Midterm Exam, CSCI 385, Fall 2013

• Please answer each question carefully and thoughtfully. You need to demonstrate that you understand the material covered.
• Use as many pages as you wish, but make sure that you label each answer with the question number.
• Please write neatly, and print if possible. If I can't read your answer, it is incorrect.
• Please make sure that you answer all parts of the questions.
• For each answer, justify you answer with a short argument or explain any mathematical expression, A simple Yes is wrong.

1. Asymptotic Notation
   1. [2 points] State the definition of f(n) ∈ O(g(n))
   2. [4 points] Show that if f₁(n) ∈ O(g₁(n)) and f₂(n) ∈ O(g₂(n)), then f₁(n) + f₂(n) ∈ O(max(g₁(n),g₂(n))
   3. [2 points] What does the above property imply in terms of algorithm analysis?
2. STOOGE_GCD: Compute the GCD of two positive integers

   STOOGE_GCD(a,b) 
   
    done ← false
    while not done do
       x ← rand() % (min(a,b))
       if a %x = 0 and b%x = 0 
          return x * STOOGE_GCD(a/x,b/x)
    return 1
    

   1. [4 points] Is STOOGE_GCD an algorithm? Why or why not.
   2. [2 points] What design pattern is STOOGE_GCD attempting to employ?
3. CLOSEST_PAIR: Given a set of n points P, find the closest two.

    
   CLOSEST_PAIR(P)
   
    d ← ∞
    for i ← 1 to n-1 do
        for j ← i+1 to n do
            if d > distance(Pi,Pj)
                  d = distance(Pi,Pj)
                  a = i, b = j
    return Pa, Pb
   
   

   1. [2 points] Does this algorithm terminate? (why or why not)
   2. [2 points] Does this algorithm work? (why or why not)
   3. [2 points] What is the critical operation for this algorithm?
   4. [4 points] What is the performance of this algorithm? (Give a mathematical expression and relate the lines of the instructions to the terms of the expression)
   5. [3 points] Is there a different best, worst and average case performance for this algorithm? If so, state the different values, if not, state why.
   6. [2 points] What design pattern was applied to develop this algorithm?
4. PATTERN_MATCH: Given a text T[1..n] and a pattern P[1..m] locate the first occurrence of P in T.

   PATTERN_MATCH(P,T)
   
    L ← []
    for i ← 1 to n do
       if T[i] = P[1]
           L.append(i)
    for i ← 0 to L.size() do
        t = T.substring(L[i])
        if t == P
          return i
    return -1
   
         

   1. [2 points] Does this algorithm terminate?
   2. [2 points] Does this algorithm work?
   3. [2 points] What is the critical operation for this algorithm?
   4. [4 points] What is the performance of this algorithm? (Give a mathematical expression and relate the lines of the instructions to the terms of the expression)
   5. [3 points] Is there a different best, worst and average case performance for this algorithm? If so, state the different values, if not, state why.
5. FLIP_IT (S): reverse a string

   FLIP_IT(S)
   
    if S = "" or S.size() = 1
        return S 
     else
        return FLIP_IT(S.substr(1,S.size()-1) + S[0]
   
      

   1. [4 points] Build a recurrence relation to represent the performance of this algorithm. Describe how your recurrence relation relates to the algorithm (use line numbers).
   2. [4 pints] Solve the recurrence relation.