AtCoder Regular Contest 076 - Exhausted?

There are M chairs arranged in a line. The coordinate of the i-th chair (1 ≤ i ≤ M) is i.
N people of the Takahashi clan played too much games, and they are all suffering from backaches. They need to sit in chairs and rest, but they are particular about which chairs they sit in. Specifically, the i-th person wishes to sit in a chair whose coordinate is not greater than L_i, or not less than R_i. Naturally, only one person can sit in the same chair.
It may not be possible for all of them to sit in their favorite chairs, if nothing is done.
Aoki, who cares for the health of the people of the Takahashi clan, decides to provide additional chairs so that all of them can sit in chairs at their favorite positions.
Additional chairs can be placed at arbitrary real coordinates. Find the minimum required number of additional chairs.

-----Constraints-----
 - 1 ≤ N,M ≤ 2 × 10^5
 - 0 ≤ L_i < R_i ≤ M + 1(1 ≤ i ≤ N)
 - All input values are integers.

-----Input-----
Input is given from Standard Input in the following format:
N M
L_1 R_1
:
L_N R_N

-----Output-----
Print the minimum required number of additional chairs.

-----Sample Input-----
4 4
0 3
2 3
1 3
3 4

-----Sample Output-----
0

The four people can sit in chairs at the coordinates 3, 2, 1 and 4, respectively, and no more chair is needed.

Source: atcoder | Difficulty: VERY_HARD


Solutions:
import sys
input = sys.stdin.readline
(N, M) = list(map(int, input().split()))
INF = 2 ** 31 - 1
LV = (M + 2 - 1).bit_length()
N0 = 2 ** LV
data = [0] * (2 * N0)
lazy = [0] * (2 * N0)

def gindex(l, r):
	L = l + N0 >> 1
	R = r + N0 >> 1
	lc = 0 if l & 1 else (L & -L).bit_length()
	rc = 0 if r & 1 else (R & -R).bit_length()
	for i in range(LV):
		if rc <= i:
			yield R
		if L < R and lc <= i:
			yield L
		L >>= 1
		R >>= 1

def propagates(*ids):
	for i in reversed(ids):
		v = lazy[i - 1]
		if not v:
			continue
		lazy[2 * i - 1] += v
		lazy[2 * i] += v
		data[2 * i - 1] += v
		data[2 * i] += v
		lazy[i - 1] = 0

def update(l, r, x):
	(*ids,) = gindex(l, r)
	propagates(*ids)
	L = N0 + l
	R = N0 + r
	while L < R:
		if R & 1:
			R -= 1
			lazy[R - 1] += x
			data[R - 1] += x
		if L & 1:
			lazy[L - 1] += x
			data[L - 1] += x
			L += 1
		L >>= 1
		R >>= 1
	for i in ids:
		data[i - 1] = min(data[2 * i - 1], data[2 * i])

def query(l, r):
	propagates(*gindex(l, r))
	L = N0 + l
	R = N0 + r
	s = INF
	while L < R:
		if R & 1:
			R -= 1
			s = min(s, data[R - 1])
		if L & 1:
			s = min(s, data[L - 1])
			L += 1
		L >>= 1
		R >>= 1
	return s
for i in range(1, M + 1):
	update(0, i + 1, 1)
add = M - N
hito = []
for i in range(N):
	(L, R) = list(map(int, input().split()))
	hito.append((L, R))
hito.sort()
for (l, r) in hito:
	update(0, r + 1, -1)
	m = query(l + 1, M + 2) + l
	add = min(m, add)
print(max(-add, 0))


from collections import deque

class LazySegtree:

	def __init__(self, n, init_val, merge_func, ide_ele):
		self.n = n
		self.ide_ele = ide_ele
		self.merge_func = merge_func
		self.val = [0 for i in range(1 << n)]
		self.merge = [0 for i in range(1 << n)]
		self.parent = [-1 for i in range(1 << n)]
		self.lazy = [0 for i in range(1 << n)]
		deq = deque([1 << n - 1])
		res = []
		while deq:
			v = deq.popleft()
			res.append(v)
			if not v & 1:
				gap = (v & -v) // 2
				self.parent[v - gap] = v
				deq.append(v - gap)
				self.parent[v + gap] = v
				deq.append(v + gap)
		for v in res[::-1]:
			if v - 1 < len(init_val):
				self.val[v - 1] = init_val[v - 1]
			else:
				self.val[v - 1] = 10 ** 18
			self.merge[v - 1] = self.val[v - 1]
			if not v & 1:
				gap = (v & -v) // 2
				self.merge[v - 1] = self.merge_func(self.merge[v - 1], self.merge[v - gap - 1], self.merge[v + gap - 1])

