Best,Worst and Average Cases

• We have already seen an algorithm where these are the same
  • Can you name it?
• But sometimes they are different.
• Consider Sequential Search

  • SEQUENTIAL_SEARCH(A, k)
    
     i ← 0
     while i < A.size() and A[i] ≠ k do
       i ← i + 1
     if  i = A.size() 
          return -1
     else
          return  i
    
    	

  • Does it work?
  • How does it work?
  • Can we argue that it works?
  • What is the measure of the input
  • What is the basic operation.
  • Can we come up with an expression T(n) which describes
    • The BEST performance, when is this?
    • The WORST performance, when is this?
    • The AVERAGE performance?
  • Sometimes best and worst are easy and average is hard.
  • In this case, assume the data is uniformly distributed,
    • Let p be the probability that the item is in the array.
    • And the array contains n items.
    • At each point the probability is p/n
    • And the probability that it does not occur in the array is (1-p)

    • Cavg = 1* p/n + 2* p/n + ...+ (n-1)* p/n  +np/n + n(1-p)
           = p/n [1+2+...+n] + n(1-p)
           = p/n * (n)(n+1)/2 +n(1-p)
           = p(n+1)/2 + n(1-p)
           = pn/2 +p/2 +n -pn
           = n-pn/2 + p/2
           = n(1-p/2) + p/2
                      

    • If p = 0, C_avg = n
    • If p = 1, C_avg = (n+1)/2
  • We are usually most concerned about worst case,
  • Then about average
  • We never think things will go well, so we are less concerned about best case.
  • When will these three be different?
• There is another type of analysis that we will investigate later.
  • Amortized analysis considers the case where an algorithm must be run repeatedly.
  • Consider push and pop in a queue.
  • But we will consider this later.