JJ found a balanced parenthesis sequence. He does not like balanced parentheses so he will make the parenthesis sequence unbalanced by swapping adjacent characters in the sequence. 

JJ is lazy and wants to know minimum number of adjacent swaps required to make the given balanced parenthesis sequence unbalanced.

A parenthesis sequence is called balanced if one can turn it into a valid mathematical expression by adding characters + and 1. Sequences such as (())() and ()() are balanced whereas sequences such as ())( and )() are unbalanced.   

------ Input Format ------ 

- The first line contains T - the number of test cases. Then the test cases follow.
- The first line of each test case contains S - a balanced parenthesis sequence.

------ Output Format ------ 

For each test case, output the minimum number of adjacent swaps required to make the parenthesis sequence unbalanced.

------ Constraints ------ 

$1 ≤ T ≤ 5\cdot10^{4}$
$2 ≤ |S| ≤ 10^{5}$
- Sum of $S$ does not exceeds $10^{6}$ over all test cases
$S$ is a balanced parentheses sequence

----- Sample Input 1 ------ 
4
()
(())
()()
((()))(())

----- Sample Output 1 ------ 
1
2
1
2

----- explanation 1 ------ 
- Test case $1$: Swap the first and second character to get the unbalanced sequence )(.
- Test case $2$: Swap the second and third character first to obtain the bracket sequence ()(). Now you can swap the first and second character to obtain the sequence )(() which is unbalanced.

Difficulty: VERY_HARD
Tags: Constructive algorithms

String algorithms

Range queries

Greedy algorithms

Data structures

Ad-hoc
Skill types: Data structures

Range queries

Greedy algorithms

Starter code:

Solutions:
from sys import stdin
input = stdin.readline

def answer():
	ans = float('inf')
	(o, c, ind) = (0, 0, 0)
	for i in range(n):
		req = o - c + 1
		while dist[-1] < i + 1:
			dist.pop()
			ind += 1
		if n // 2 - ind < req:
			break
		cal = prefix[ind + req] - prefix[ind] - req * i - req * (req + 1) // 2
		ans = min(ans, cal)
		if a[i] == '(':
			o += 1
		else:
			c += 1
	return ans
for T in range(int(input())):
	a = input().strip()
	n = len(a)
	dist = []
	for i in range(n):
		if a[i] == ')':
			dist.append(i + 1)
	prefix = [0]
	for i in range(len(dist)):
		prefix.append(prefix[-1] + dist[i])
	dist = dist[::-1]
	print(answer())


for _ in range(int(input())):
	s = input()
	l = len(s)
	pos = [0]
	for i in range(l):
		if s[i] == ')':
			pos.append(pos[-1] + i)
	ans = 10000007
	(o, c) = (0, 0)
	for i in range(l):
		if s[i] == '(':
			o += 1
			t = o - c
			try:
				temp = pos[o] - pos[c] - i * t - t * (t - 1) // 2
				ans = min(ans, temp)
			except:
				break
		else:
			c += 1
	print(ans)


t = int(input())
for tc in range(t):
	s = input()
	positions = [0]
	l = len(s)
	for i in range(l):
		if s[i] == ')':
			positions.append(positions[-1] + i)
	ans = 1000000007
	(o, c) = (0, 0)
	for i in range(l):
		if s[i] == '(':
			o += 1
			x = o - c
			try:
				tmp = positions[o] - positions[c] - i * x - x * (x - 1) // 2
				ans = min(ans, tmp)
			except:
				break
		else:
			c += 1
	print(ans)


for _ in range(int(input())):
	s = input()
	n = len(s)
	l = [0 for _x in range(n)]
	l[0] = 1
	for i in range(1, n):
		if s[i] == '(':
			l[i] = l[i - 1] + 1
		else:
			l[i] = l[i - 1] - 1
	temp = [0]
	ans = float('inf')
	for i in range(n - 1, -1, -1):
		if len(temp) > l[i] + 1:
			t = temp[-1] - temp[-1 * l[i] - 2] - (l[i] + 1) * (l[i] + 2) // 2 - i * (l[i] + 1)
			ans = min(ans, t)
		if s[i] == ')':
			temp.append(temp[-1] + i)
	ans = min(ans, temp[-1] - temp[-2])
	print(ans)


