Problem:


Score : 500 points

Problem Statement

There is an infinitely long street that runs west to east, which we consider as a number line.

There are N roadworks scheduled on this street.
The i-th roadwork blocks the point at coordinate X_i from time S_i - 0.5 to time T_i - 0.5.

Q people are standing at coordinate 0. The i-th person will start the coordinate 0 at time D_i, continue to walk with speed 1 in the positive direction and stop walking when reaching a blocked point.

Find the distance each of the Q people will walk.

Constraints

All values in input are integers.

1 \leq N, Q \leq 2 \times 10^5

0 \leq S_i < T_i \leq 10^9

1 \leq X_i \leq 10^9

0 \leq D_1 < D_2 < ... < D_Q \leq 10^9

If i \neq j and X_i = X_j, the intervals [S_i, T_i) and [S_j, T_j) do not overlap.

Input

Input is given from Standard Input in the following format:

N Q
S_1 T_1 X_1
:
S_N T_N X_N
D_1
:
D_Q

Output

Print Q lines. The i-th line should contain the distance the i-th person will walk or -1 if that person walks forever.

Sample Input 1

4 6
1 3 2
7 13 10
18 20 13
3 4 2
0
1
2
3
5
8

Sample Output 1

2
2
10
-1
13
-1

The first person starts coordinate 0 at time 0 and stops walking at coordinate 2 when reaching a point blocked by the first roadwork at time 2.

The second person starts coordinate 0 at time 1 and reaches coordinate 2 at time 3. The first roadwork has ended, but the fourth roadwork has begun, so this person also stops walking at coordinate 2.

The fourth and sixth persons encounter no roadworks while walking, so they walk forever. The output for these cases is -1.



Sample input:
4 6
1 3 2
7 13 10
18 20 13
3 4 2
0
1
2
3
5
8


Sample output:
2
2
10
-1
13
-1

Submission status: Accepted

Solution code:
from heapq import *

class MultisetBIT:
    """
    最大値が小さい数について、その重複を許した集合を管理する。
    最大値 maxvalue を受け取り、[0, maxvalue] を管理する。
    counter は辞書型かリスト型で、counter[i] が i の個数を持つことを仮定する。
    counter が与えられなかった場合は、全て個数 0 で初期化される。
    基本的に閉区間で管理する。
    内部では受け取った数を +1 したものをindexとして持っているので注意。
    """

    __slots__ = ["n", "k", "data"]

    def __init__(self, maxvalue, counter = None):
        self.n = maxvalue + 1
        self.k = 1 << ((self.n + 1).bit_length() - 1)
        self.data = [0] * (self.n + 1)
        if counter is not None:
            self.update(counter)
        
    def update(self, counter):
        """
        counter を受け取って、それを反映させる。
        counter は辞書型かリスト型で、counter[i] が i の個数を持つことを仮定する。
        計算量は、n を要素の最大値として、 O(n) となる。
        """
        if isinstance(counter, list):
            self.data = [0] + counter + [0] * (self.n - len(counter))
        else:
            self.data = [0] * (self.n + 1)
            for k, v in counter.items():
                self.data[k + 1] = v
        for i in range(1, self.n + 1):
            if i + (i & -i) <= self.n:
                self.data[i + (i & -i)] += self.data[i]

    def add(self, value):
        """
        与えられた引数を Multiset に加える。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        value += 1
        while value <= self.n:
            self.data[value] += 1
            value += value & -value

    def pop(self, value):
        """
        Multiset から与えられた引数を取り除く。
        与えられた引数が Multiset に入っているかのチェックは行わなず、
        単にその個数を 1 減らすだけなので注意。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        value += 1
        while value <= self.n:
            self.data[value] -= 1
            value += value & -value

    def count_le(self, value):
        """
        Multiset 内の要素 elem のうち、0 <= elem <= value を満たすものを数える。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        value += 1
        ret = 0
        while value > 0:
            ret += self.data[value]
            value -= value & -value
        return ret

    def count(self, first, last):
        """
        Multiset 内の要素 elem のうち、first <= elem <= last を満たすものを数える。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        last += 1
        ret = 0
        while first < last:
            ret += self.data[last]
            last -= last & -last
        while last < first:
            ret -= self.data[first]
            first -= first & -first
        return ret

    def bisect(self, count):
        """
        Multiset 内の要素 elem のうち、count <= count_le(elem) を満たす最小のelemを返す。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        ret = 0
        k = self.k
        while k > 0:
            if ret + k <= self.n and self.data[ret + k] < count:
                count -= self.data[ret + k]
                ret += k
            k //= 2
        return ret

    def lower_bound(self, value):
        """
        Multiset 内の要素 elem のうち、value <= elem を満たす最小のelemを返す。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        return self.bisect(self.count_le(value - 1) + 1)

    def upper_bound(self, value):
        """
        Multiset 内の要素 elem のうち、value < elem を満たす最小のelemを返す。
        計算量は、n を要素の最大値として、 O(log n) となる。
        """
        return self.bisect(self.count_le(value) + 1)


