[python刷题模板] 珂朵莉树 ODT （基于支持随机访问的跳表
[python刷题模板] 珂朵莉树 ODT （基于支持随机访问的跳表）
-
- 一、 算法&数据结构
-
- 1. 描述
- 2. 复杂度分析
- 3. 常见应用
- 4. 常用优化
- 二、 模板代码
-
- 0. 区间推平(lg)，单点询问(lg) CF292E. Copying Data
- 1. 区间推平，区间询问最小值
- 2. 区间推平，区间查询最大值
- 3. 不存在区间推平，只有区间加，区间询问最大值(不该用珂朵莉)
- 5. 区间推平，区间询问最大值
- 6. 单点赋值，合并区间（不该用珂朵莉）
- 三、其他
- 四、更多例题
- 五、参考链接
一、 算法&数据结构
1. 描述
我曾写过一篇基于SortedList的珂朵莉模板，在CF上宣告弃用。 [python刷题模板] 珂朵莉树 ODT （20220928弃用）
- 原因是做了CF上一道题，过不了。究其原因是assign操作时del tree[begin:end]的复杂度太高。
- SortedList实现是每块最大2000的分块List，当每次只删1个值时，块内操作需要O(2000)，块外是O(n/2000)。
- 因此我试了下手撕跳表，这里需要注意：要实现跳表的随机化访问和删除（即快速访问第k大的数）
- 以下部分是原文，把SortedList部分改掉，如果要看基于SortedList实现，可以通过上边的链接去看旧文。
- 而且SortedList在很多OJ上不支持，可以选择手动把源码拷贝到代码里，挺麻烦。
- 目前测试来看，跳表写法在力扣上的表现不如SortedList，应该是数据较弱的原因。
- 另外缺点是码量较高，毕竟手撕跳表+珂朵莉，好在可以直接复制模板，两三百行
- 在网络上搜了很久没怎么找到珂朵莉树的python实现，自己写一个。
- 由于python没有有序容器（除SortedList，del复杂度不满足需求），因此自己实现一个。
- 这里选择了码量较少，可以支持随机访问的跳表。
- 珂朵莉树是建立在数据随机的情况下的一个乱搞算法。
- 当操作中有大量
区间赋值动作，尤其是大区间赋值，会把这个区间的值推平，并且将原本这里边的很多小区间合并。 - 如果数据随机，可以预见合并区间后，保留下来的区间不会很多。
- 那么查询区间如果不大，只需要处理几个区间即可。
- 底层用有序列表储存，c++set是红黑树，所以珂朵莉树也算树，
但python的sortedlist是list套list。。。可能名不副实了。不再使用SortedList，改为手撕随机化访问跳表。- 如果有心造数据卡珂朵莉树是可能实现的，数据不保证随机，大范围赋值少，查询多。
- 但这种的可以做特判，因为如果大范围赋值少，那数据范围应该小，没准可以暴力。
2. 复杂度分析
- 在随机数据下，珂朵莉树可达到O(nloglogn)
- split，珂朵莉熟的核心操作，在pos位置切割区间，返回以pos为左端点的区间。O(lgn).