for _ in range(int(input())):
	s = input().strip()
	n = len(s)
	a = [0]
	for i in range(n):
		if s[i] == ')':
			a.append(a[-1] + i)
	op = 0
	cl = 0
	mini = a[1]
	for i in range(n):
		if s[i] == '(':
			op += 1
			if op >= len(a):
				break
			ans = a[op] - a[cl] - i * (op - cl) - (op - cl) * (op - cl - 1) // 2
			mini = min(mini, ans)
		else:
			cl += 1
	print(mini)


tt = 1
tt = int(input())
for _ in range(tt):
	s = input()
	n = len(s)
	end = []
	prefix = [0]
	ans = n // 2
	for (i, ch) in enumerate(s):
		if ch == '(':
			ans = min(ans, n - i - 1)
		else:
			end.append(i)
			ans = min(ans, i)
			prefix.append(prefix[-1] + i)
	end.append(n)
	current = 0
	index = 1
	m = len(end) - 1
	for (i, ch) in enumerate(s):
		if ch == '(':
			current += 1
		else:
			current -= 1
		if end[index] == i:
			index += 1
		if index + current < m:
			d = prefix[index + current + 1] - prefix[index]
			d -= ((i + current + 1) * (i + current + 2) - i * (i + 1)) // 2
			ans = min(ans, d)
	print(ans)


def divisors(M):
	d = []
	i = 1
	while M >= i ** 2:
		if M % i == 0:
			d.append(i)
			if i ** 2 != M:
				d.append(M // i)
		i = i + 1
	return d

def popcount(x):
	x = x - (x >> 1 & 1431655765)
	x = (x & 858993459) + (x >> 2 & 858993459)
	x = x + (x >> 4) & 252645135
	x = x + (x >> 8)
	x = x + (x >> 16)
	return x & 127

def eratosthenes(n):
	res = [0 for i in range(n + 1)]
	prime = set([])
	for i in range(2, n + 1):
		if not res[i]:
			prime.add(i)
			for j in range(1, n // i + 1):
				res[i * j] = 1
	return prime

def factorization(n):
	res = []
	for p in prime:
		if n % p == 0:
			while n % p == 0:
				n //= p
			res.append(p)
	if n != 1:
		res.append(n)
	return res

def euler_phi(n):
	res = n
	for x in range(2, n + 1):
		if x ** 2 > n:
			break
		if n % x == 0:
			res = res // x * (x - 1)
			while n % x == 0:
				n //= x
	if n != 1:
		res = res // n * (n - 1)
	return res

def ind(b, n):
	res = 0
	while n % b == 0:
		res += 1
		n //= b
	return res

def isPrimeMR(n):
	d = n - 1
	d = d // (d & -d)
	L = [2, 3, 5, 7, 11, 13, 17]
	for a in L:
		t = d
		y = pow(a, t, n)
		if y == 1:
			continue
		while y != n - 1:
			y = y * y % n
			if y == 1 or t == n - 1:
				return 0
			t <<= 1
	return 1

def findFactorRho(n):
	from math import gcd
	m = 1 << n.bit_length() // 8
	for c in range(1, 99):
		f = lambda x: (x * x + c) % n
		(y, r, q, g) = (2, 1, 1, 1)
		while g == 1:
			x = y
			for i in range(r):
				y = f(y)
			k = 0
			while k < r and g == 1:
				ys = y
				for i in range(min(m, r - k)):
					y = f(y)
					q = q * abs(x - y) % n
				g = gcd(q, n)
				k += m
			r <<= 1
		if g == n:
			g = 1
			while g == 1:
				ys = f(ys)
				g = gcd(abs(x - ys), n)
		if g < n:
			if isPrimeMR(g):
				return g
			elif isPrimeMR(n // g):
				return n // g
			return findFactorRho(g)