N, Q = map(int, input().split())
STXs = [tuple(map(int, input().split())) for _ in range(N)]
Ds = [int(input()) for _ in range(Q)]

i2X = sorted(map(lambda t: t[2], STXs))
X2i = {X:i for i, X in enumerate(i2X)}
max_i = len(i2X) - 1
mbit = MultisetBIT(max_i)

events = []
for S, T, X in STXs:
    events.append( ((S - X) << 32) + (X2i[X] << 1)     )
    events.append( ((T - X) << 32) + (X2i[X] << 1) + 1 )
heapify(events)

mask = (1 << 32) - 1
for D in Ds:
    while events:
        if events[0] % 2 == 0:
            if (events[0] >> 32) <= D:
                e = heappop(events)
                mbit.add((e & mask) >> 1)
            else:
                break
        else:
            if (events[0] >> 32) <= D:
                e = heappop(events)
                mbit.pop((e & mask) >> 1)
            else:
                break
    # print([mbit.count(i, i) for i in range(max_i + 1)])
    ans_i = mbit.lower_bound(0)
    print(-1 if ans_i > max_i else i2X[ans_i])

Submission status: Accepted

Solution code:
class AVLtree:
    """
    参考：http://www.nct9.ne.jp/m_hiroi/light/pyalgo12.html
    insert : 挿入
    erase : 削除
    lower_bound(x) : x <= v なる最小のv、なければ-float("inf")、最小値はlower(-float("inf"))
    upper_bound(x) : v < x なる最大のv、なければfloat("inf")、最大値はupper(float("inf"))
    print(avltree) : デバッグ用
    """
 
    def __init__(self):
        self.root = None

        
    def insert(self, x):
        if self.root is None:
            self.root = Node(x)
            return
        q = []
        node = self.root
        while True:
            if node.val == x:
                return
            elif x < node.val:
                q.append((node, 1))
                if node.left is None:
                    node.left = Node(x)
                    break
                node = node.left
            else:
                q.append((node, -1))
                if node.right is None:
                    node.right = Node(x)
                    break
                node = node.right
                
        # バランス調整
        new_node = None
        while q:
            node, key = q.pop()
            node.bal += key
            bal = node.bal
            if not bal:
                return
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    new_node.bal = 0
                    node.bal = 0
                break
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    new_node.bal = 0
                    node.bal = 0
                break
        if q:
            node, key = q.pop()
            if key == 1:
                node.left = new_node
            else:
                node.right = new_node
        elif new_node is not None:
            self.root = new_node

            
    def erase(self, x):
        if self.root is None:
            return
        q = []
        node = self.root
        while node is not None:
            if node.val == x:
                break
            if x < node.val:
                q.append((node, 1))
                node = node.left
            else:
                q.append((node, -1))
                node = node.right
        else:
            return
        if node.left is not None and node.right is not None:
            q.append((node, -1))
            min_node = node.right
            while min_node.left is not None:
                q.append((min_node, 1))
                min_node = min_node.left
            node.val = min_node.val
            node = min_node
        if q:
            pre, key = q[-1]
        else:
            pre = None
        if node.left is None:
            cnode = node.right
        else:
            cnode = node.left
        if pre is None:
            self.root = cnode
            return
        if key == 1:
            pre.left = cnode
        else:
            pre.right = cnode
        
        # バランス調整
        while q:
            new_node = None
            node, key = q.pop()
            node.bal -= key
            bal = node.bal
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    if not new_node.bal:
                        new_node.bal = -1
                        node.bal = 1
                    else:
                        new_node.bal = 0
                        node.bal = 0
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    if not new_node.bal:
                        new_node.bal = 1
                        node.bal = -1
                    else:
                        new_node.bal= 0
                        node.bal = 0
            elif bal:
                break
            if new_node is not None:
                if not q:
                    self.root = new_node
                    return
                node, key = q[-1]
                if key == 1:
                    node.left = new_node
                else:
                    node.right = new_node
                if new_node.bal:
                    break

