A *算法中的启发式值

Question

I am learning A* algorithm and dijkstra algorithm. And found out the only difference is the Heuristic value it used by A* algorithm. But how can I get these Heuristic value in my graph?. I found a example graph for A* Algorithm(From A to J). Can you guys help me how these Heuristic value are calculated.

The RED numbers denotes Heuristic value.

My current problem is in creating maze escape.

Answer 1

The heuristic is an estimate of the additional distance you would have to traverse to get to your destination.

It is problem specific and appears in different forms for different problems. For your graph , a good heuristic could be: the actual distance from the node to destination, measured by an inch tape or centimeter scale. Funny right but thats exactly how my college professor did it. He took an inch tape on black board and came up with very good heuristic.

So h(A) could be 10 units means the length measured by a measuring scale physically from A to J.

Of course for your algorithm to work the heuristic must be admissible , if not it could give you wrong answer.

Answer 2

In order to get a heuristic that estimates (lower bounds) the minimum path cost between two nodes there are two possibilities (that I know of):

Knowledge about the underlying space the graph is part of

As an example assume the nodes are points on a plane (with x and y coordinate) and the cost of each edge is the euclidean distance between the corresponding nodes. In this case you can estimate (lower bound) the path cost from node U to node V by calculating the euclidean distance between U.position and V.position .

Another example would be a road network where you know its lying on the earth surface. The cost on the edges might represent travel times in minutes. In order to estimate the path cost from node U to node V you can calculate the great-circle distance between the two and divide it by the maximum travel speed possible.

Graph Embedding

Another possibility is to embed your graph in a space where you can estimate the path distance between two nodes efficiently. This approach does not make any assumptions on the underlying space but requires precomputation.

For example you could define a landmark L in your graph. Then you precalculate the distance between each node of the graph to your landmark and safe this distance at the node. In order to estimate the path distance during A* search you can now use the precalculated distances as follows: The path distance between node U and V is lower bounded by |dist(U, L) - dist(V,L)| .You can improve this heuristic by using more than one landmark.

For your graph you could use node A and node H as landmarks, which will give you the graph embedding as shown in the image below. You would have to precompute the shortest paths between the nodes A and H and all other nodes beforehand in order to compute this embedding. When you want to estimate for example the distance between two nodes B and J you can compute the distance in each of the two dimensions and use the maximum of the two distances as estimation. This corresponds to the L-infinity norm .