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272. 最长公共上升子序列

• 题意

如题

• 思路

若按这个思路的话，代码为$O(n^3)$

for (int i = 1; i <= n; i ++ )
{for (int j = 1; j <= n; j ++ ){f[i][j] = f[i - 1][j];if (a[i] == b[j]){int maxv = 1;for (int k = 1; k < j; k ++ )if (b[i] > b[k])maxv = max(maxv, f[i - 1][k] + 1);f[i][j] = max(f[i][j], maxv);}}
}

然后我们发现，b[i] > b[k] 中的 b[i] 可以替换为 a[i]，也就是求比a[i]小的b[k]的值有什么，这个地方就会被重复计算，因为a[i]是固定的，b中比a[i]小的值也是固定的
所以设置maxv是满足a[i] > b[k]的f[i - 1][k] + 1的 前缀最大值
也就是 if (a[i] > b[j]) maxv = max(maxv, f[i - 1][j]);
因此可以直接将maxv提到第一层循环外面，减少重复计算，此时只剩下两重循环。

最终答案枚举子序列结尾取最大值即可。

• 代码

'''
Author: NEFU AB-IN
Date: 2024-07-03 20:45:57
FilePath: \Acwing\272\272.py
LastEditTime: 2024-07-03 22:35:46
'''
# 3.8.19 import
from bisect import bisect_left, bisect_right
from collections import Counter, defaultdict, deque
from datetime import datetime, timedelta
from functools import lru_cache
from heapq import heapify, heappop, heappush, nlargest, nsmallest
from itertools import combinations, compress, permutations, starmap, tee
from math import ceil, fabs, floor, gcd, log, sqrt
from string import ascii_lowercase, ascii_uppercase
from sys import exit, setrecursionlimit, stdin, stdout
from typing import Any, Dict, Generic, List, TypeVar, UnionTYPE = TypeVar('TYPE')# Data structureclass SA(Generic[TYPE]):def __init__(self, x: TYPE, y: TYPE):self.x = xself.y = ydef __lt__(self, other: 'SA[TYPE]') -> bool:return self.x < other.xdef __eq__(self, other: 'SA[TYPE]') -> bool:return self.x == other.x and self.y == other.ydef __repr__(self) -> str:return f'SA(x={self.x}, y={self.y})'# Constants
N = int(2e5 + 10)  # If using AR, modify accordingly
M = int(20)  # If using AR, modify accordingly
INF = int(2e9)
OFFSET = int(100)# Set recursion limit
setrecursionlimit(INF)# Readdef input(): return stdin.readline().rstrip("\r\n")  # Remove when Mutiple data
def read(): return map(int, input().split())
def read_list(): return list(read())# Func
class std:letter_to_num = staticmethod(lambda x: ord(x.upper()) - 65)  # A -> 0num_to_letter = staticmethod(lambda x: ascii_uppercase[x])  # 0 -> Aarray = staticmethod(lambda x=0, size=N: [x] * size)array2d = staticmethod(lambda x=0, rows=N, cols=M: [std.array(x, cols) for _ in range(rows)])graph = staticmethod(lambda size=N: [[] for _ in range(size)])max = staticmethod(lambda a, b: a if a > b else b)min = staticmethod(lambda a, b: a if a < b else b)removeprefix = staticmethod(lambda s, prefix: s[len(prefix):] if s.startswith(prefix) else s)removesuffix = staticmethod(lambda s, suffix: s[:-len(suffix)] if s.endswith(suffix) else s)@staticmethoddef find(container: Union[List[TYPE], str], value: TYPE):"""Returns the index of value in container or -1 if value is not found."""if isinstance(container, list):try:return container.index(value)except ValueError:return -1elif isinstance(container, str):return container.find(value)@staticmethoddef pairwise(iterable):"""Return successive overlapping pairs taken from the input iterable."""a, b = tee(iterable)next(b, None)return zip(a, b)# ————————————————————— Division line ——————————————————————n, = read()
a = read_list()
b = read_list()dp = std.array2d(0, n, n)for i in range(n):dp[0][i] = 1 if b[i] == a[0] else 0for i in range(n):dp[i][0] = 1 if b[0] == a[i] else 0for i in range(1, n):mx = 1for j in range(1, n):dp[i][j] = dp[i - 1][j]if a[i] == b[j]:dp[i][j] = std.max(dp[i][j], mx)if a[i] > b[j]:mx = std.max(mx, dp[i - 1][j] + 1)res = 0
for i in range(n):res = std.max(dp[n - 1][i], res)print(res)