結果

問題 No.977 アリス仕掛けの摩天楼
ユーザー None
提出日時 2021-03-13 17:34:54
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 542 ms / 2,000 ms
コード長 7,979 bytes
コンパイル時間 270 ms
コンパイル使用メモリ 82,388 KB
実行使用メモリ 114,436 KB
最終ジャッジ日時 2024-10-15 08:12:42
合計ジャッジ時間 6,601 ms
ジャッジサーバーID
（参考情報） 		judge4 / judge3
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ファイルパターン 結果
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ソースコード

diff # 

#####################################################################################################
##### Low Link (二重辺連結成分分解)
#####################################################################################################

"""

計算量
O(V+E)

二重辺連結成分分解
閉路のリストに分解する

参考
https://algo-logic.info/bridge-lowlink/

ベンチマーク
https://atcoder.jp/contests/abc075/submissions/15110847

"""

class Graph:
    def __init__(self, n, directed=False, decrement=True, destroy=False, edges=[]):
        self.n = n
        self.directed = directed
        self.decrement = decrement
        self.destroy = destroy
        self.edges = [[] for _ in range(self.n)]
        self.deg = [0] * n
        for x, y in edges:
            self.add_edge(x,y)

    def add_edge(self, u, v):
        self.edges[u - self.decrement].append(v - self.decrement)
        self.edges[v - self.decrement].append(u - self.decrement)
        self.deg[u - self.decrement] += 1
        self.deg[v - self.decrement] += 1

    def get_multi_edge(self):
        res = set()
        for i, g in enumerate(self.edges):
            tmp = set((i, v) for v in set(g) if g.count(v) > 1)
            res |= tmp
        return res

    def bfs(self, s, t=None):
        dist = [None] * self.n
        dist[s] = 0
        prev = [None] * self.n
        queue = deque([s])
        while queue:
            v = queue.popleft()
            for adj in self.edges[v]:
                if dist[adj] is not None:continue
                dist[adj] = dist[v] + 1
                queue.append(adj)
        if t is None: return dist
        path = [t]
        while t != s:
            t = prev[t]
            path.append(t)
        return dist, path[::-1]

    def lowlink(self):
        self.ord = [None] * self.n
        self.low = [None] * self.n
        self.par = [None] * self.n
        self.tree = [[] for _ in range(self.n)]
        group = [None] * self.n
        medge = self.get_multi_edge()
        vis = []
        ord = 0
        cnt = 0
        for s in range(self.n):
            if group[s] is not None: continue
            stack = [(s, None)]
            while stack:
                v, p = stack.pop()
                if group[v] is not None: continue
                self.ord[v] = ord
                group[v] = cnt
                vis.append(v)
                ord += 1
                if p is not None:
                    self.par[v] = p
                    self.tree[p].append(v)
                for adj in self.edges[v]:
                    if group[adj] is not None: continue
                    stack.append((adj, v))
            cnt += 1
        for v in vis[::-1]:
            self.low[v] = self.ord[v]
            for adj in self.edges[v]:
                if self.par[adj] == v:
                    self.low[v] = min(self.low[v], self.low[adj])
                elif self.par[v] == adj and (v, adj) not in medge:
                    continue
                else:
                    self.low[v] = min(self.low[v], self.ord[adj])

    def articulation(self):
        res = []
        idx=self.decrement
        for v in range(self.n):
            if self.par[v] is None and len(self.tree[v]) > 1:
                res.append(v)
            if self.par[v] is None: continue
            for adj in self.tree[v]:
                if self.ord[v] <= self.low[adj]:
                    res.append(v+idx)
                    break
        return res

    def bridge(self):
        res = []
        idx=self.decrement
        for v in range(self.n):
            for adj in self.tree[v]:
                if self.ord[v] < self.low[adj]:
                    res.append((v+idx, adj+idx))
        return res

    def two_edge_connected(self, bridge):
        """ decompose graph into list of cycles """
        idx=self.decrement
        bridge = set(bridge)
        group = [None] * self.n
        cnt = 0
        for s in range(self.n):
            if group[s] is not None: continue
            group[s] = cnt
            stack = [s]
            while stack:
                v = stack.pop()
                for adj in self.edges[v]:
                    if group[adj] is not None or (v+idx, adj+idx) in bridge or (adj+idx, v+idx) in bridge:
                        continue
                    group[adj] = cnt
                    stack.append(adj)
            cnt += 1
        res = [[] for _ in range(cnt)]
        for i, g in enumerate(group):
            res[g].append(i+idx)
        return res

    def draw(self):
        """
        :return: グラフを可視化
        """
        import matplotlib.pyplot as plt
        import networkx as nx

        if self.directed:
            G = nx.DiGraph()
        else:
            G = nx.Graph()
        for x in range(self.n):
            for y in self.edges[x]:
                G.add_edge(x + self.decrement, y + self.decrement)

        pos = nx.spring_layout(G)
        nx.draw_networkx(G, pos, connectionstyle='arc3, rad = 0.1')
        plt.axis("off")
        plt.show()

class UnionFind:
    def __init__(self, n):
        self.n = n
        self.parents = [-1] * n

    def find(self, x):
        """ 根を見つける関数を定義(同時にxを直接根にくっつける操作も行う)"""
        tmp = []
        parents = self.parents
        while parents[x] >= 0:
            tmp.append(x)
            x = parents[x]
        for y in tmp:
            parents[y] = x
        return x

    def union(self, x, y):
        """ 二つの木をくっつける(子を多く持つ方を根とした親子関係)。これは破壊的操作を行う。"""
        x = self.find(x)
        y = self.find(y)
        if x == y:
            return
        if self.parents[x] > self.parents[y]:
            x, y = y, x
        self.parents[x] += self.parents[y]
        self.parents[y] = x

    def same(self, x, y):
        """ xとyが同じ根の子かを判定 """
        return self.find(x) == self.find(y)

    def size(self, x):
        """ xの根のparent(= 要素数)を返す """
        return -self.parents[self.find(x)]

    def members(self, x):
        """ xが属するグループの要素をリストとして返す O(N)"""
        root = self.find(x)
        return [i for i in range(self.n) if self.find(i) == root]

    def roots(self):
        """ 全ての根の要素をリストとして返す O(N)"""
        return [i for i, x in enumerate(self.parents) if x < 0]

    def group_count(self):
        """ グループの数を返す O(N)"""
        return len(self.roots())

    def size_list(self):
        """ 各グループの要素数のリストを返す(根の番号は返さない) O(N)"""
        return [-x for x in self.parents if x < 0]

    def all_group_members(self):
        """ {根:[根の子(根を含む)のリスト],...}を辞書で返す O(N)"""
        res = [[] for _ in range(self.n)]
        for i in range(self.n):
            x = self.find(i)
            res[x].append(i)
        return {r: res[r] for r in self.roots()}

    def __str__(self):
        return '\n'.join('{}: {}'.format(r, self.members(r)) for r in self.roots())



###############################################################################################
def example():
    global input
    example = iter(
        """
6
1 2
2 3
3 1
4 5
5 6

        """
            .strip().split("\n"))
    input = lambda: next(example)
###############################################################################################
import sys
input = sys.stdin.readline
from collections import deque

# example()

N=int(input())
graph = Graph(N, directed=False, decrement=False, destroy=False)
U=UnionFind(N)
for _ in range(N-1):
    x, y = map(int, input().split())
    graph.add_edge(x, y)
    U.union(x,y)

graph.lowlink()

island=U.group_count()
bridge = len(graph.bridge())


if bridge>=1:
    bridge-=1
    island+=1
island-=1

print("Alice" if island>1 else "Bob")
0