Decrease and Conquer

• Find a solution by solve one part of the problem and then solve the smallr problem with that part removed.
• He gives a nice write up in 4.1, but let's begin with some examples
  • Compute aⁿ
    • Can you give me a brute force method?
    • How about

      • POWER(a,n)
        
         if n = 0 then 
              return 1
         elseif n%2=0 then
              tmp ← POWER(a,n/2)
              return tmp * tmp
         else
              tmp ← POWER(a,(n-1)/2)
              return tmp * tmp * a
        
        		  

      • This is really not just the brute force approach.
      • Size of the next problem is 1/2 of the size of the current problem.
      • Five questions
  • Sorting, again Insertion Sort

    • INSERTION_SORT(A)
      
       for i ← 1 to n-1 do
           v ← A[i]
           j ← i-1
            while  j ≥ 0  and  A[j] > v do
              [j+1] ← A[j]
              j ← j - 1
          A[j+1] ← v
      
      	  

    • | sorted array | unsorted array|
    • Decrease the size of the unsorted array by 1 each time.
    • Standard Questions.
    • Is this sort stable?
    • Best Case?
    • Worst Case.