(self, k, v):
		if not self._tree:
			self._tree = TreeNode(k, v)
			return
		node = self._add(k, v)
		if node:
			self._rebalance(node)

	def _add(self, k, v):
		node = self._tree
		while node:
			if k < node.key:
				if node.left:
					node = node.left
				else:
					node.left = TreeNode(k, v)
					node.left.parent = node
					return node.left
			elif node.key < k:
				if node.right:
					node = node.right
				else:
					node.right = TreeNode(k, v)
					node.right.parent = node
					return node.right
			else:
				node.value = v
				return

	@staticmethod
	def get_height(x):
		return x.height if x else 0

	@staticmethod
	def get_num_total(x):
		return x.num_total if x else 0

	def _rebalance(self, node):
		n = node
		while n:
			lh = self.get_height(n.left)
			rh = self.get_height(n.right)
			n.height = max(lh, rh) + 1
			balance_factor = lh - rh
			n.num_total = 1 + self.get_num_total(n.left) + self.get_num_total(n.right)
			n.num_left = 1 + self.get_num_total(n.left)
			if balance_factor > 1:
				if self.get_height(n.left.left) < self.get_height(n.left.right):
					self._rotate_left(n.left)
				self._rotate_right(n)
			elif balance_factor < -1:
				if self.get_height(n.right.right) < self.get_height(n.right.left):
					self._rotate_right(n.right)
				self._rotate_left(n)
			else:
				n = n.parent

	def _remove_one(self, node):
		replacement = node.left or node.right
		if node.parent:
			if AvlTree._is_left(node):
				node.parent.left = replacement
			else:
				node.parent.right = replacement
			replacement.parent = node.parent
			node.parent = None
		else:
			self._tree = replacement
			replacement.parent = None
		node.left = None
		node.right = None
		node.parent = None
		self._rebalance(replacement)

	def _remove_leaf(self, node):
		if node.parent:
			if AvlTree._is_left(node):
				node.parent.left = None
			else:
				node.parent.right = None
			self._rebalance(node.parent)
		else:
			self._tree = None
		node.parent = None
		node.left = None
		node.right = None

	def remove(self, k):
		node = self._get_node(k)
		if not node:
			return
		if AvlTree._is_leaf(node):
			self._remove_leaf(node)
			return
		if node.left and node.right:
			nxt = AvlTree._get_next(node)
			node.key = nxt.key
			node.value = nxt.value
			if self._is_leaf(nxt):
				self._remove_leaf(nxt)
			else:
				self._remove_one(nxt)
			self._rebalance(node)
		else:
			self._remove_one(node)

	def get(self, k):
		node = self._get_node(k)
		return node.value if node else -1

	def _get_node(self, k):
		if not self._tree:
			return None
		node = self._tree
		while node:
			if k < node.key:
				node = node.left
			elif node.key < k:
				node = node.right
			else:
				return node
		return None

	def get_at(self, pos):
		x = pos + 1
		node = self._tree
		while node:
			if x < node.num_left:
				node = node.left
			elif node.num_left < x:
				x -= node.num_left
				node = node.right
			else:
				return (node.key, node.value)
		raise IndexError('Out of ranges')

	@staticmethod
	def _is_left(node):
		return node.parent.left and node.parent.left == node

	@staticmethod
	def _is_leaf(node):
		return node.left is None and node.right is None

	def _rotate_right(self, node):
		if not node.parent:
			self._tree = node.left
			node.left.parent = None
		elif AvlTree._is_left(node):
			node.parent.left = node.left
			node.left.parent = node.parent
		else:
			node.parent.right = node.left
			node.left.parent = node.parent
		bk = node.left.right
		node.left.right = node
		node.parent = node.left
		node.left = bk
		if bk:
			bk.parent = node
		node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1
		node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)
		node.num_left = 1 + self.get_num_total(node.left)

	def _rotate_left(self, node):
		if not node.parent:
			self._tree = node.right
			node.right.parent = None
		elif AvlTree._is_left(node):
			node.parent.left = node.right
			node.right.parent = node.parent
		else:
			node.parent.right = node.right
			node.right.parent = node.parent
		bk = node.right.left
		node.right.left = node
		node.parent = node.right
		node.right = bk
		if bk:
			bk.parent = node
		node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1
		node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)
		node.num_left = 1 + self.get_num_total(node.left)

