A company has n employees with a unique ID for each employee from 0 to n - 1. The head of the company has is the one with headID.
Each employee has one direct manager given in the manager array where manager[i] is the direct manager of the i-th employee, manager[headID] = -1. Also it's guaranteed that the subordination relationships have a tree structure.
The head of the company wants to inform all the employees of the company of an urgent piece of news. He will inform his direct subordinates and they will inform their subordinates and so on until all employees know about the urgent news.
The i-th employee needs informTime[i] minutes to inform all of his direct subordinates (i.e After informTime[i] minutes, all his direct subordinates can start spreading the news).
Return the number of minutes needed to inform all the employees about the urgent news.
 
Example 1:
Input: n = 1, headID = 0, manager = [-1], informTime = [0]
Output: 0
Explanation: The head of the company is the only employee in the company.

Example 2:

Input: n = 6, headID = 2, manager = [2,2,-1,2,2,2], informTime = [0,0,1,0,0,0]
Output: 1
Explanation: The head of the company with id = 2 is the direct manager of all the employees in the company and needs 1 minute to inform them all.
The tree structure of the employees in the company is shown.

Example 3:

Input: n = 7, headID = 6, manager = [1,2,3,4,5,6,-1], informTime = [0,6,5,4,3,2,1]
Output: 21
Explanation: The head has id = 6. He will inform employee with id = 5 in 1 minute.
The employee with id = 5 will inform the employee with id = 4 in 2 minutes.
The employee with id = 4 will inform the employee with id = 3 in 3 minutes.
The employee with id = 3 will inform the employee with id = 2 in 4 minutes.
The employee with id = 2 will inform the employee with id = 1 in 5 minutes.
The employee with id = 1 will inform the employee with id = 0 in 6 minutes.
Needed time = 1 + 2 + 3 + 4 + 5 + 6 = 21.

Example 4:
Input: n = 15, headID = 0, manager = [-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6], informTime = [1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
Output: 3
Explanation: The first minute the head will inform employees 1 and 2.
The second minute they will inform employees 3, 4, 5 and 6.
The third minute they will inform the rest of employees.

Example 5:
Input: n = 4, headID = 2, manager = [3,3,-1,2], informTime = [0,0,162,914]
Output: 1076

 
Constraints:

1 <= n <= 10^5
0 <= headID < n
manager.length == n
0 <= manager[i] < n
manager[headID] == -1
informTime.length == n
0 <= informTime[i] <= 1000
informTime[i] == 0 if employee i has no subordinates.
It is guaranteed that all the employees can be informed.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "# class Node:\n#     def __init__(self, id: int, inform_t: int):\n#         self.id = id\n#         self.children = []\n#         self.inform_t = inform_t\n    \n#     def __repr__(self):\n#         return f'<{self.id} ({self.inform_t}): {len(self.children)}>'\n    \n#     def total_inform_time(self):\n#         t = self.inform_t\n#         if self.children:\n#             t += max(ch.total_inform_time() for ch in self.children)\n#         return t\n\n# class Solution:\n    \n#     def build_tree(self, manager: List[int], inform_time: List[int]) -> Node:\n#         nodes = [None] * len(manager)\n#         root = None\n#         for id, man_id in enumerate(manager):\n#             if nodes[id] is None:\n#                 nodes[id] = Node(id, inform_time[id])\n            \n#             if man_id >= 0:\n#                 if nodes[man_id] is None:\n#                     nodes[man_id] = Node(man_id, inform_time[man_id])\n\n#                 nodes[man_id].children.append(nodes[id])\n#             else:\n#                 root = nodes[id]\n#         return root\n    \n    \n#     def numOfMinutes(self, n: int, head_id: int, manager: List[int], inform_time: List[int]) -> int:\n#         root = self.build_tree(manager, inform_time)\n#         return root.total_inform_time()\n    \n\n# Top down\n# class Solution:\n#     def numOfMinutes(self, n: int, head_id: int, manager: List[int], inform_time: List[int]) -> int:\n#         children = [[] for _ in range(n)]\n#         for id, man_id in enumerate(manager):\n#             if man_id >= 0:\n#                 children[man_id].append(id)\n        \n#         def dfs(id):\n#             t = inform_time[id]\n#             if children[id]:\n#                 t += max(dfs(ch) for ch in children[id])\n#             return t\n        \n#         return dfs(head_id)\n \n    \n# Bottom up\n# class Solution:\n#     def numOfMinutes(self, n: int, head_id: int, manager: List[int], inform_time: List[int]) -> int:\n        \n#         def dfs(id):\n#             if manager[id] >= 0:\n#                 inform_time[id] += dfs(manager[id])\n#                 manager[id] = -1\n#             return inform_time[id]\n        \n#         return max(dfs(id) for id in range(n))\n\n\n# Bottom up with cache\nclass Solution:\n    def numOfMinutes(self, n: int, head_id: int, manager: List[int], inform_time: List[int]) -> int:\n        cache = [None] * n\n        \n        def dfs(id):\n            if cache[id]:\n                return cache[id]\n            \n            t = inform_time[id]\n            if manager[id] >= 0:\n                t += dfs(manager[id])\n            \n            cache[id] = t\n            return t\n        \n        return max(dfs(id) for id in range(n))\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numOfMinutes'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6378