def primeFactor(n):
	i = 2
	ret = {}
	rhoFlg = 0
	while i * i <= n:
		k = 0
		while n % i == 0:
			n //= i
			k += 1
		if k:
			ret[i] = k
		i += 1 + i % 2
		if i == 101 and n >= 2 ** 20:
			while n > 1:
				if isPrimeMR(n):
					(ret[n], n) = (1, 1)
				else:
					rhoFlg = 1
					j = findFactorRho(n)
					k = 0
					while n % j == 0:
						n //= j
						k += 1
					ret[j] = k
	if n > 1:
		ret[n] = 1
	if rhoFlg:
		ret = {x: ret[x] for x in sorted(ret)}
	return ret

def divisors(n):
	res = [1]
	prime = primeFactor(n)
	for p in prime:
		newres = []
		for d in res:
			for j in range(prime[p] + 1):
				newres.append(d * p ** j)
		res = newres
	res.sort()
	return res

def xorfactorial(num):
	if num == 0:
		return 0
	elif num == 1:
		return 1
	elif num == 2:
		return 3
	elif num == 3:
		return 0
	else:
		x = baseorder(num)
		return 2 ** x * ((num - 2 ** x + 1) % 2) + function(num - 2 ** x)

def xorconv(n, X, Y):
	if n == 0:
		res = [X[0] * Y[0] % mod]
		return res
	x = [X[i] + X[i + 2 ** (n - 1)] for i in range(2 ** (n - 1))]
	y = [Y[i] + Y[i + 2 ** (n - 1)] for i in range(2 ** (n - 1))]
	z = [X[i] - X[i + 2 ** (n - 1)] for i in range(2 ** (n - 1))]
	w = [Y[i] - Y[i + 2 ** (n - 1)] for i in range(2 ** (n - 1))]
	res1 = xorconv(n - 1, x, y)
	res2 = xorconv(n - 1, z, w)
	former = [(res1[i] + res2[i]) * inv for i in range(2 ** (n - 1))]
	latter = [(res1[i] - res2[i]) * inv for i in range(2 ** (n - 1))]
	former = list(map(lambda x: x % mod, former))
	latter = list(map(lambda x: x % mod, latter))
	return former + latter

def merge_sort(A, B):
	(pos_A, pos_B) = (0, 0)
	(n, m) = (len(A), len(B))
	res = []
	while pos_A < n and pos_B < m:
		(a, b) = (A[pos_A], B[pos_B])
		if a < b:
			res.append(a)
			pos_A += 1
		else:
			res.append(b)
			pos_B += 1
	res += A[pos_A:]
	res += B[pos_B:]
	return res

class UnionFindVerSize:

	def __init__(self, N):
		self._parent = [n for n in range(0, N)]
		self._size = [1] * N
		self.group = N

	def find_root(self, x):
		if self._parent[x] == x:
			return x
		self._parent[x] = self.find_root(self._parent[x])
		stack = [x]
		while self._parent[stack[-1]] != stack[-1]:
			stack.append(self._parent[stack[-1]])
		for v in stack:
			self._parent[v] = stack[-1]
		return self._parent[x]

	def unite(self, x, y):
		gx = self.find_root(x)
		gy = self.find_root(y)
		if gx == gy:
			return
		self.group -= 1
		if self._size[gx] < self._size[gy]:
			self._parent[gx] = gy
			self._size[gy] += self._size[gx]
		else:
			self._parent[gy] = gx
			self._size[gx] += self._size[gy]

	def get_size(self, x):
		return self._size[self.find_root(x)]

	def is_same_group(self, x, y):
		return self.find_root(x) == self.find_root(y)

class WeightedUnionFind:

	def __init__(self, N):
		self.parent = [i for i in range(N)]
		self.size = [1 for i in range(N)]
		self.val = [0 for i in range(N)]
		self.flag = True
		self.edge = [[] for i in range(N)]

	def dfs(self, v, pv):
		stack = [(v, pv)]
		new_parent = self.parent[pv]
		while stack:
			(v, pv) = stack.pop()
			self.parent[v] = new_parent
			for (nv, w) in self.edge[v]:
				if nv != pv:
					self.val[nv] = self.val[v] + w
					stack.append((nv, v))

	def unite(self, x, y, w):
		if not self.flag:
			return
		if self.parent[x] == self.parent[y]:
			self.flag = self.val[x] - self.val[y] == w
			return
		if self.size[self.parent[x]] > self.size[self.parent[y]]:
			self.edge[x].append((y, -w))
			self.edge[y].append((x, w))
			self.size[x] += self.size[y]
			self.val[y] = self.val[x] - w
			self.dfs(y, x)
		else:
			self.edge[x].append((y, -w))
			self.edge[y].append((x, w))
			self.size[y] += self.size[x]
			self.val[x] = self.val[y] + w
			self.dfs(x, y)

class Dijkstra:

	class Edge:

		def __init__(self, _to, _cost):
			self.to = _to
			self.cost = _cost

	def __init__(self, V):
		self.G = [[] for i in range(V)]
		self._E = 0
		self._V = V

	@property
	def E(self):
		return self._E

	@property
	def V(self):
		return self._V

	def add_edge(self, _from, _to, _cost):
		self.G[_from].append(self.Edge(_to, _cost))
		self._E += 1

	def shortest_path(self, s):
		import heapq
		que = []
		d = [10 ** 15] * self.V
		d[s] = 0
		heapq.heappush(que, (0, s))
		while len(que) != 0:
			(cost, v) = heapq.heappop(que)
			if d[v] < cost:
				continue
			for i in range(len(self.G[v])):
				e = self.G[v][i]
				if d[e.to] > d[v] + e.cost:
					d[e.to] = d[v] + e.cost
					heapq.heappush(que, (d[e.to], e.to))
		return d

def Z_algorithm(s):
	N = len(s)
	Z_alg = [0] * N
	Z_alg[0] = N
	i = 1
	j = 0
	while i < N:
		while i + j < N and s[j] == s[i + j]:
			j += 1
		Z_alg[i] = j
		if j == 0:
			i += 1
			continue
		k = 1
		while i + k < N and k + Z_alg[k] < j:
			Z_alg[i + k] = Z_alg[k]
			k += 1
		i += k
		j -= k
	return Z_alg

class BIT:

	def __init__(self, n, mod=0):
		self.BIT = [0] * (n + 1)
		self.num = n
		self.mod = mod

	def query(self, idx):
		res_sum = 0
		mod = self.mod
		while idx > 0:
			res_sum += self.BIT[idx]
			if mod:
				res_sum %= mod
			idx -= idx & -idx
		return res_sum

	def update(self, idx, x):
		mod = self.mod
		while idx <= self.num:
			self.BIT[idx] += x
			if mod:
				self.BIT[idx] %= mod
			idx += idx & -idx
		return

class dancinglink:

	def __init__(self, n, debug=False):
		self.n = n
		self.debug = debug
		self._left = [i - 1 for i in range(n)]
		self._right = [i + 1 for i in range(n)]
		self.exist = [True for i in range(n)]

	def pop(self, k):
		if self.debug:
			assert self.exist[k]
		L = self._left[k]
		R = self._right[k]
		if L != -1:
			if R != self.n:
				(self._right[L], self._left[R]) = (R, L)
			else:
				self._right[L] = self.n
		elif R != self.n:
			self._left[R] = -1
		self.exist[k] = False

	def left(self, idx, k=1):
		if self.debug:
			assert self.exist[idx]
		res = idx
		while k:
			res = self._left[res]
			if res == -1:
				break
			k -= 1
		return res