                    
    def lower_bound(self, x):
        res = float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                res = node.val
                node = node.left
            else:
                node = node.right
        return res
    
    
    def upper_bound(self, x):
        res = -float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                node = node.left
            else:
                res = node.val
                node = node.right
        return res
    
    
    def __str__(self):
        if self.root is None:
            return "[]"
        buff = "["
        for x in self.root._dfs():
            buff += str(x)+", "
        buff = buff.rstrip(", ")
        buff += "]"
        return buff


class Node:
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None
        self.bal = 0 #1:左1, -1:右1
    
    
    def _rotateR(self):
        nodeL = self.left
        self.left = nodeL.right
        nodeL.right = self
        return nodeL
    
    
    def _rotateL(self):
        nodeR = self.right
        self.right = nodeR.left
        nodeR.left = self
        return nodeR
    
    
    def _update_bal(self):
        if self.bal == 1:
            self.right.bal = -1
            self.left.bal = 0
        elif self.bal == -1:
            self.right.bal = 0
            self.left.bal = 1
        else:
            self.right.bal = 0
            self.left.bal = 0
        self.bal = 0
    
    
    def _dfs(self):
        if self is not None:
            if self.left is not None:
                for x in self.left._dfs():
                    yield x
            yield self.val
            if self.right is not None:
                for x in self.right._dfs():
                    yield x

n,q = map(int, input().split())
e = []
for i in range(n):
  s,t,x = map(int, input().split())
  e.append((s-x,x,i))
  e.append((t-x,x,i))
e.sort(key=lambda x:x[0])
D = [int(input()) for _ in range(q)]
avl = AVLtree()
ans = []
f = [0]*n
j = 0
N = 2*n
for d in D:
  while j < N and e[j][0] <= d:
    _,x,i = e[j]
    if f[i]:
      avl.erase((x,i))
    else:
      avl.insert((x,i))
      f[i] = 1
    j += 1
  m = avl.lower_bound((-float("inf"),0))
  ans.append(m[0] if m != float("inf") else -1)

for i in ans:
  print(i)

Submission status: Accepted

Solution code:
class AVLtree:
    """
    参考：http://www.nct9.ne.jp/m_hiroi/light/pyalgo12.html
    insert : 挿入
    erase : 削除
    lower_bound(x) : x <= v なる最小のv、なければ-float("inf")、最小値はlower(-float("inf"))
    upper_bound(x) : v < x なる最大のv、なければfloat("inf")、最大値はupper(float("inf"))
    print(avltree) : デバッグ用
    """
 
    def __init__(self):
        self.root = None
        self.num = 0
        
    def insert(self, x):
        if self.root is None:
            self.root = Node(x)
            self.num += 1
            return
        q = []
        node = self.root
        while True:
            if node.val == x:
                return
            elif x < node.val:
                q.append((node, 1))
                if node.left is None:
                    node.left = Node(x)
                    break
                node = node.left
            else:
                q.append((node, -1))
                if node.right is None:
                    node.right = Node(x)
                    break
                node = node.right
        self.num += 1
        
        # バランス調整
        new_node = None
        while q:
            node, key = q.pop()
            node.bal += key
            node.size += 1
            bal = node.bal
            if not bal:
                while q:
                    node, key = q.pop()
                    node.size += 1
                return
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    new_node.bal = 0
                    node.bal = 0
                break
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    new_node.bal = 0
                    node.bal = 0
                break
        if q:
            node, key = q.pop()
            node.size += 1
            if key == 1:
                node.left = new_node
            else:
                node.right = new_node
        elif new_node is not None:
            self.root = new_node
        while q:
            node, key = q.pop()
            node.size += 1

            
    def erase(self, x):
        if self.root is None:
            return
        q = []
        node = self.root
        while node is not None:
            if node.val == x:
                break
            if x < node.val:
                q.append((node, 1))
                node = node.left
            else:
                q.append((node, -1))
                node = node.right
        else:
            return

        self.num -= 1
        if node.left is not None and node.right is not None:
            q.append((node, -1))
            min_node = node.right
            while min_node.left is not None:
                q.append((min_node, 1))
                min_node = min_node.left
            node.val = min_node.val
            node = min_node
        if q:
            pre, key = q[-1]
        else:
            pre = None
        if node.left is None:
            cnode = node.right
        else:
            cnode = node.left
        if pre is None:
            self.root = cnode
            return
        if key == 1:
            pre.left = cnode
        else:
            pre.right = cnode
        
