CS 141 Homework Assignment #1
                           Set on Jan. 14, 2008 
                           Due on Jan. 28, 2008 


1. [15 pts]  Analyze the worst-case time complexity of the following
   algorithms, and give tight bounds using the Theta notation. You need
   show the key steps involved in the analyses.

   a)      procedure matrixvector(n: integer);
               var  i,j: integer;
               begin
                  for i <- 1 to n do begin
		      B[i] = 0;
		      C[i] = 0;
                      for j <- 1 to i do 
                          B[i] <- B[i] + A[i,j];
                      for j <- n downto i+1 do 
                          C[i] <- C[i] + A[i,j]
	          end
               end;

   b)      Algorithm MysteryPrint(A[1..n]);
               begin
	          if n = 1 then
		       print A[1]
		  else begin
		     print A[n];
		     call MysteryPrint(A[1..n-1]);
		     call MysteryPrint(A[2..n])
		  end
               end;

   c)      procedure whileloop(n: integer);
               var   i,count: integer;
               begin
                   count <- 0;
                   i <- 1;
                   while i < n do begin
                       i <- 3*i;
                       count <- count + 1
                   end
               end; 


2. [10 pts]  [Levitin, p. 60; same for the 1st ed] Question 7 (a, c). 

3. [10 pts]  Insertion sort can be expressed as a recursive procedure 
   as follows. Given A[1..n], we recursively sort A[1..n-1] and then
   insert element A[n] into the sorted array A[1..n-1]. Write a 
   recurrence relation for the running time of this recursive version
   of insertion sort.

   Feel free to use the big-O notation and assume the worst case. Can 
   you also solve the recurrence relation? Is your result the same as 
   the one obtained by the summation analysis in the textbook (pp. 160-162)?


4. [15 pts]  Give a tight asymptotic bound for T(n) in each of
   the following recurrences. Assume that T(n) is a constant for n <= 2. 
   Justify your answers.

   a.  T(n) = 16*T(n/4) + n^2
   b.  T(n) = 7*T(n/3) + n^2
   c.  T(n) = T(n-1) + n^2

   You may solve these recurrences using either the Master Theorem 
   or the backward substitution method.

5. [15 pts]  Describe a Theta(nlog n) time algorithm that, given a set S
   of n integers and another integer x, determines if or not there exist
   two elements in S whose sum is exactly x.

   Analyze the running time of your algorithm to make sure that it
   is really Theta(nlog n) (it would be sufficient to show that the
   running time is O(nlog n)).

   Before you present the pseudo-code, use a few words to describe
   the basic ideas in your algorithm to help the reader understand
   your algorithm.

   Hint: Sorting and binary search.