- 之后所有动作都要包含两个split
- 更新assign，O(log2n)
- 查询query, O(lgn)+O©,c是区间内节点数,但c应该小所以速度很客观。
- 所有的query都是暴力操作。
- 找到这个区间所有节点，然后再算。
具体复杂度分析见知乎-珂朵莉树的复杂度分析
3. 常见应用
注意，这类题通常正解是线段树，但珂朵莉在特定情况下吊打正解。
这个特定情况是指：操作中存在大量assign碾平操作。
如果操作中不存在assign，请尽量不要用珂朵莉。（但数据弱的话也没准）
- 区间赋值，区间询问最大最小值。
- 区间赋值，区间询问第K小。
- 区间赋值，区间询问求和
- 区间赋值，区间询问n次方和（一般会有mod）。
这些操作全部暴力处理，因为我们认为：- 在随机数据下，大量区间被合并，询问的区间里不会有太多节点。
4. 常用优化
最大的优化是从SortedList改为手撕跳表。
- 由于实现了下标访问，但每次下标访问都是一次lg,因此在跳表中实现了
bisect_left_kvp方法，直接返回需要下标和两个相邻节点，避免多次lg。 - 从元组变成结构体，这样就可以直接修改。而且在结构体里实现小于运算。
- split时，本应删除原节点，加入两个节点。但实际上插入的左节点和原节点只差了一个右边界，因此可以直接修改。
- split生成时调用节点的解包函数，好写还快一点。
- 由于存在大量node,slot必须写；main中加入gc.disable()，实测不加TLE。
- 跳表的max_level设置成20即可，因为nlgn的题一般n<=2e5
- 下标取值时实现切片，方便操作。del暂时没写切片删除，理论上在均摊下每次操作应该是lg，因此先不写了。如果实现的话，单次切片删除最坏应该是n+lgn,但均摊依然是lg
二、 模板代码
0. 区间推平(lg)，单点询问(lg) CF292E. Copying Data
CF292E. Copying Data
![[python刷题模板] 珂朵莉树 ODT （基于支持随机访问的跳表_第1张图片](http://img.e-com-net.com/image/info8/94583d5a922448aca825e8f3ba908990.jpg)
这题要区间推平，我只会线段树和珂朵莉，而线段树常数太大需要zkw。
import sys
from collections import *
from itertools import *
from math import *
from array import *
from functools import lru_cache
import heapq
import bisect
import random
import io, os
from bisect import *
if sys.hexversion == 50924784:
sys.stdin = open('cfinput.txt')
RI = lambda: map(int, sys.stdin.buffer.readline().split())
RS = lambda: map(bytes.decode, sys.stdin.buffer.readline().strip().split())
RILST = lambda: list(RI())
MOD = 10 ** 9 + 7
"""https://codeforces.com/problemset/problem/292/E
输入 n(≤1e5) 和 m (≤1e5)，两个长度都为 n 的数组 a 和 b（元素范围在 [-1e9,1e9] 内，下标从 1 开始）。
然后输入 m 个操作：
操作 1 形如 1 x y k，表示把 a 的区间 [x,x+k-1] 的元素拷贝到 b 的区间 [y,y+k-1] 上（输入保证下标不越界）。
操作 2 形如 2 x，输出 b[x]。
输入
5 10
1 2 0 -1 3
3 1 5 -2 0
2 5
1 3 3 3
2 5
2 4
2 1
1 2 1 4
2 1
2 4
1 4 2 1
2 2
输出
0
3
-1
3
2
3
-1
"""
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
# 1714 ms
def solve(n, m, a, b, qs):
INF = inf
st = ODT(0, 10 ** 10, INF)
ans = []
for q in qs:
if q[0] == 1:
x, y, k = q[1] - 1, q[2] - 1, q[3]
st.assign(y, y + k - 1, x - y)
elif q[0] == 2:
i = q[1] - 1
d = st.query_point(i)
if d < INF:
ans.append(a[i + d])
else:
ans.append((b[i]))
print('\n'.join(map(str, ans)))
if __name__ == '__main__':
import gc; # 注意gc不写会TLE
gc.disable()
n, m = RI()
a = RILST()
b = RILST()
q = []
for _ in range(m):
q.append(RILST())
solve(n, m, a, b, q)
1. 区间推平，区间询问最小值
例题: 715. Range 模块
本题实测，基于跳表不如基于SortedList，可能是数据弱吧。但是比线段树快一些。
这题当然可以线段树，套一下板就行。
珂朵莉树实测比线段树快3倍。
当然最佳方案是有序列表合并拆分线段。
import gc;gc.disable()
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 lgn """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
def query_min(self,l,r):
""" 查找x,y区间的最小值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return min(node.v for node in self.tree[begin:end])
class RangeModule:
def __init__(self):
self.tree = ODT(1,10**9,0)
def addRange(self, left: int, right: int) -> None:
self.tree.assign(left,right-1,1)
def queryRange(self, left: int, right: int) -> bool:
return 1 == self.tree.query_min(left,right-1)
def removeRange(self, left: int, right: int) -> None:
self.tree.assign(left,right-1,0)
2. 区间推平，区间查询最大值
链接: 729. 我的日程安排表 I
book是给区间全赋值1，区间操作前检查是否这个区间有非0的值，sum或者max都可以。
线段树也可以，珂朵莉树快一点，表现依然弱于SortedList。
import gc;gc.disable()
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 lgn """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
def query_min(self,l,r):
