Recursion trees can be useful for gaining intuition about the closed form of a recurrence, but they are not a proof
Recursion Tree for the recurrence
can be shown as,
Starting the root as, n, then expnading out to the first level be
Similarly expanding the above tree for further more levels gives,
Note that the tree here is not balanced: the longest path is the rightmost one, and its length is log5/3 n.
Hence our guess for the closed form of complexity of this recurrence is O(n log n).
(15 pts) 1. Create the recursion tree for the recurrence T(n)-T(2n/5)T3n/5) O(n). Show total complexity
Solve the recurrence relation using a recursion tree AND substitution method: T(n) = T(n-1) + 10n
Solve the recurrence relation using a recursion tree AND substitution method: T(n) = 2T(n - 1) + 10n.
(a) Use the recursion tree method to guess tight 5 asymptotic bounds for the recurrence T(n)-4T(n/2)+n. Use substitution method to prove it.
Consider the Catalan numbers P(n) described here Show a recursion-dependence tree for P(5) according to the recurrence relation. Design a bottom-up dynamic programming algorithm to compute P(n) based on the recurrence relation (without using the formula P(n) = (2n−2)!/(n!(n−1)!)). Analyze its asymptotic runtime efficiency. Design a top-down memoized algorithm to compute P(n) based on the recurrence relation (without using the formula P(n) = (2n−2)!/(n!(n−1)!)). Analyze its asymptotic runtime efficiency.
draw the first 3 levels of a recursion tree for the recurrence T(n) = 4T(n/2) + n. How many levels does it have? Find a summation for the running time and solve for it.
Consider the recurrence T (n) = 3 · T (n/2) + n. • Use the recursion tree method to guess an asymptotic upper bound for T(n). Show your work. • Prove the correctness of your guess by induction. Assume that values of n are powers of 2.
(5 pts.) (b) Use a recursion tree to determine a good asymptotic upper bound on the recurrence T(n) = 6T ([n/4]) + 11n. Verify your bound by the substitution method.
Consider the following recurrence T(n) = 3T(n/5) + c What is the complexity of T(n) in Big O? ___________.
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)...
(Weight: 3090) Use substitution, summation, or recursion tree method to solve the f ollowi recurrence relations. (a) T(n) = 2T(n/2) + nign (b) T(n) 2T(n-1)+5" 7(0) = 8