	@staticmethod
	def _get_next(node):
		if not node.right:
			return node.parent
		n = node.right
		while n.left:
			n = n.left
		return n

class SegmentTree1:

	def __init__(self, data, default=2 ** 51, func=lambda a, b: a & b):
		self._default = default
		self._func = func
		self._len = len(data)
		self._size = _size = 1 << (self._len - 1).bit_length()
		self.data = [default] * (2 * _size)
		self.data[_size:_size + self._len] = data
		for i in reversed(range(_size)):
			self.data[i] = func(self.data[i + i], self.data[i + i + 1])

	def __delitem__(self, idx):
		self[idx] = self._default

	def __getitem__(self, idx):
		return self.data[idx + self._size]

	def __setitem__(self, idx, value):
		idx += self._size
		self.data[idx] = value
		idx >>= 1
		while idx:
			self.data[idx] = self._func(self.data[2 * idx], self.data[2 * idx + 1])
			idx >>= 1

	def __len__(self):
		return self._len

	def query(self, start, stop):
		if start == stop:
			return self.__getitem__(start)
		stop += 1
		start += self._size
		stop += self._size
		res = self._default
		while start < stop:
			if start & 1:
				res = self._func(res, self.data[start])
				start += 1
			if stop & 1:
				stop -= 1
				res = self._func(res, self.data[stop])
			start >>= 1
			stop >>= 1
		return res

	def __repr__(self):
		return 'SegmentTree({0})'.format(self.data)

class SegmentTree:

	def __init__(self, data, default=0, func=lambda a, b: max(a, b)):
		self._default = default
		self._func = func
		self._len = len(data)
		self._size = _size = 1 << (self._len - 1).bit_length()
		self.data = [default] * (2 * _size)
		self.data[_size:_size + self._len] = data
		for i in reversed(range(_size)):
			self.data[i] = func(self.data[i + i], self.data[i + i + 1])

	def __delitem__(self, idx):
		self[idx] = self._default

	def __getitem__(self, idx):
		return self.data[idx + self._size]

	def __setitem__(self, idx, value):
		idx += self._size
		self.data[idx] = value
		idx >>= 1
		while idx:
			self.data[idx] = self._func(self.data[2 * idx], self.data[2 * idx + 1])
			idx >>= 1

	def __len__(self):
		return self._len

	def query(self, start, stop):
		if start == stop:
			return self.__getitem__(start)
		stop += 1
		start += self._size
		stop += self._size
		res = self._default
		while start < stop:
			if start & 1:
				res = self._func(res, self.data[start])
				start += 1
			if stop & 1:
				stop -= 1
				res = self._func(res, self.data[stop])
			start >>= 1
			stop >>= 1
		return res

	def __repr__(self):
		return 'SegmentTree({0})'.format(self.data)

class Factorial:

	def __init__(self, MOD):
		self.MOD = MOD
		self.factorials = [1, 1]
		self.invModulos = [0, 1]
		self.invFactorial_ = [1, 1]

	def calc(self, n):
		if n <= -1:
			print('Invalid argument to calculate n!')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		if n < len(self.factorials):
			return self.factorials[n]
		nextArr = [0] * (n + 1 - len(self.factorials))
		initialI = len(self.factorials)
		prev = self.factorials[-1]
		m = self.MOD
		for i in range(initialI, n + 1):
			prev = nextArr[i - initialI] = prev * i % m
		self.factorials += nextArr
		return self.factorials[n]

	def inv(self, n):
		if n <= -1:
			print('Invalid argument to calculate n^(-1)')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		p = self.MOD
		pi = n % p
		if pi < len(self.invModulos):
			return self.invModulos[pi]
		nextArr = [0] * (n + 1 - len(self.invModulos))
		initialI = len(self.invModulos)
		for i in range(initialI, min(p, n + 1)):
			next = -self.invModulos[p % i] * (p // i) % p
			self.invModulos.append(next)
		return self.invModulos[pi]

	def invFactorial(self, n):
		if n <= -1:
			print('Invalid argument to calculate (n^(-1))!')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		if n < len(self.invFactorial_):
			return self.invFactorial_[n]
		self.inv(n)
		nextArr = [0] * (n + 1 - len(self.invFactorial_))
		initialI = len(self.invFactorial_)
		prev = self.invFactorial_[-1]
		p = self.MOD
		for i in range(initialI, n + 1):
			prev = nextArr[i - initialI] = prev * self.invModulos[i % p] % p
		self.invFactorial_ += nextArr
		return self.invFactorial_[n]