""" 查找x,y区间的最小值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return min(node.v for node in self.tree[begin:end])
def query_max(self,l,r):
""" 查找x,y区间的最大值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return max(node.v for node in self.tree[begin:end])
class MyCalendar:
def __init__(self):
self.odt = ODT(0,10**9,0)
def book(self, start: int, end: int) -> bool:
if self.odt.query_max(start,end-1) == 1:
return False
self.odt.assign(start,end-1,1)
return True
3. 不存在区间推平，只有区间加，区间询问最大值(不该用珂朵莉)
链接: 731. 我的日程安排表 II
这题没有assign本不该用珂朵莉做，但这个数据比较弱，确实能过，而且吊打线段树。
由于是求区间内一个超过1的数，因此可以区间查询时，提前退出。
![[python刷题模板] 珂朵莉树 ODT （基于支持随机访问的跳表_第2张图片](http://img.e-com-net.com/image/info8/7bccdc52556e4177a815f733193709f2.jpg)
import gc;gc.disable()
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 lgn """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
def query_min(self,l,r):
""" 查找x,y区间的最小值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return min(node.v for node in self.tree[begin:end])
def query_max(self,l,r):
""" 查找x,y区间的最大值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return max(node.v for node in self.tree[begin:end])
# 以下操作全是暴力，寄希望于这里边元素不多。
def add_interval(self,l,r,val):
"""区间挨个加
"""
tree = self.tree
begin = self.split(l)
end = self.split(r+1)
for i in range(begin,end):
tree[i].v += val
def query_has_greater_than(self,l,r,val):
"""
查找x,y区间是否有大于val的数
"""
begin = self.split(l)
end = self.split(r+1)
return any(node.v>val for node in self.tree[begin:end])
class MyCalendarTwo:
def __init__(self):
self.odt = ODT(0,10**9,0)
def book(self, start: int, end: int) -> bool:
if self.odt.query_has_greater_than(start,end-1,1) :
return False
self.odt.add_interval(start,end-1,1)
return True
5. 区间推平，区间询问最大值
链接: 699. 掉落的方块
优化维护最大值思路类似上题。
但是存在赋值，因此可以珂朵莉，实测效率和线段树差不多。
import gc;gc.disable()
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 lgn """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
# 以下操作全是暴力，寄希望于这里边元素不多。
def query_min(self,l,r):
""" 查找x,y区间的最小值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return min(node.v for node in self.tree[begin:end])
def query_max(self,l,r):
""" 查找x,y区间的最大值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return max(node.v for node in self.tree[begin:end])
def add_interval(self,l,r,val):
"""区间挨个加
"""
tree = self.tree
begin = self.split(l)
end = self.split(r+1)
for i in range(begin,end):
tree[i].v += val
def query_has_greater_than(self,l,r,val):
"""
查找x,y区间是否有大于val的数
"""
begin = self.split(l)
end = self.split(r+1)
return any(node.v>val for node in self.tree[begin:end])
class Solution:
def fallingSquares(self, positions: List[List[int]]) -> List[int]:
odt = ODT(1,10**9,0)
ans = []
m = 0
for l,d in positions:
h = odt.query_max(l,l+d-1)
odt.assign(l,l+d-1,h+d)
m = max(m,h+d)
ans.append(m)
return ans
6. 单点赋值，合并区间（不该用珂朵莉）
链接: 352. 将数据流变为多个不相交区间
实际上是线段合并。
注意合并时，访问了整个跳表里所有数据，注意写法是切片。
import gc;gc.disable()
MAX_LEVEL = 20 # 建议设置成lg(n),n是数据长度，20满足1e6,32满足1e10。由于跳表复杂度lgn，所以通常n<2e5,20就够
P_FACTOR = 0.25
def random_level() -> int:
lv = 1
while lv < MAX_LEVEL and random.random() < P_FACTOR:
lv += 1
return lv
class SkiplistNode:
__slots__ = 'val', 'forward', 'nb_len'
def __init__(self, val: int, max_level=MAX_LEVEL):
self.val = val
self.forward = [None] * max_level
self.nb_len = [1] * max_level
def __str__(self):
return str(f'{self.val},{self.forward},{self.nb_len}')