	def right(self, idx, k=1):
		if self.debug:
			assert self.exist[idx]
		res = idx
		while k:
			res = self._right[res]
			if res == self.n:
				break
			k -= 1
		return res

class SparseTable:

	def __init__(self, A, merge_func, ide_ele):
		N = len(A)
		self.merge_func = merge_func
		self.lg = [0] * (N + 1)
		for i in range(2, N + 1):
			self.lg[i] = self.lg[i >> 1] + 1
		self.pow_2 = [pow(2, i) for i in range(20)]
		self.table = [None] * (self.lg[N] + 1)
		st0 = self.table[0] = [a for a in A]
		b = 1
		for i in range(self.lg[N]):
			st0 = self.table[i + 1] = [self.merge_func(u, v) for (u, v) in zip(st0, st0[b:])]
			b <<= 1

	def query(self, s, t):
		b = t - s + 1
		m = self.lg[b]
		return self.merge_func(self.table[m][s], self.table[m][t - self.pow_2[m] + 1])

class BinaryTrie:

	class node:

		def __init__(self, val):
			self.left = None
			self.right = None
			self.max = val

	def __init__(self):
		self.root = self.node(-10 ** 15)

	def append(self, key, val):
		pos = self.root
		for i in range(29, -1, -1):
			pos.max = max(pos.max, val)
			if key >> i & 1:
				if pos.right is None:
					pos.right = self.node(val)
					pos = pos.right
				else:
					pos = pos.right
			elif pos.left is None:
				pos.left = self.node(val)
				pos = pos.left
			else:
				pos = pos.left
		pos.max = max(pos.max, val)

	def search(self, M, xor):
		res = -10 ** 15
		pos = self.root
		for i in range(29, -1, -1):
			if pos is None:
				break
			if M >> i & 1:
				if xor >> i & 1:
					if pos.right:
						res = max(res, pos.right.max)
					pos = pos.left
				else:
					if pos.left:
						res = max(res, pos.left.max)
					pos = pos.right
			elif xor >> i & 1:
				pos = pos.right
			else:
				pos = pos.left
		if pos:
			res = max(res, pos.max)
		return res

def solveequation(edge, ans, n, m):

	def dfs(v):
		used[v] = True
		r = ans[v]
		for (to, dire, id) in edge[v]:
			if used[to]:
				continue
			y = dfs(to)
			if dire == -1:
				x[id] = y
			else:
				x[id] = -y
			r += y
		return r
	x = [0] * m
	used = [False] * n
	for v in range(n):
		if used[v]:
			continue
		y = dfs(v)
		if y != 0:
			return False
	return x

class slope_trick:

	def __init__(self):
		self.L = [10 ** 17]
		self.R = [10 ** 17]
		self.min_f = 0
		self.x_left = 0
		self.x_right = 0

	def add_right(self, a):
		a -= self.x_left
		l0 = -self.L[0]
		self.min_f = self.min_f + max(0, l0 - a)
		if l0 <= a:
			a += self.x_left
			a -= self.x_right
			heappush(self.R, a)
		else:
			heappush(self.L, -a)
			a = -heappop(self.L)
			a += self.x_left
			a -= self.x_right
			heappush(self.R, a)

	def add_left(self, a):
		a -= self.x_right
		r0 = self.R[0]
		self.min_f = self.min_f + max(0, a - r0)
		if a <= r0:
			a += self.x_right
			a -= self.x_left
			heappush(self.L, -a)
		else:
			heappush(self.R, a)
			a = heappop(self.R)
			a += self.x_right
			a -= self.x_left
			heappush(self.L, -a)

	def add_abs(self, a):
		self.add_left(a)
		self.add_right(a)

