CS141 Lab 9

Main task for this lab is to finish the In-lab assignment of Lab 8.
For those studetns who have finished lab 8, we have bonus questions for you !!

Assignment

Continue doing last lab -- Finding the shortest path using Dijkstra's Algorithm

We added another test case
Try your best to finish it !!

Bonus Questions

Shortest path with at least B bankwidth -- in Lab 8 ( 2 bonus points !! )

Think of a solution based on simple modification of Dijkstra's algorithm
Chang your code to implement it
Show me your running results of the test cases.

You can get 1 bonus point if you only show me your idea of the solution, without finishing your code.

Find the minimum cost path with less than K hops. ( 2 bonus points !! )

            This is problem does not have a polynomial time solution. But you can come up with heuristics methods (approximation algorithms) ie:
            * Find shorstest path, if length < k, then try to improve the cost (is there another path with less cost? and still less than K hops)

            So the question becomes how can I improve the cost of the current path:
            Again heuristics:

Remove the most costly (or any) edge of the path and try to find another path hopefully with less cost.

To further improve things, after I "replace" an edge try to replace another edge and so on.

Stop when I can not replace any edge (including the new ones I introduced)

            Question:
                    Will this guarantee the best solution?
                    Please think of an example if you think that it will not find the asbolutely best solution.

Euler cycle ( 2 bonus points !! )

            Euler's Problem:
                asks whether there is a path through a graph which traverses each edge only once.
                It can be drawn with a single line or there exists a cycle that passes from every edge exactly once.
                The problem can be solved polynomially: given the starting point you just need to visit all edges.

                See: http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/hard.html#euler
                (only the first part)

          Questions:
              How to check whether there exists an Euler cycle or not in a graph, using polynomial running time?

Suppose the starting point is given
What you can do if the starting point is not given?