def __repr__(self):
return str(f'{self.val},,{self.nb_len}')
class Skiplist:
def __init__(self, nums=None):
self.head = SkiplistNode(-1)
self.level = 0
self._len = 0
if nums:
for i in nums:
self.add(i)
def __len__(self):
return self._len
def find(self, target: int) -> bool: # 判断target是否存在
cur = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
cur = cur.forward[i]
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
return cur is not None and cur.val == target
def index(self, target: int) -> int: # 在数据中找到target第一次出现的位置，找不到就返回-1
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while cur.forward[i] and cur.forward[i].val < target:
idx += cur.nb_len[i]
cur = cur.forward[i]
# idx -= 1
cur = cur.forward[0]
# 检测当前元素的值是否等于 target
if cur is None or cur.val != target:
return -1
return idx
def bisect_left(self, target) -> int: # 返回第一个插入位置
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_right(self, target) -> int: # 返回最后一个插入位置（本方法未经严格测试
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val <= target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx
def bisect_left_kvp(self, target): # 返回下标，当前下标的值，前一个下标的值
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 target 的元素
while curr.forward[i] and curr.forward[i].val < target:
idx += curr.nb_len[i]
curr = curr.forward[i]
return idx, curr.forward[0].val if idx < self._len else None, curr.val
def __getitem__(self, index):
if isinstance(index, slice):
start, stop, step = index.indices(self._len)
if step == 1 and start < stop:
ans = []
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= start:
idx += cur.nb_len[i]
cur = cur.forward[i]
p = start
while p < stop and cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
p += 1
return ans
if start < stop:
return self.__getitem__[slice(start, stop, 1)][index]
if stop < start:
return self.__getitem__[slice(start + 1, stop + 1, -step)][::-1]
else:
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range: {index}')
if index < 0:
index += self._len
cur = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当本层位置+前向长度小于目标位置时，在同层右移
while cur.forward[i] and idx + cur.nb_len[i] <= index:
idx += cur.nb_len[i]
cur = cur.forward[i]
return cur.forward[0].val
def __delitem__(self, index):
if index >= self._len or index + self._len < 0:
raise Exception(f'index out of range {index}')
if index < 0:
index += self._len
update = [None] * MAX_LEVEL
curr = self.head
idx = 0
for i in range(self.level - 1, -1, -1):
# 当同层位置+前向长度小于目标位置时，在同层右移
while curr.forward[i] and idx + curr.nb_len[i] <= index:
idx += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
def add(self, num) -> None:
self._len += 1
update = [self.head] * MAX_LEVEL
cur_idx = [0] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
if i < self.level - 1 and not cur_idx[i]:
cur_idx[i] = cur_idx[i + 1]
while curr.forward[i] and curr.forward[i].val < num:
# cur_idx[i] += curr.forward[i].nb_len[i]
cur_idx[i] += curr.nb_len[i]
curr = curr.forward[i]
update[i] = curr
idx = cur_idx[0] + 1
# print(cur_idx,idx)
lv = random_level()
self.level = max(self.level, lv)
new_node = SkiplistNode(num, lv)
for i in range(lv):
# 对第 i 层的状态进行更新，将当前元素的 forward 指向新的节点
old = update[i].nb_len[i] + 1
update[i].nb_len[i] = idx - cur_idx[i]
new_node.nb_len[i] = old - update[i].nb_len[i]
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
for i in range(lv, self.level):