	def lower_kth_update(self, nd, k, x):
		if k == -1:
			return
		maxi = nd - (nd & -nd) + 1 + k
		ids = self.idgetter(nd, maxi)
		for pos in ids:
			gap = (pos & -pos) // 2
			self.propagate(pos)
			if pos <= maxi:
				self.val[pos - 1] += x
				self.lazy[pos - gap - 1] += x * bool(gap)
		ids = ids[::-1]
		if pos & 1:
			self.merge[pos - 1] = self.val[pos - 1]
		else:
			gap = (pos & -pos) // 2
			self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
		for pos in ids[1:]:
			gap = (pos & -pos) // 2
			self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])

	def upper_kth_update(self, nd, k, x):
		if k == -1:
			return
		mini = nd + (nd & -nd) - 1 - k
		ids = self.idgetter(nd, mini)
		for pos in ids:
			gap = (pos & -pos) // 2
			self.propagate(pos)
			if pos >= mini:
				self.val[pos - 1] += x
				self.lazy[pos + gap - 1] += x * bool(gap)
		ids = ids[::-1]
		if pos & 1:
			self.merge[pos - 1] = self.val[pos - 1]
		else:
			gap = (pos & -pos) // 2
			self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
		for pos in ids[1:]:
			gap = (pos & -pos) // 2
			self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])

	def update(self, l, r, x):
		pos = 1 << self.n - 1
		while True:
			gap = (pos & -pos) // 2
			self.propagate(pos)
			if pos - 1 < l:
				pos += (pos & -pos) // 2
			elif pos - 1 > r:
				pos -= (pos & -pos) // 2
			else:
				self.val[pos - 1] += x
				self.upper_kth_update(pos - gap, pos - 1 - l - 1, x)
				self.lower_kth_update(pos + gap, r - pos, x)
				break
		while self.parent[pos] != -1:
			if pos & 1:
				self.merge[pos - 1] = self.val[pos - 1]
			else:
				gap = (pos & -pos) // 2
				self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
			pos = self.parent[pos]

	def lower_kth_merge(self, nd, k, debug=False):
		res = self.ide_ele
		if k == -1:
			return res
		maxi = nd - (nd & -nd) + 1 + k
		ids = self.idgetter(nd, maxi)
		for pos in ids:
			self.propagate(pos)
		stack = [self.ide_ele]
		for pos in ids[::-1]:
			gap = (pos & -pos) // 2
			if pos <= maxi:
				if not pos & 1:
					self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1])
					stack.append(self.merge[pos - 1])
					self.merge[pos - 1] = self.merge_func(self.merge[pos - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
				else:
					self.merge[pos - 1] = self.val[pos - 1]
					stack.append(self.merge[pos - 1])
			elif not pos & 1:
				self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
			else:
				self.merge[pos - 1] = self.val[pos - 1]
		return self.merge_func(stack)

	def upper_kth_merge(self, nd, k):
		res = self.ide_ele
		if k == -1:
			return res
		mini = nd + (nd & -nd) - 1 - k
		ids = self.idgetter(nd, mini)
		for pos in ids:
			self.propagate(pos)
		stack = [self.ide_ele]
		for pos in ids[::-1]:
			gap = (pos & -pos) // 2
			if pos >= mini:
				if not pos & 1:
					self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
					stack.append(self.merge[pos - 1])
					self.merge[pos - 1] = self.merge_func(self.merge[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1])
				else:
					self.merge[pos - 1] = self.val[pos - 1]
					stack.append(self.merge[pos - 1])
			elif not pos & 1:
				self.merge[pos - 1] = self.merge_func(self.val[pos - 1], self.merge[pos - gap - 1] + self.lazy[pos - gap - 1], self.merge[pos + gap - 1] + self.lazy[pos + gap - 1])
			else:
				self.merge[pos - 1] = self.val[pos - 1]
		return self.merge_func(stack)

	def query(self, l, r):
		pos = 1 << self.n - 1
		res = self.ide_ele
		while True:
			gap = (pos & -pos) // 2
			self.propagate(pos)
			if pos - 1 < l:
				pos += (pos & -pos) // 2
			elif pos - 1 > r:
				pos -= (pos & -pos) // 2
			else:
				left = self.upper_kth_merge(pos - gap, pos - 1 - l - 1)
				right = self.lower_kth_merge(pos + gap, r - pos)
				res = self.merge_func(left, right, self.val[pos - 1])
				return res

	def propagate(self, pos):
		if self.lazy[pos - 1]:
			self.val[pos - 1] += self.lazy[pos - 1]
			self.merge[pos - 1] += self.lazy[pos - 1]
			if not pos & 1:
				gap = (pos & -pos) // 2
				self.lazy[pos - gap - 1] += self.lazy[pos - 1]
				self.lazy[pos + gap - 1] += self.lazy[pos - 1]
			self.lazy[pos - 1] = 0
		return

