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| '''
Author: NEFU AB-IN
Date: 2024-08-15 23:57:49
FilePath: \GPLT\A1087\A1087.py
LastEditTime: 2024-08-17 21:06:27
'''
import random
from collections import Counter, defaultdict, deque
from datetime import datetime, timedelta
from functools import lru_cache, reduce
from heapq import heapify, heappop, heappush, nlargest, nsmallest
from itertools import combinations, compress, permutations, starmap, tee
from math import ceil, comb, fabs, floor, gcd, hypot, log, perm, sqrt
from string import ascii_lowercase, ascii_uppercase
from sys import exit, setrecursionlimit, stdin
from typing import Any, Callable, Dict, List, Optional, Tuple, TypeVar, Union
TYPE = TypeVar('TYPE')
N = int(2e5 + 10)
M = int(20)
INF = int(1e12)
OFFSET = int(100)
MOD = int(1e9 + 7)
setrecursionlimit(int(2e9))
class Arr:
array = staticmethod(lambda x=0, size=N: [x() if callable(x) else x for _ in range(size)])
array2d = staticmethod(lambda x=0, rows=N, cols=M: [Arr.array(x, cols) for _ in range(rows)])
graph = staticmethod(lambda size=N: [[] for _ in range(size)])
class Math:
max = staticmethod(lambda a, b: a if a > b else b)
min = staticmethod(lambda a, b: a if a < b else b)
class IO:
input = staticmethod(lambda: stdin.readline().strip())
read = staticmethod(lambda: map(int, IO.input().split()))
read_list = staticmethod(lambda: list(IO.read()))
read_mixed = staticmethod(lambda *types: [t(v) for t, v in zip(types, IO.input().split())])
class Std:
class Dijkstra:
"""Dijkstra's algorithm for finding the shortest path in a weighted graph, designed to compute various properties related to the shortest paths from a source node."""
def __init__(self, n: int, val_: List):
self.n = n
self.val_ = val_
self.g_ = Arr.graph(n)
self.dist_ = Arr.array(INF, n)
self.sum_ = Arr.array(0, n)
self.cnt_ = Arr.array(0, n)
self.pre_ = Arr.array(0, n)
self.num_ = Arr.array(0, n)
def add_edge(self, u: int, v: int, w: int):
"""Add an edge to the graph."""
self.g_[u].append((v, w))
def dijkstra(self, s: int):
"""Dijkstra's algorithm for finding the shortest path in a graph.
This method calculates the shortest distances, maximizes the sum of node values (`val_`) along the paths, and minimizes the number of edges (`num_`) used in the paths. Additionally, it counts the number of distinct shortest paths to each node."""
st_ = Arr.array(0, self.n)
q = []
self.dist_[s] = 0
self.sum_[s] = self.val_[s]
self.cnt_[s] = self.num_[s] = 1
heappush(q, (0, s))
while q:
_, u = heappop(q)
if st_[u]:
continue
st_[u] = 1
for v, w in self.g_[u]:
if self.dist_[v] > self.dist_[u] + w:
self.dist_[v] = self.dist_[u] + w
self.pre_[v] = u
self.sum_[v] = self.sum_[u] + self.val_[v]
self.num_[v] = self.num_[u] + 1
self.cnt_[v] = self.cnt_[u]
heappush(q, (self.dist_[v], v))
elif self.dist_[v] == self.dist_[u] + w:
self.cnt_[v] += self.cnt_[u]
if self.sum_[u] > self.sum_[self.pre_[v]]:
self.pre_[v] = u
self.sum_[v] = self.sum_[u] + self.val_[v]
self.num_[v] = self.num_[u] + 1
heappush(q, (self.dist_[v], v))
elif self.sum_[u] == self.sum_[self.pre_[v]] and self.num_[v] > self.num_[u] + 1:
self.pre_[v] = u
self.num_[v] = self.num_[u] + 1
heappush(q, (self.dist_[v], v))
class TrieNodeGraph:
"""TrieNode class can convert each string into an integer identifier, useful in graph theory."""
_sid_cnt = 0
_sid_to_word_ = {}
def __init__(self):
"""Initialize children dictionary and cost. The trie tree is a 26-ary tree."""
self._children_ = {}
self._is_end_of_word = False
self._sid = -1
def add(self, word: str) -> int:
"""Add a word to the trie and return a unique ID."""
node = self
for c in word:
if c not in node._children_:
node._children_[c] = Std.TrieNodeGraph()
node = node._children_[c]
node._is_end_of_word = True
if node._sid < 0:
node._sid = self._sid_cnt
self._sid_cnt += 1
self._sid_to_word_[node._sid] = word
return node._sid
def _search(self, word: str) -> int:
"""Search for the exact word in the trie and return its unique ID, else -1."""
node = self
for c in word:
if c not in node._children_:
return -1
node = node._children_[c]
return node._sid if node._is_end_of_word else -1
def get_id(self, word: str) -> int:
"""Retrieve the unique ID for a given word."""
return self._search(word)
def get_str(self, sid: int) -> str:
"""Retrieve the original string associated with a given unique ID."""
return word if (word := self._sid_to_word_.get(sid)) else "-1"
n, k, st = IO.read_mixed(int, int, str)
trie = Std.TrieNodeGraph()
dj = Std.Dijkstra(n, Arr.array(0, n))
trie.add(st)
for i in range(n - 1):
u, d = IO.read_mixed(str, int)
index = trie.add(u)
dj.val_[index] = d
for i in range(k):
u, v, d = IO.read_mixed(str, str, int)
dj.add_edge(trie.get_id(u), trie.get_id(v), d)
dj.add_edge(trie.get_id(v), trie.get_id(u), d)
dj.dijkstra(trie.get_id(st))
ed_id = trie.get_id("ROM")
print(dj.cnt_[ed_id], dj.dist_[ed_id], dj.sum_[ed_id], dj.sum_[ed_id] // (dj.num_[ed_id] - 1))
stack_ = []
while ed_id != dj.pre_[ed_id]:
stack_.append(ed_id)
ed_id = dj.pre_[ed_id]
print(st, end="")
while stack_:
print(f"->{trie.get_str(stack_.pop())}", end="")
|