Homework 4, What does this algorithm do.

Short Description:

1. Trace an algorithm.
2. Describe the task accomplished by this algorithm.
3. Argue that the algorithm is correct and discuss performance

This assignment is worth 40 points.

Goals

• Improved your skills at algorithm analysis. (Objective 4)

Formal Description

Unknown(G = (V,E), s)
// G is a directed graph.
// s ∈ V
// H is a Heap the key is cost.

 For each vertex in  V
     v.parent = unknown
     v.cost = ∞
     v.status = unknown
     H.Insert(v)
 EndFor
 s.cost = 0
 H.DecreaseKey(s)
 s.parent = null;
 s.status = discovered
 While H.Size() > 0
    u = H.ExtractMin()
    u.status = finished
    For each v $\in$ U.Adj() 
        If v.status ≠ finished 
           If v.cost > u.cost + w(u,v)
              v.parent = u
              v.cost = u.cost+w(u,v);
              H.DecreaseKey(v)
              v.status = discovered
           EndIf
        EndIf
     EndFor
 EndWHile 
 

1. [10 points] Trace this algorithm with the following undirected weighted graph pictured below.
   • Use multiple copies of this graph (one for each iteration of loop h)
   • Show the parent pointers at each step
   • Show the cost at each step.
   • Indicate the node "u" at each step (from line 12)
   • 
2. [10 points] What does this algorithm do?
   • Do this step before you move on.
   • Consider the reverse of the parent pointers. What is this tree?
     • This might be hard to determine at first, tracing the algorithm the following graph might help.
     • 
     • If that doesn't help, try adding up the values of all the edges selected.
3. [10 points] The purpose of this algorithm is to find a tree which connects all nodes where the sum of the weights of the edges of the tree is minimized.
   • Argue that this algorithm accomplishes this task.
4. [10 points] What is the performance of this algorithm?
   • Provide an argument to support your claim.

Discussion

Required Files

Submission