A*等路径规划算法可100%保证最短路径，如何手动判定加权图中给定路径是否为最短路径？

Great question! When you’ve got a direct path like node 1 -> node 2 and want to confirm it’s the shortest path without a computer, there are several straightforward, manual approaches depending on the size and complexity of your graph. Here’s how to go about it:

1. Enumerate and Compare All Possible Paths #

For small graphs with only a handful of nodes and edges, this is the most direct method:

• List every possible path from node 1 to node 2 (you can skip paths with loops, since adding a loop will only increase the total weight and can’t be shorter than a loop-free path).
• Calculate the total weight of each path by summing the weights of its individual edges.
• Compare the total weight of your target path (node 1 -> node 2) to all others. If its weight is the smallest (or tied for the smallest, if multiple shortest paths exist), then it’s indeed a shortest path.

Note: This only works for tiny graphs—once you have more than 4-5 nodes, the number of possible paths grows exponentially, making this impractical.

2. Use the Triangle Inequality Check #

If your graph has intermediate nodes but isn’t too large, you can leverage the triangle inequality (a core concept in shortest path algorithms) to verify:

• For every intermediate node X that forms a valid path between node 1 and node 2, check if:

  weight(node1 -> node2) ≤ weight(node1 -> X) + weight(X -> node2)
  

• If this holds true for all possible intermediate nodes X, then the direct path can’t be beaten by any path that goes through another node. Because if a shorter path through X existed, the inequality would be violated (the right-hand side would be smaller than the left).

Example: Suppose node1->node2 has a weight of 5. If node1->node3 is 3 and node3->node2 is 3 (total 6), and node1->node4 is 2 and node4->node2 is 4 (total 6), then 5 ≤ 6 and 5 ≤ 6—so the direct path is shorter.

3. Manual Dijkstra’s Algorithm #

For larger graphs where enumerating paths isn’t feasible, you can run a simplified, manual version of Dijkstra’s algorithm to compute the shortest distance to node 2:

• Start by labeling node 1 with a distance of 0 (since that’s your starting point). Label all other nodes with infinity (a stand-in for "unknown" or "very large").
• Look at all neighbors of node 1. For each neighbor (including node 2), update their labeled distance to be the weight of the edge from node 1 to them (since that’s the shortest path we know so far to those nodes).
• Next, pick the node with the smallest labeled distance that you haven’t processed yet. For each of its neighbors, check if going through this node gives a shorter path than their current labeled distance. If so, update their distance.
• Repeat this process until you’ve processed node 2. The final labeled distance for node 2 should match the weight of your direct path if it’s the shortest.

This method is systematic and works for graphs with more nodes, as it avoids checking every possible path directly.

内容的提问来源于stack exchange，提问作者stunicks