Greedy Techniques

• A greedy technique solves a problem by making a localy greedy choice.
• Consider the Continuous Knapsack Problem
  • Also known as the fractional knapsack problem.
  • You have a knapsack of a fixed size.
  • You have a collections of objects with two values
    • The unit value
    • The number of units available.
  • in this case, the solution is

    FNP(A,w)
    
     Sort A by value largest to smallest
     total ← 0
     i ← 0
     while total < w and i < A.size do
        if A[i].size ≤ w-total
           Take all of the objects of this value
           total ← total + A[i].size 
        else
           Take w-total objects of this value
           total ← w  
        i ← i + 1
    
    	 

  • Normal questions.
• The greedy choice is only good for solving optimization problems.
• To employ this technique, at each step the following must hold
  • The choice must be feasible, it must satisfy the problems constraints
  • The choice must be locally optimal: it his to be test best choice among all othe other choices at that step.
  • It must be irrevocable: once you make this choice, you never undo it.
• These definintions can be put into mathematical terms and problems can be proved to be optimally solved by greedy choices.
• Can you give an algorithm for making optimal change (The fewest number of coins?)
• A spanning tree of a graph G=(V,E) is an acyclic subgraph that contains all of the vertices.
• If the weight of the edges of the tree is the least of any such tree, then it is a minimal spanning tree.
• Prim's Minimal Spanning Tree Algorithm
  • Apparently discovered several times.

  • PRIM(G=(V,E),w,r)
    
     for each v ∈ V do
         key[u] ← ∞
         π[u] ← NIL
     key[r] ← 0
     Q ← V
     while Q ≠ ∅ do
         u ← Q.extract_min()
         for each v ∈ Adj(u) do
               if v∈ Q and w(u,v) < key[v]  then
                    π[v] ← u
                    key[v] ← w(u,v)
    
    	  

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  • Normal Questions.
  • If we employ a min-heap to keep the edges
    • Put the start nodes edges in the min heap.
    • Each time we add a new node, add the edges for that node to the min heap.
    • This will be a total (worst case) of O(|E|lg|V|) (Line 9/10)