	def idgetter(self, start, goal):
		res = []
		pos = goal
		while pos != start:
			res.append(pos)
			pos = self.parent[pos]
		res.append(start)
		return res[::-1]
import sys
input = sys.stdin.buffer.readline
ide_ele = 10 ** 18
(N, M) = map(int, input().split())
init = [0 for i in range(M + 2)]
for i in range(1, M + 1):
	init[0] += 1
	init[i + 1] -= 1
for i in range(1, M + 1):
	init[i] += init[i - 1]
init[-1] = 0
LST = LazySegtree((M + 2).bit_length(), init, merge_func=min, ide_ele=10 ** 18)
hito = [tuple(map(int, input().split())) for i in range(N)]
hito.sort()
add = M - N
for (l, r) in hito:
	LST.update(0, r, -1)
	m = LST.query(l + 1, M + 1) + l
	add = min(m, add)
print(max(-add, 0))


from operator import itemgetter
from heapq import heappush, heappop
(N, M) = map(int, input().split())
LR = [list(map(int, input().split())) for _ in range(N)]
LR.sort(key=itemgetter(1))
A = []
ans_left = 0
idx = 1
for (_, r) in LR:
	if r == M + 1 or idx == M + 1:
		break
	ans_left += 1
	idx = max(idx + 1, r + 1)
idx_LR = 0
q = []
for i in range(M + 1 - ans_left, M + 1):
	while idx_LR < N and LR[idx_LR][1] <= i:
		(l, _) = LR[idx_LR]
		heappush(q, l)
		idx_LR += 1
	heappop(q)
while idx_LR < N:
	(l, _) = LR[idx_LR]
	q.append(l)
	idx_LR += 1
q.sort(reverse=True)
ans_right = 0
idx = M
for l in q:
	if l == 0 or idx == 0 or ans_right + ans_left == M:
		break
	ans_right += 1
	idx = min(l - 1, idx - 1)
ans = N - ans_left - ans_right
print(ans)


from operator import add

class LazySegmentTree:

	def __init__(self, n, init, merge=min, merge_unit=10 ** 18, operate=add, operate_unit=0):
		self.merge = merge
		self.merge_unit = merge_unit
		self.operate = operate
		self.operate_unit = operate_unit
		self.n = (n - 1).bit_length()
		self.data = [0 for i in range(1 << self.n + 1)]
		self.lazy = [0 for i in range(1 << self.n + 1)]
		for i in range(n):
			self.data[2 ** self.n + i] = init[i]
		for i in range(2 ** self.n - 1, 0, -1):
			self.data[i] = self.merge(self.data[2 * i], self.data[2 * i + 1])

	def propagate_above(self, i):
		m = i.bit_length() - 1
		for bit in range(m, 0, -1):
			v = i >> bit
			add = self.lazy[v]
			self.lazy[v] = 0
			self.data[2 * v] = self.operate(self.data[2 * v], add)
			self.data[2 * v + 1] = self.operate(self.data[2 * v + 1], add)
			self.lazy[2 * v] = self.operate(self.lazy[2 * v], add)
			self.lazy[2 * v + 1] = self.operate(self.lazy[2 * v + 1], add)

	def remerge_above(self, i):
		while i:
			i >>= 1
			self.data[i] = self.operate(self.merge(self.data[2 * i], self.data[2 * i + 1]), self.lazy[i])

	def update(self, l, r, x):
		l += 1 << self.n
		r += 1 << self.n
		l0 = l // (l & -l)
		r0 = r // (r & -r) - 1
		while l < r:
			if l & 1:
				self.data[l] = self.operate(self.data[l], x)
				self.lazy[l] = self.operate(self.lazy[l], x)
				l += 1
			if r & 1:
				self.data[r - 1] = self.operate(self.data[r - 1], x)
				self.lazy[r - 1] = self.operate(self.lazy[r - 1], x)
			l >>= 1
			r >>= 1
		self.remerge_above(l0)
		self.remerge_above(r0)

	def query(self, l, r):
		l += 1 << self.n
		r += 1 << self.n
		l0 = l // (l & -l)
		r0 = r // (r & -r) - 1
		self.propagate_above(l0)
		self.propagate_above(r0)
		res = self.merge_unit
		while l < r:
			if l & 1:
				res = self.merge(res, self.data[l])
				l += 1
			if r & 1:
				res = self.merge(res, self.data[r - 1])
			l >>= 1
			r >>= 1
		return res
import sys
input = sys.stdin.buffer.readline
ide_ele = 10 ** 18
(N, M) = map(int, input().split())
init = [0 for i in range(M + 2)]
for i in range(1, M + 1):
	init[0] += 1
	init[i + 1] -= 1
for i in range(1, M + 1):
	init[i] += init[i - 1]
init[-1] = 0
LST = LazySegmentTree(M + 2, init)
hito = [tuple(map(int, input().split())) for i in range(N)]
hito.sort()
add = M - N
for (l, r) in hito:
	LST.update(0, r + 1, -1)
	m = LST.query(l + 1, M + 2) + l
	add = min(m, add)
print(max(-add, 0))