플로이드-워셜: 모든 쌍의 최단 경로

모든 쌍 최단 경로

핵심 아이디어: 중간 노드

거리 행렬 초기화

def floyd_warshall(V, edges):
    INF = float('inf')
    dist = [[INF]*V for _ in range(V)]
    for i in range(V):
        dist[i][i] = 0
    for u, v, w in edges:
        dist[u][v] = w  # directed graph
    
    for k in range(V):       # intermediate node
        for i in range(V):
            for j in range(V):
                if dist[i][k] + dist[k][j] < dist[i][j]:
                    dist[i][j] = dist[i][k] + dist[k][j]
    
    return dist

예제를 포함한 전체 구현

def floyd_warshall(V, edges):
    INF = float('inf')
    dist = [[INF]*V for _ in range(V)]
    for i in range(V):
        dist[i][i] = 0
    for u, v, w in edges:
        dist[u][v] = w
    for k in range(V):
        for i in range(V):
            for j in range(V):
                if dist[i][k] != INF and dist[k][j] != INF:
                    dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
    return dist

V = 4
edges = [(0,1,3),(0,2,7),(1,2,1),(1,3,5),(2,3,2)]
dist = floyd_warshall(V, edges)
for row in dist:
    print([x if x != float('inf') else 'INF' for x in row])

음수 사이클 감지

def has_negative_cycle_fw(V, edges):
    dist = floyd_warshall(V, edges)
    for i in range(V):
        if dist[i][i] < 0:
            return True  # negative cycle through node i
    return False

# Negative cycle: 0->1->2->0 with weights 1,-3,1 (sum=-1)
edges_neg = [(0,1,1),(1,2,-3),(2,0,1)]
print(has_negative_cycle_fw(3, edges_neg))  # True

경로 복원

def fw_with_path(V, edges):
    INF = float('inf')
    dist = [[INF]*V for _ in range(V)]
    nxt = [[None]*V for _ in range(V)]
    for i in range(V): dist[i][i] = 0
    for u, v, w in edges:
        dist[u][v] = w; nxt[u][v] = v
    for k in range(V):
        for i in range(V):
            for j in range(V):
                if dist[i][k] + dist[k][j] < dist[i][j]:
                    dist[i][j] = dist[i][k] + dist[k][j]
                    nxt[i][j] = nxt[i][k]
    return dist, nxt

def get_path(nxt, i, j):
    if nxt[i][j] is None: return []
    path = [i]
    while i != j:
        i = nxt[i][j]; path.append(i)
    return path

전이 폐쇄

def transitive_closure(V, edges):
    reach = [[False]*V for _ in range(V)]
    for i in range(V):
        reach[i][i] = True
    for u, v, _ in edges:
        reach[u][v] = True
    for k in range(V):
        for i in range(V):
            for j in range(V):
                reach[i][j] = reach[i][j] or (reach[i][k] and reach[k][j])
    return reach

edges = [(0,1,1),(1,2,1)]
R = transitive_closure(3, edges)
print(R[0][2])  # True (0 can reach 2 via 0->1->2)

복잡도와 사용 시점

모든 정점 쌍 사이의 최소 홉 수

def min_hops_all_pairs(V, adj_list):
    INF = float('inf')
    dist = [[INF]*V for _ in range(V)]
    for i in range(V):
        dist[i][i] = 0
        for j in adj_list[i]:
            dist[i][j] = 1
    for k in range(V):
        for i in range(V):
            for j in range(V):
                dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
    return dist

adj = [[1,2],[2],[3],[],[]]
print(min_hops_all_pairs(5, adj)[0])  # [0, 1, 1, 2, INF]

면접 맥락: 면접관이 플로이드-워셜에 관해 질문할 때

무방향 그래프에서의 플로이드-워셜

def fw_undirected(V, edges):
    INF = float('inf')
    dist = [[INF]*V for _ in range(V)]
    for i in range(V): dist[i][i] = 0
    for u, v, w in edges:
        dist[u][v] = w
        dist[v][u] = w  # both directions for undirected
    for k in range(V):
        for i in range(V):
            for j in range(V):
                if dist[i][k] + dist[k][j] < dist[i][j]:
                    dist[i][j] = dist[i][k] + dist[k][j]
    return dist

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