        # バランス調整
        while q:
            new_node = None
            node, key = q.pop()
            node.bal -= key
            node.size -= 1
            bal = node.bal
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    if not new_node.bal:
                        new_node.bal = -1
                        node.bal = 1
                    else:
                        new_node.bal = 0
                        node.bal = 0
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    if not new_node.bal:
                        new_node.bal = 1
                        node.bal = -1
                    else:
                        new_node.bal= 0
                        node.bal = 0
            elif bal:
                break
            if new_node is not None:
                if not q:
                    self.root = new_node
                    return
                node, key = q[-1]
                if key == 1:
                    node.left = new_node
                else:
                    node.right = new_node
                if new_node.bal:
                    break
        while q:
            node, key = q.pop()
            node.size -= 1
                    
    def lower_bound(self, x):
        res = float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                res = node.val
                node = node.left
            else:
                node = node.right
        return res
    
    
    def upper_bound(self, x):
        res = -float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                node = node.left
            else:
                res = node.val
                node = node.right
        return res
    
    def __len__(self):
        return self.num

    def __getitem__(self, key):
        if key < 0:
            key += self.num
        key += 1
        node = self.root
        while True:
            idx = node.left.size + 1 if node.left is not None else 1
            if idx <= key:
                key -= idx
                if not key:
                    break
                node = node.right
            else:
                node = node.left
        return node.val

    
    def __str__(self):
        if self.root is None:
            return "[]"
        buff = "["
        for x in self.root._dfs():
            buff += str(x)+", "
        buff = buff.rstrip(", ")
        buff += "]"
        return buff


class Node:
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None
        self.bal = 0 #1:左1, -1:右1
        self.size = 1
    
    def _rotateR(self):
        nodeL = self.left
        nodeLL_size = nodeL.left.size if nodeL.left is not None else 0
        nodeR_size = self.right.size if self.right is not None else 0
        self.size -= nodeLL_size + 1
        nodeL.size += nodeR_size + 1
        self.left = nodeL.right
        nodeL.right = self
        return nodeL
    
    
    def _rotateL(self):
        nodeR = self.right
        nodeRR_size = nodeR.right.size if nodeR.right is not None else 0
        nodeL_size = self.left.size if self.left is not None else 0
        self.size -= nodeRR_size + 1
        nodeR.size += nodeL_size + 1
        self.right = nodeR.left
        nodeR.left = self
        return nodeR
    
    
    def _update_bal(self):
        if self.bal == 1:
            self.right.bal = -1
            self.left.bal = 0
        elif self.bal == -1:
            self.right.bal = 0
            self.left.bal = 1
        else:
            self.right.bal = 0
            self.left.bal = 0
        self.bal = 0
    
    
    def _dfs(self):
        if self is not None:
            if self.left is not None:
                for x in self.left._dfs():
                    yield x
            yield self.val
            if self.right is not None:
                for x in self.right._dfs():
                    yield x
                    

n,q = map(int, input().split())
e = []
for i in range(n):
  s,t,x = map(int, input().split())
  e.append((s-x,x,i))
  e.append((t-x,x,i))
e.sort(key=lambda x:x[0])
D = [int(input()) for _ in range(q)]
avl = AVLtree()
ans = []
f = [0]*n
j = 0
N = 2*n
for d in D:
  while j < N and e[j][0] <= d:
    _,x,i = e[j]
    if f[i]:
      avl.erase((x,i))
    else:
      avl.insert((x,i))
      f[i] = 1
    j += 1
  m = avl.lower_bound((-float("inf"),0))
  ans.append(m[0] if m != float("inf") else -1)

for i in ans:
  print(i)

Submission status: Accepted

Solution code:
class AVLtree:
    """
    参考：http://www.nct9.ne.jp/m_hiroi/light/pyalgo12.html
    insert : 挿入
    erase : 削除
    lower_bound(x) : x <= v なる最小のv、なければ-float("inf")、最小値はlower(-float("inf"))
    upper_bound(x) : v < x なる最大のv、なければfloat("inf")、最大値はupper(float("inf"))
    print(avltree) : デバッグ用
    """
 
    def __init__(self):
        self.root = None

        
    def insert(self, x):
        if self.root is None:
            self.root = Node(x)
            return
        q = []
        node = self.root
        while True:
            if node.val == x:
                return
            elif x < node.val:
                q.append((node, 1))
                if node.left is None:
                    node.left = Node(x)
                    break
                node = node.left
            else:
                q.append((node, -1))
                if node.right is None:
                    node.right = Node(x)
                    break
                node = node.right
                
