GRL_1_A - Single Shortest Path / PythonでAizu Online Judgeを解く

    Posted on 2018/10/08

    概要

    重み付きの有向グラフ \(G(V, E)\) が与えられます。頂点 \(r\) を始点としたときの各頂点までの最短経路の長さを出力してください。

    制約

    • \(1 \leq |V| \leq 100000\)
    • \(0 \leq d_i \leq 10000\)
    • \(0 \leq |E| \leq 500000\)
    • \(G\)に平行辺はない
    • \(G\)に自己ループはない

    Input

    %python
    def input(f):
        global v, e, r, s, t, d, g
        
        v, e, r = map(int, f.readline().split())
        s = []
        t = []
        d = []
        for _ in range(e):
            s_, t_, d_ = map(int, f.readline().split())
            s.append(s_)
            t.append(t_)
            d.append(d_)
            
        g = [list() for _ in range(v)]
        for i in range(e):
            g[s[i]].append(i)

    Example Input

    %sh
    cat << EOF > /tmp/input1
    4 5 0
    0 1 1
    0 2 4
    1 2 2
    2 3 1
    1 3 5
    EOF
    
    cat << EOF > /tmp/input2
    4 6 1
    0 1 1
    0 2 4
    2 0 1
    1 2 2
    3 1 1
    3 2 5
    EOF

    Python: networkxで可視化してみる

    %sh
    pip install networkx
    Collecting networkx
      Downloading https://files.pythonhosted.org/packages/f3/f4/7e20ef40b118478191cec0b58c3192f822cace858c19505c7670961b76b2/networkx-2.2.zip (1.7MB)
    Collecting decorator>=4.3.0 (from networkx)
      Downloading https://files.pythonhosted.org/packages/bc/bb/a24838832ba35baf52f32ab1a49b906b5f82fb7c76b2f6a7e35e140bac30/decorator-4.3.0-py2.py3-none-any.whl
    Building wheels for collected packages: networkx
      Running setup.py bdist_wheel for networkx: started
      Running setup.py bdist_wheel for networkx: finished with status 'done'
      Stored in directory: /root/.cache/pip/wheels/68/f8/29/b53346a112a07d30a5a84d53f19aeadaa1a474897c0423af91
    Successfully built networkx
    Installing collected packages: decorator, networkx
    Successfully installed decorator-4.3.0 networkx-2.2
    You are using pip version 9.0.1, however version 18.1 is available.
    You should consider upgrading via the 'pip install --upgrade pip' command.
    

    %python
    import os
    
    if os.environ.get('ZEPPELIN_HOME'):
        import networkx as nx
        import matplotlib.pyplot as plt
        
        def visualize():
            G = nx.DiGraph()
            for i in range(v):
                G.add_node(i)
            for i in range(e):
                G.add_edge(s[i], t[i], t=d[i])
            
            plt.figure()
            plt.style.use('seaborn')
            plt.axis("off")
            pos = nx.spring_layout(G)
            nodes = nx.draw_networkx_nodes(G, pos, node_color='w', linewidths=2)
            nx.draw_networkx_labels(G, pos)
            nodes.set_edgecolor('black')
            nx.draw_networkx_edges(G, pos)
            nx.draw_networkx_edge_labels(G, pos)
            plt.show()

    %python
    if os.environ.get('ZEPPELIN_HOME'):
        with open('/tmp/input1') as f:
            print('Example #1')
            input(f)
            visualize()
    Example #1
    

    %python
    if os.environ.get('ZEPPELIN_HOME'):
        with open('/tmp/input2') as f:
            print('Example #2')
            input(f)
            visualize()
    Example #2
    

    考え方

    隣接する頂点について幅優先探索していきます。Pythonではcollectionsモジュールのdequeを使うと良さそうでした。

    %python
    import sys
    import collections
    
    def solve():
        inf = sys.maxint
        q = collections.deque()
        q.append(r)
        mincost = [inf]*v
        mincost[r] = 0
        while len(q) > 0:
            cur = q.popleft()
            for e in g[cur]:
                if mincost[t[e]] == inf or mincost[t[e]] > mincost[cur] + d[e]:
                    mincost[t[e]] = mincost[cur] + d[e]
                    q.append(t[e])
        
        for i in range(v):
            if mincost[i] == inf:
                print('INF')
            else:
                print(mincost[i])

    Example Output #1

    %python
    if os.environ.get('ZEPPELIN_HOME'):
        os.environ['INPUT'] = '/tmp/input1'
    
    with open(os.environ.get('INPUT') or '/dev/stdin') as f:
        input(f)
        solve()
    0
    1
    3
    4
    

    Example Output #2

    %python
    if os.environ.get('ZEPPELIN_HOME'):
        os.environ['INPUT'] = '/tmp/input2'
    
        with open(os.environ.get('INPUT') or '/dev/stdin') as f:
            input(f)
            solve()
    3
    0
    2
    INF