算法导论（3版）第4章习题的部分解答

Introduction to algorithms (3rd editon)

第四章部分解答        by zevolo

4.3-1

Show that the solution of T(n) = T(n-1) + n is O(n^2).
proof:
assume  T(m) <= c * m^2 ( m < n)
T(n) = T(n-1) + n <= c (n-1)^2 + n = c (n^2 - 2n + 1) + n
         = cn^2 - 2cn + c + n
         = cn^2 + (1 - 2c)n + c
         <= cn^2 (when c > 1 && c < 2 && n > 1)

    so modify the assumption to T(m) <= c m^2 (1 < m < 2)
for n = 1
   T(n) = T(1) = 1
   c n^2 = c (1)^2 = c
   satisfy the T(1) <= c (1)^2

so the assumption is correct


4.3-2
show that the solution of T(n) = T(ceiling(n/2)) + 1 is O(lgn)

proof:
assume that T(m) <= c * lg(m) (m < n), k = ceiling(n/2)
    T(n) <= c * k + 1
         <= c * lg((n+1)/2) + 1
         =  c * lg(n+1) - c + 1
         =  c * lg(n) + c * lg((n+1)/n) - c + 1
         <= c * lg(n) + c * lg((2+1)/2) - c + 1 (when n > 1)
         =  c * lg(n) + c * lg(3) - 2c + 1 (when n > 1)
         <= c * lg(n) (when n > 1 && c >= (1/lg(4/3)) )
              note: c * lg(3) - 2c + 1 <=0 ==>  c >= (1/lg(4/3))

the bound