1 =1 ifn -HS).lft! 6. For the recurrence T(n)-n = 37 1 +1 ifn>1 show that...
Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy; the lower bound T(n) = Ω(logk +1 n) is harder.) Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy;...
1. Solve the recurrence relation T(n) = 2T(n/2) + n, T(1) = 1 and prove your result is correct by induction. What is the order of growth? 2. I will give you a shortcut for solving recurrence relations like the previous problem called the Master Theorem. Suppose T(n) = aT(n/b) + f(n) where f(n) = Θ(n d ) with d≥0. Then T(n) is: • Θ(n d ) if a < bd • Θ(n d lg n) if a = b...
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.
Consider the recurrence equation defining T(n): T(n)=1 if n-1; T(n)=T(n-1)+2^n Otherwise . Show by induction T(n)=2^(n+1)-1
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or whatever your inequality by induction. Show the base case. Then show the how that T( n)= 0(n lg ig n). First, clearly indicate the inequality that you wish to hen proceed to prove the inductive hypothesis inductive case, and clearly...
Consider recurrence T(n) = 2T () +n Ign. Assume T (1) = : 0(1) Draw its recursion tree using your favorite tool. Follow the instructions (regarding the tree, step 1~3) to format your tree. Level Tree Node Per-Level Cost . 1 O Step 1: Draw the "head" of the tree. Step 2: Start at level 0, draw the tree downto level 2. 2 cn 1X CP = CP Tw/2 (wa), T(1/2) 1 cn/2 cn/2 28 cm/2 = 0 T( W22)...
Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into 5 sub-instances of size n/3, and the dividing and combining steps take a time in Θ(n n). Write a recurrence equation for the running time T (n) , and solve the equation for T (n) 2. Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into 5 sub-instances of size n/3, and the dividing...
Formal Definitions of Big-Oh, Big-Theta and Big-Omega: 1. Use the formal definition of Big-Oh to prove that if f(n) is a decreasing function, then f(n) = 0(1). A decreasing function is one in which f(x1) f(r2) if and only if xi 5 r2. You may assume that f(n) is positive evervwhere Hint: drawing a picture might make the proof for this problem more obvious 2. Use the formal definition of Big-Oh to prove that if f(n) = 0(g(n)) and g(n)...
An m×n array A of real numbers is a Monge array if for all i,j,k, and l such that 1≤i<k≤m and 1≤j<l≤n , we have >A[i,j]+a[k,l]≤A[i,l]+A[k,j]> In other words, whenever we pick two rows and two columns of a Monge array and consider the four elements at the intersections of the rows and columns, the sum of the upper-left and lower-right elements is less than or equal to the sum of the lower-left and upper-right elements. For example, the following...
only f and g please 1. The motion of a vibrating string of length T, with fixed endpoints, immersed in a fluid (such as air) can be modeled by Fu 2&u 21 0rT t>0 at Ot2 (P1) u(0,t)u, t) 0 t20 Qu is a damping term, modelling the effect of at where c,>0. The term proportional to air resistance on the string. (a) Explain why the damping term has a minus sign (2 points) (4 points) (b) Consider the separable...