CS 280, HW3, due Friday 26th Sept

In the following, when asked about algorithms, you should use the notation from 
the text (chapter 1) and estimate their running time.

1.  Construct and analyse an algorithm to solve 5.2 from HW2.

2.  A recursive version of insertion sort could be expressed as sorting an 
array of length n by sorting the first (n-1) terms and then inserting 
inserting the n-th term.  Construct and analyse this algorithm.

3.  If A[n] is an array of n distinct numbers, then we define a pair (i,j) 
to be an "inversion" of A if  (i < j ) and (A[i] > A[j]).
   (i)   give an array with numbers from the set {1,2,...,n} having the 
        most inversions and state how many inversions it has.
   (ii)  what is the relationship between the running time of the usual 
        form of insertion sort and the number of inversions in the input 
        array?
   (iii) Construct and analyse an algorithm to determine the number of 
        inversions in any permutation on n elements in O( n log n ) 
        worst-case time.

The following are induction proof questions.

4.  Prove each of the following arithmetic statements by induction:
   (i)   question 2.2.16 from the textbook, i.e., discover and prove a 
        formula for the n-th power of the 2x2 matrix  / 1 1 \
                                                      \ 0 1 / .
   (ii)  (n-r)!.r! divides n!  for all n and for all r (with 0<=r<=n).
   (iii) (a+b)^n = a^n + n.a^(n-1).b + ... + nCr.a^(n-r).b^r + ... + nCn.b^n ,
        where  nCr := ( n! / ((n-r)!.r!) ) ,
        hint: show first that nCr = nC(n-r) and nCr + nC(r-1) = (n+1)Cr.
   (iv)  nC0 + nC1 + ... + nCn = 2^n . 

5.  Devise, state clearly, and prove by induction, a result analogous to 
the Euclidean algorithm for polynomials in one real variable and with real 
coefficients.