update[i].nb_len[i] += 1
def discard(self, num: int) -> bool:
update = [None] * MAX_LEVEL
curr = self.head
for i in range(self.level - 1, -1, -1):
# 找到第 i 层小于且最接近 num 的元素
while curr.forward[i] and curr.forward[i].val < num:
curr = curr.forward[i]
update[i] = curr
curr = curr.forward[0]
if curr is None or curr.val != num: # 值不存在
return False
self._len -= 1
for i in range(self.level):
if update[i].forward[i] != curr:
update[i].nb_len[i] -= 1
# break
else:
# 对第 i 层的状态进行更新，将 forward 指向被删除节点的下一跳
update[i].nb_len[i] += curr.nb_len[i] - 1
update[i].forward[i] = curr.forward[i]
# 更新当前的 level
while self.level > 1 and self.head.forward[self.level - 1] is None:
self.level -= 1
return True
def __str__(self):
cur = self.head
ans = []
while cur.forward[0]:
ans.append(cur.forward[0].val)
cur = cur.forward[0]
return str(ans)
def __repr__(self):
return self.__str__()
class ODTNode:
__slots__ = ['l', 'r', 'v']
def __init__(self, l, r, v):
self.l, self.r, self.v = l, r, v
def __lt__(self, other):
return self.l < other.l
def jiebao(self):
return self.l, self.r, self.v
def __str__(self):
return str(self.jiebao())
def __repr__(self):
return str(self.jiebao())
class ODT:
def __init__(self, l=0, r=10 ** 9, v=0):
# 珂朵莉树，声明闭区间[l,r]上所有的值都是v
# l可以从0开始，r可以不建议大于1e18，因为py会变成大数运算
# 注意v赋初值
# 以后都用手撕的跳表做了
self.tree = Skiplist([ODTNode(l, r, v)])
def __str__(self):
return str(self.tree)
def split(self, pos):
""" 在pos位置切分，返回左边界l为pos的线段下标 lgn """
tree = self.tree
p, v1, v2 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v1.l == pos:
return p
l, r, v = v2.jiebao()
v2.r = pos - 1
# tree.pop(p)
# tree.add(ODTNode(l,pos-1,v))
tree.add(ODTNode(pos, r, v))
return p
def assign(self, l, r, v):
""" 把[l,r]区域全变成val lgn """
tree = self.tree
begin = self.split(l)
end = self.split(r + 1)
for _ in range(begin, end):
del tree[begin]
tree.add(ODTNode(l, r, v))
def query_point(self, pos):
""" 单点查询pos位置的值 lgn """
tree = self.tree
p, v, v1 = tree.bisect_left_kvp(ODTNode(pos, 0, 0))
if p != len(tree) and v.l == pos:
return v.v
return v1.v
# 以下操作全是暴力，寄希望于这里边元素不多。
def query_min(self,l,r):
""" 查找x,y区间的最小值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return min(node.v for node in self.tree[begin:end])
def query_max(self,l,r):
""" 查找x,y区间的最大值 暴力lgn+lgn+ lgnlgn(期望) """
begin = self.split(l)
end = self.split(r+1)
return max(node.v for node in self.tree[begin:end])
def add_interval(self,l,r,val):
"""区间挨个加
"""
tree = self.tree
begin = self.split(l)
end = self.split(r+1)
for i in range(begin,end):
tree[i].v += val
def query_has_greater_than(self,l,r,val):
"""
查找x,y区间是否有大于val的数
"""
begin = self.split(l)
end = self.split(r+1)
return any(node.v>val for node in self.tree[begin:end])
# 以下操作全是暴力，寄希望于这里边元素不多。
def query_all_intervals(self):
tree = self.tree
lines = []
l = r = -1
for node in tree[:] :
if node.v == 0:
if l != -1:
lines.append([l,r])
l = -1
else:
if l == -1:
l = node.l
r = node.r
return lines
class SummaryRanges:
def __init__(self):
self.odt = ODT(0,10**4+1,0)
def addNum(self, val: int) -> None:
self.odt.assign(val,val,1)
def getIntervals(self) -> List[List[int]]:
return self.odt.query_all_intervals()
# Your SummaryRanges object will be instantiated and called as such:
# obj = SummaryRanges()
# obj.addNum(val)
# param_2 = obj.getIntervals()
三、其他
- lc上的数据很弱，跳表写法不如SortedList，但CF不是。
四、更多例题
- 待补充
五、参考链接
- 链接: 知乎-算法学习笔记(15): 珂朵莉树