# Recent questions tagged recurrence

1
For constants $a \geq 1$ and $b>1$, consider the following recurrence defined on the non-negative integers: $T(n) = a \cdot T \left(\dfrac{n}{b} \right) + f(n)$ Which one of the following options is correct about the recurrence $T(n)$? If $f(n)$ is $n \log_2(n)$, then $T(n)$ ... $\Theta(n^{\log_b(a)})$ If $f(n)$ is $\Theta(n^{\log_b(a)})$, then $T(n)$ is $\Theta(n^{\log_b(a)})$
2
Consider the following recurrence relation. $T\left ( n \right )=\left\{\begin{array} {lcl} T(n ∕ 2)+T(2n∕5)+7n & \text{if} \; n>0\\1 & \text{if}\; n=0 \end{array}\right.$ Which one of the following options is correct? $T(n)=\Theta (n^{5/2})$ $T(n)=\Theta (n\log n)$ $T(n)=\Theta (n)$ $T(n)=\Theta ((\log n)^{5/2})$
3
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
4
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?
5
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
6
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
1 vote
7
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
8
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
9
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
10
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
11
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
12
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
13
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$
14
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$-bit integers using the fast multiplication algorithm.
15
Express the fast multiplication algorithm in pseudocode.
16
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.
17
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?
18
How many comparisons are needed for a binary search in a set of $64$ elements?
1 vote
19
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... is $m,$ there is a particular solution of the form $n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$
20
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$ has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ with multiplicities $m_{1},m_{2},\dots,m_{t},$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$
21
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]
It can be shown that Cn, the average number of comparisons made by the quick sort algorithm (described in preamble to question $50$ in exercise $5.4),$ when sorting $n$ ... $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$
Some linear recurrence relations that do not have constant coefficients can be systematically solved. This is the case for recurrence relations of the form $f (n)a_{n} = g(n)a_{n-1} + h(n).$ Exercises $48-50$ ... relation to obtain $a_{n} = \dfrac{C +\displaystyle{} \sum_{i = 1}^{n}Q(i)h(i)}{g(n + 1)Q(n + 1)}$