        # バランス調整
        new_node = None
        while q:
            node, key = q.pop()
            node.bal += key
            bal = node.bal
            if not bal:
                return
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    new_node.bal = 0
                    node.bal = 0
                break
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    new_node.bal = 0
                    node.bal = 0
                break
        if q:
            node, key = q.pop()
            if key == 1:
                node.left = new_node
            else:
                node.right = new_node
        elif new_node is not None:
            self.root = new_node

            
    def erase(self, x):
        if self.root is None:
            return
        q = []
        node = self.root
        while node is not None:
            if node.val == x:
                break
            if x < node.val:
                q.append((node, 1))
                node = node.left
            else:
                q.append((node, -1))
                node = node.right
        else:
            return
        if node.left is not None and node.right is not None:
            q.append((node, -1))
            min_node = node.right
            while min_node.left is not None:
                q.append((min_node, 1))
                min_node = min_node.left
            node.val = min_node.val
            node = min_node
        if q:
            pre, key = q[-1]
        else:
            pre = None
        if node.left is None:
            cnode = node.right
        else:
            cnode = node.left
        if pre is None:
            self.root = cnode
            return
        if key == 1:
            pre.left = cnode
        else:
            pre.right = cnode
        
        # バランス調整
        while q:
            new_node = None
            node, key = q.pop()
            node.bal -= key
            bal = node.bal
            if bal == 2:
                if node.left.bal == -1:
                    node.left = node.left._rotateL()
                    new_node = node._rotateR()
                    new_node._update_bal()
                else:
                    new_node = node._rotateR()
                    if not new_node.bal:
                        new_node.bal = -1
                        node.bal = 1
                    else:
                        new_node.bal = 0
                        node.bal = 0
            elif bal == -2:
                if node.right.bal == 1:
                    node.right = node.right._rotateR()
                    new_node = node._rotateL()
                    new_node._update_bal()
                else:
                    new_node = node._rotateL()
                    if not new_node.bal:
                        new_node.bal = 1
                        node.bal = -1
                    else:
                        new_node.bal= 0
                        node.bal = 0
            elif bal:
                break
            if new_node is not None:
                if not q:
                    self.root = new_node
                    return
                node, key = q[-1]
                if key == 1:
                    node.left = new_node
                else:
                    node.right = new_node
                if new_node.bal:
                    break

                    
    def lower_bound(self, x):
        res = float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                res = node.val
                node = node.left
            else:
                node = node.right
        return res
    
    
    def upper_bound(self, x):
        res = -float("inf")
        node = self.root
        while node is not None:
            if x <= node.val:
                node = node.left
            else:
                res = node.val
                node = node.right
        return res
    

class Node:
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None
        self.bal = 0 #1:左1, -1:右1
    
    
    def _rotateR(self):
        nodeL = self.left
        self.left = nodeL.right
        nodeL.right = self
        return nodeL
    
    
    def _rotateL(self):
        nodeR = self.right
        self.right = nodeR.left
        nodeR.left = self
        return nodeR
    
    
    def _update_bal(self):
        if self.bal == 1:
            self.right.bal = -1
            self.left.bal = 0
        elif self.bal == -1:
            self.right.bal = 0
            self.left.bal = 1
        else:
            self.right.bal = 0
            self.left.bal = 0
        self.bal = 0
    
    
n,q = map(int, input().split())
e = []
for i in range(n):
  s,t,x = map(int, input().split())
  e.append((s-x,x,i))
  e.append((t-x,x,i))
e.sort(key=lambda x:x[0])
D = [int(input()) for _ in range(q)]
avl = AVLtree()
ans = []
f = [0]*n
j = 0
N = 2*n
for d in D:
  while j < N and e[j][0] <= d:
    _,x,i = e[j]
    if f[i]:
      avl.erase((x,i))
    else:
      avl.insert((x,i))
      f[i] = 1
    j += 1
  m = avl.lower_bound((-float("inf"),0))
  ans.append(m[0] if m != float("inf") else -1)

for i in ans:
  print(i)