	def change_min_slide(self, a, b):
		self.x_left += a
		self.x_right += b

	def get_val(self, x):
		L = [-l + self.x_left for l in self.L]
		L.sort()
		R = [r + self.x_right for r in self.R]
		R.sort()
		res = self.min_f
		if 0 < L[-1]:
			L = L[::-1]
			n = len(L)
			for i in range(n):
				c0 = L[i]
				c1 = L[i + 1]
				if c1 <= x <= c0:
					res += (i + 1) * (c0 - x)
					break
				else:
					res += (i + 1) * (c0 - c1)
			return res
		elif L[-1] <= x <= R[0]:
			return res
		else:
			n = len(R)
			for i in range(n):
				c0 = R[i]
				c1 = R[i + 1]
				if c0 <= x <= c1:
					res += (i + 1) * (x - c0)
					break
				else:
					res += (i + 1) * (c1 - c0)
			return res

class SegmentTree:

	def __init__(self, init_val, segfunc, ide_ele):
		n = len(init_val)
		self.segfunc = segfunc
		self.ide_ele = ide_ele
		self.num = 1 << (n - 1).bit_length()
		self.tree = [ide_ele] * 2 * self.num
		self.size = n
		for i in range(n):
			self.tree[self.num + i] = init_val[i]
		for i in range(self.num - 1, 0, -1):
			self.tree[i] = self.segfunc(self.tree[2 * i], self.tree[2 * i + 1])

	def update(self, k, x):
		k += self.num
		self.tree[k] = x
		while k > 1:
			k >>= 1
			self.tree[k] = self.segfunc(self.tree[2 * k], self.tree[2 * k + 1])

	def query(self, l, r):
		if r == self.size:
			r = self.num
		res = self.ide_ele
		l += self.num
		r += self.num
		right = []
		while l < r:
			if l & 1:
				res = self.segfunc(res, self.tree[l])
				l += 1
			if r & 1:
				right.append(self.tree[r - 1])
			l >>= 1
			r >>= 1
		for e in right[::-1]:
			res = self.segfunc(res, e)
		return res

	def bisect_l(self, l, r, x):
		l += self.num
		r += self.num
		Lmin = -1
		Rmin = -1
		while l < r:
			if l & 1:
				if self.tree[l] <= x and Lmin == -1:
					Lmin = l
				l += 1
			if r & 1:
				if self.tree[r - 1] <= x:
					Rmin = r - 1
			l >>= 1
			r >>= 1
		if Lmin != -1:
			pos = Lmin
			while pos < self.num:
				if self.tree[2 * pos] <= x:
					pos = 2 * pos
				else:
					pos = 2 * pos + 1
			return pos - self.num
		elif Rmin != -1:
			pos = Rmin
			while pos < self.num:
				if self.tree[2 * pos] <= x:
					pos = 2 * pos
				else:
					pos = 2 * pos + 1
			return pos - self.num
		else:
			return -1
import sys, random, bisect
from collections import deque, defaultdict
from heapq import heapify, heappop, heappush
from itertools import permutations
from math import gcd, log
input = lambda : sys.stdin.readline().rstrip()
mi = lambda : map(int, input().split())
li = lambda : list(mi())
for _ in range(int(input())):
	S = input()
	N = len(S)
	imos_left = [0 for i in range(N)]
	imos_right = [0 for i in range(N)]
	for i in range(N):
		if S[i] == '(':
			imos_left[i] += i
		else:
			imos_right[i] += i
		if i != 0:
			imos_left[i] += imos_left[i - 1]
			imos_right[i] += imos_right[i - 1]
	left = []
	right = [i for i in range(N) if S[i] == ')'][::-1]
	res = N * 100
	s = 0
	for i in range(N):
		if S[i] == '(':
			s += 1
			left.append(i)
		else:
			right.pop()
			s -= 1
		need = (s + 2) // 2
		if min(len(left), len(right)) < need:
			continue
		tmp = need * i - (imos_left[left[-1]] - imos_left[left[-need]] + left[-need]) + (imos_right[right[-need]] - imos_right[right[-1]] + right[-1]) - need * (i + 1) + need
		res = min(res, tmp)
	print(res)