Question

I’m working through previous years ACM Programming Competition problems trying to get better at solving Graph problems.

The one I’m working on now is I’m given an arbitrary number of undirected graph nodes, their neighbors and the distances for the edges connecting the nodes. What I NEED is the distance between the two farthest nodes from eachother (the weight distance, not by # of nodes away).

Now, I do have Dijkstra’s algorithm in the form of:

// Dijkstra's Single-Source Algorithm private int cheapest(double[] distances, boolean[] visited) {         int best = -1;         for (int i = 0; i < size(); i++)         {                 if (!visited[i] && ((best < 0) || (distances[i] < distances[best])))                 {                         best = i;                 }         }         return best; }  // Dijkstra's Continued public double[] distancesFrom(int source) {         double[] result = new double[size()];         java.util.Arrays.fill(result, Double.POSITIVE_INFINITY);         result[source] = 0; // zero distance from itself         boolean[] visited = new boolean[size()];         for (int i = 0; i < size(); i++)         {                 int node = cheapest(result, visited);                 visited[node] = true;                 for (int j = 0; j < size(); j++)                 {                         result[j] = Math.min(result[j], result[node] + getCost(node, j));                 }          }         return result; } 

With this implementation I can give it a particular node and it will give me a list of all the distances from that node. So, I could grab the largest distance in that list of distances but I can’t be sure that any particular node is one of the two furthest ones at either end.

So the only solution I can think of is to run this Dijkstra’s algorithm on every node, go through each returned list of distances and looking for the largest distance. After exhausting each node returning it’s list of distances I should have the value of the largest distance between any two nodes (the ‘road’ distance between the two most widely seperated villages). There has got to be an easier way to do this because this seems really computationally expensive. The problem says that there could be sample inputs with up to 500 nodes so I wouldn’t want it to take prohibitively long. Is this how I should do it?

Here is a sample input for the problem:

Total Nodes: 5

Edges:
Nodes 2 – Connect – Node 4. Distance/Weight 25
Nodes 2 – Connect – Node 5. Distance/Weight 26
Nodes 3 – Connect – Node 4. Distance/Weight 16
Nodes 1 – Connect – Node 4. Distance/Weight 14

The answer to this sample input is ’67 miles’. Which is the length of the road between the two most widely separated villages.

So should I do it how I described or is there a much simpler and much less computationally expensive way?

Answer 1

It looks like you can use either of:

• Floyd Warshall algorithm
• Johnson’s algorithm.

I can’t give you much guidance about them though – I’m no expert.