Transform and Conquer

• Several Techniques
  • Transform to a simpler or more convenient version of the same problem.
  • Transform to a different representation of the same instance
  • Transform into a different problem.
• We need to be concerned about some things
  • How expensive is it to transform to the new problem?
  • How expensive is it to solve the new problem?
  • Can we turn the answer to the answer to our question?
  • How expensive is it to return to the real answer?
• Sorting By Convex Hull
  • Assume you have a list A of n real numbers you wish to sort.
  • Consider the Following

     
    HULL_SORT(A)
    
     for i ← 0 to n-1 do 
         pi ← (A[i],(A[i])2)
     C = ConvexHull(P)
     i ← 0
     for p ← the leftmost point in the hull to the rightmost point do
           A[i] ← p.x
           i ← i + 1
     return A
    
    	

  • Try this with the points A = {4, 2, 6, 1, 3}
  • For any set of numbers this will produce points on the parabola.
  • The convex hull will contain all of these points.
  • How long will the transformation take? (Each way)
• The Graham Scan is an example of a presorting algorithm.
  • This is a very powerful class of algorithms which begin by sorting.
    • Notice we reduced the problem from O(n³) or even O(n²) to O(n lg(n)).
    • And the real work even further.
  • The author suggests finding duplicates in an array is a transform and conquer problem of this type.
    • How would you find duplicates in an unordered array?
    • How could sorting help?
  • How about computing statistics on elements of an array
    • Median? Q₁, Q₃, ...?
    • Mode?
  • Actually, you know about one of these already... Binary Search.
  • Sorting is not the solution to everything...
    • But there are times when you want to pay that price
• Topological Sort
  • He covered this problem much earlier in the book section 4.2.
  • Given a directed acyclic graph (DAG), find a linear ordering between the vertexes.
  • This has applications to scheduling
    • Deciding the order in which to perform a series of interrelated tasks.
    • Deciding what order to take courses with pre-requisites.
  • The algorithm, from CLRS

  • TOPOLOGICAL_SORT(G)
    
     DFS(G), to compute f[v] for each vertex.
     as each vertex is finished, insert it into the front of a linked list
     return this list of vertexes.
    
    	 

  • Clrs gives an example of getting dressed.
  • 
  • In this case, the numbers in red are the finish times for the graph in DFS.
    • Note, these times may change depending on which node is selected.
    • The watch can go at any time.
    • 
    • Note, we can use DFS to detect back edges as well (Is G a DAG)
      • A tree edge will go to a white note.
      • A cross edge will go to a black node
        • All the ancestors of the current node are gray
        • All black nodes have returned already
      • A back will go to a gray node.