class Combination:

	def __init__(self, MOD):
		self.MOD = MOD
		self.factorial = Factorial(MOD)

	def ncr(self, n, k):
		if k < 0 or n < k:
			return 0
		k = min(k, n - k)
		f = self.factorial
		return f.calc(n) * f.invFactorial(max(n - k, k)) * f.invFactorial(min(k, n - k)) % self.MOD

def powm(a, n, m):
	if a == 1 or n == 0:
		return 1
	if n % 2 == 0:
		s = powm(a, n // 2, m)
		return s * s % m
	else:
		return a * powm(a, n - 1, m) % m

def sort_list(list1, list2):
	zipped_pairs = zip(list2, list1)
	z = [x for (_, x) in sorted(zipped_pairs)]
	return z

def product(l):
	por = 1
	for i in range(len(l)):
		por *= l[i]
	return por

def binarySearchCount(arr, n, key):
	left = 0
	right = n - 1
	count = 0
	while left <= right:
		mid = int((right + left) / 2)
		if arr[mid] < key:
			count = mid + 1
			left = mid + 1
		else:
			right = mid - 1
	return count

def countdig(n):
	c = 0
	while n > 0:
		n //= 10
		c += 1
	return c

def binary(x, length):
	y = bin(x)[2:]
	return y if len(y) >= length else '0' * (length - len(y)) + y

def countGreater(arr, n, k):
	l = 0
	r = n - 1
	leftGreater = n
	while l <= r:
		m = int(l + (r - l) / 2)
		if arr[m] >= k:
			leftGreater = m
			r = m - 1
		else:
			l = m + 1
	return n - leftGreater
n = int(input())
s = list(input())
aster = []
p = []
for i in range(n):
	if s[i] == '*':
		aster.append(i)
	elif s[i] == 'P':
		p.append(i)

def find(x):
	t = 0
	for i in range(len(p)):
		time = 0
		if aster[t] < p[i]:
			time += abs(p[i] - aster[t])
		if time > x:
			return False
		else:
			while aster[t] < p[i]:
				t += 1
				if t == len(aster):
					break
			if t == len(aster):
				break
			cur = time
			t -= 1
			time = 0
			while time * 2 + cur <= x or cur * 2 + time <= x:
				t += 1
				if t == len(aster):
					break
				time = aster[t] - p[i]
		if t == len(aster):
			break
	if t != len(aster):
		return False
	return True
st = 1
end = 2 * n
ans = 2 * n
while st <= end:
	mid = (st + end) // 2
	if find(mid) == True:
		ans = mid
		end = mid - 1
	else:
		st = mid + 1
print(ans)


from typing import List

def eatable_in_time(food: List[int], packmen: List[int], time: int) -> bool:
	food_pointer = 0
	food_amt = len(food)
	for p in packmen:
		have_to_eat = []
		while food_pointer < food_amt and food[food_pointer] < p:
			have_to_eat.append(food[food_pointer])
			food_pointer += 1
		if not have_to_eat:
			while food_pointer < food_amt and food[food_pointer] - p <= time:
				food_pointer += 1
		else:
			if p - have_to_eat[0] > time:
				return False
			left_time = p - have_to_eat[0]
			right_free_time = time - 2 * left_time
			left_first = -1
			while food_pointer + left_first + 1 < food_amt and food[food_pointer + left_first + 1] - p <= right_free_time:
				left_first += 1
			right_free_time = time - left_time
			right_first = -1
			while food_pointer + right_first + 1 < food_amt and 2 * (food[food_pointer + right_first + 1] - p) <= right_free_time:
				right_first += 1
			food_pointer += max(left_first, right_first) + 1
		if food_pointer >= food_amt:
			return True
	return food_pointer >= food_amt
input()
field = input().upper()
all_food = []
all_packmen = []
for (v, c) in enumerate(field):
	if c == 'P':
		all_packmen.append(v)
	elif c == '*':
		all_food.append(v)
lo = 0
hi = len(field) * 2
valid = -1
while lo <= hi:
	mid = (lo + hi) // 2
	if eatable_in_time(all_food, all_packmen, mid):
		valid = mid
		hi = mid - 1
	else:
		lo = mid + 1
print(valid)