Using Dijkstra’s Algorithm to Find All Shortest Paths in a Graph

I always had trouble spelling Dijkstra correctly until someone pointed out “It’s just D, followed by i-j-k, then stra”. Once it was pointed out that i-j-k is represented sequentially in alphabetical order, spelling Dijkstra becomes trival - almost fun!

Dijkstra’s algorithm is one of the most well-known and studied graph algorithms for finding the shortest path between a set of vertices. For a specified starting node, the algorithm finds the shortest path between the source vertex and all other vertices in the graph. Dijkstra’s cannot handle graphs with negative edge weights: For graphs with negative weight edges, Floyd-Warshall or Bellman Ford can be used. But for graphs with positive edge weights, Dijkstra’s has better worst case performance than more general alternatives (O((n + m)log(n)) for Dijkstra vs. O(mn) for Bellman-Ford and O(n^3) for Floyd-Warshall).

However, Dijkstra’s only returns a single shortest path. In many cases a single shortest path is enough. But there are plenty of applications in which knowledge of alternate minimum weight paths can be useful. If a graph has multiple shortest paths, it is necessary to add supplementary logic to identify and return all such paths. The modified routine to return all shortest paths from graph G is described below:

1. Run Dijkstra’s starting from start node a. Returns shortest path and distance to every node from vertex a.

2. Run Dijkstra’s starting from end node f. Returns shortest path and distance to every node from vertex f.

3. Let dist_a2f = A shortest path from vertex a to vertex f.

4. For (u, v) in G.edges:

   Let w_uv = Weight of edge (u, v).

   Let dist_a2u = Distance from start vertex a to vertex u.

   Let dist_f2v = Distance from end vertex f to vertex v.

   If dist_a2u + w_uv + dist_f2v == dist_a2f:

   Then edge (u, v) is on some minimal weight path.

Lets use networkx to create a graph with multiple shortest paths:

# Create network with multiple minimal weight paths.
import matplotlib.pyplot as plt
import networkx as nx

seed = 518
G = nx.Graph()

G.add_edge("a", "b", weight=4)
G.add_edge("b", "c", weight=2)
G.add_edge("a", "c", weight=6)
G.add_edge("c", "d", weight=1)
G.add_edge("d", "e", weight=3)
G.add_edge("e", "f", weight=2)
G.add_edge("c", "f", weight=6)
G.add_edge("a", "d", weight=9)
G.add_edge("f", "d", weight=10)

edges = [(u, v) for (u, v, d) in G.edges(data=True)]

color_map = []
for g in G:
    if g=="a":
        color_map.append("green")
    elif g=="f":
        color_map.append("red")
    else:
        color_map.append("blue")


# edge weight labels.
edge_labels = nx.get_edge_attributes(G, "weight")


fig, ax = plt.subplots(1, 1, figsize=(8, 6), tight_layout=True)
pos = nx.spring_layout(G, seed=seed)
nx.draw_networkx_nodes(G, pos, node_size=800, node_color=color_map)
nx.draw_networkx_edges(G, pos, edgelist=edges, width=2, alpha=0.5, edge_color="grey", style="solid")
nx.draw_networkx_labels(G, pos, font_size=16, font_color="white", font_weight="bold", font_family="sans-serif")
nx.draw_networkx_edge_labels(G, pos, edge_labels, font_size=12)
plt.axis("off")
plt.show();

G has three shortest paths of weight/distance 12:

• a - c - f
• a - b - c - f
• a - c - d - e - f

Running single_source_dijkstra from networkx returns the distance to each vertex in a graph from a specified starting vertex (in our example, vertex a). single_source_dijkstra returns two dictionaries: The first stores distances to each vertex; the second stores the shortest path from the start node to all other vertices (d2v0 and p2v0 in the code below):

d2v0, p2v0 = nx.single_source_dijkstra(G, "a")
print(f"d2v0: {d2v0}")
print(f"p2v0: {p2v0}")

d2v0: {'a': 0, 'b': 4, 'c': 6, 'd': 7, 'e': 10, 'f': 12}
p2v0: {'a': ['a'], 'b': ['a', 'b'], 'c': ['a', 'c'], 'd': ['a', 'c', 'd'], 'f': ['a', 'c', 'f'], 'e': ['a', 'c', 'd', 'e']}

single_source_dijkstra only returns a - c - f. To also identify a - b - c - f and a - c - d - e - f, we implement the steps from our pseudocode:

# Idenitfy all shortests paths in graph G.
all_shortest_paths = set()

# d2v: Distance to vertex (from start node a).
# p2v: Shortest path to vertext (from start node a).
d2v0, p2v0 = nx.single_source_dijkstra(G, "a")
d2v1, p2v1 = nx.single_source_dijkstra(G, "f")
all_shortest_paths.add(tuple(p2v0["f"]))

# Iterate through all edges. Check for additional shortest paths. 

# Distance of a shortest path from a to f. 
dist_a2f = d2v0["f"]

for u, v, dw in G.edges(data=True):
    
    # Get weight of current edge spanning (u, v)
    w_uv = dw["weight"]
    
    # Get distance from start vertex to u.
    dist_a2u = d2v0[u]
    
    # Get distance from end vertex to v.
    dist_f2v = d2v1[v]
    
    if dist_a2u + w_uv + dist_f2v == dist_a2f:
        
        # Edge uv is on some minimal weight path. Append to all_shortest_paths.
        pp = tuple(p2v0[u] + p2v1[v][::-1])
        all_shortest_paths.add(pp)

Shortest paths are stored in a set of ordered tuples, with each tuple representing the vertices of some path of minimum weight. Printing the contents of all_shortest_paths yields:

all_shortest_paths

{('a', 'b', 'c', 'f'), ('a', 'c', 'd', 'e', 'f'), ('a', 'c', 'f')}

The time complexity for two separate invocations of Dijkstra’s is 2 * O((n + m)log(n)), and O(m + n) to iterate through the adjacency list checking for intermediate edges falling on a path of minimum weight.