Find the shortest path with the least number of edges

Finding the Shortest Path with the Fewest Edges

Explore algorithms to determine the shortest path between two nodes in a graph, prioritizing paths with the minimum number of edges when multiple shortest paths exist.

In graph theory, finding the shortest path between two nodes is a fundamental problem with applications ranging from network routing to logistics. While algorithms like Dijkstra's typically find the path with the minimum total weight, sometimes an additional constraint is needed: if multiple paths share the same minimum total weight, we want the one with the fewest edges. This article delves into how to adapt standard shortest path algorithms to satisfy this secondary criterion.

Understanding the Problem: Shortest Path vs. Fewest Edges

Consider a scenario where each edge has a weight (e.g., cost, distance, time). A standard shortest path algorithm, such as Dijkstra's, will find the path where the sum of these weights is minimized. However, it doesn't inherently distinguish between two paths that have the same minimum total weight but differ in the number of edges. For instance, a path of length 10 with 2 edges might be preferred over a path of length 10 with 5 edges, especially in contexts where each 'hop' or 'connection' incurs an additional, unweighted cost (e.g., network latency per router, administrative overhead per transaction).

graph TD
    A[Start] --> B(Node B)
    A --> C(Node C)
    B -- 5 --> D(Node D)
    C -- 2 --> E(Node E)
    D -- 5 --> F(Node F)
    E -- 8 --> F
    subgraph Path 1
        A -- 1 --> B
        B -- 1 --> D
        D -- 1 --> F
    end
    subgraph Path 2
        A -- 1 --> C
        C -- 1 --> E
        E -- 1 --> F
    end
    style Path 1 fill:#f9f,stroke:#333,stroke-width:2px
    style Path 2 fill:#bbf,stroke:#333,stroke-width:2px
    A -- "Weight: 1, Edges: 3" --> F
    A -- "Weight: 1, Edges: 3" --> F

Example graph illustrating two paths with the same total weight but different edge counts (conceptual).

Adapting Dijkstra's Algorithm

Dijkstra's algorithm can be modified to find the shortest path with the fewest edges. The key is to redefine the 'distance' metric stored for each node. Instead of just storing the minimum total weight, we store a pair: (total_weight, num_edges). When comparing two paths to a node, we first compare their total_weight. If the total_weight is less, that path is preferred. If the total_weight is equal, then we compare num_edges, preferring the path with fewer edges. This lexicographical comparison ensures that the primary goal (minimum weight) is met, and the secondary goal (minimum edges) is met for ties.

import heapq

def dijkstra_fewest_edges(graph, start):
    # dist[node] = (total_weight, num_edges)
    dist = {node: (float('inf'), float('inf')) for node in graph}
    dist[start] = (0, 0)
    priority_queue = [(0, 0, start)] # (total_weight, num_edges, node)

    while priority_queue:
        current_weight, current_edges, u = heapq.heappop(priority_queue)

        # If we found a better path to u already, skip
        if (current_weight, current_edges) > dist[u]:
            continue

        for v, weight in graph[u].items():
            new_weight = current_weight + weight
            new_edges = current_edges + 1

            # Lexicographical comparison
            if (new_weight, new_edges) < dist[v]:
                dist[v] = (new_weight, new_edges)
                heapq.heappush(priority_queue, (new_weight, new_edges, v))
    return dist

# Example Usage:
# graph = {
#     'A': {'B': 1, 'C': 5},
#     'B': {'D': 2},
#     'C': {'D': 1, 'E': 1},
#     'D': {'F': 3},
#     'E': {'F': 2},
#     'F': {}
# }
# print(dijkstra_fewest_edges(graph, 'A'))

Python implementation of Dijkstra's algorithm adapted for fewest edges.

Alternative: Modifying Edge Weights

Another approach, particularly useful if you cannot directly modify the comparison logic of an existing shortest path library, is to modify the edge weights. You can assign a very small epsilon value to each edge, effectively making each edge 'cost' a tiny bit more. For example, if M is the maximum possible total weight in the graph, you could set the new weight of an edge (u, v) with original weight w to w * (N + 1) + 1, where N is the maximum number of edges in any path. This ensures that adding an edge always increases the total path weight by at least 1, and the original weights are scaled up to prevent the 'epsilon' from overriding them. However, this method requires careful selection of the scaling factor to avoid overflow or precision issues, and it can be less intuitive than the lexicographical approach.

flowchart LR
    A[Original Edge Weight] --> B{Multiply by Large Factor (e.g., Max Edges + 1)}
    B --> C{Add 1 (for each edge)}
    C --> D[New Modified Edge Weight]
    D --> E[Run Standard Dijkstra]
    E --> F[Result: Shortest Path with Fewest Edges]

Flowchart illustrating the edge weight modification strategy.