Homework Answers
The mathematician Carl Friedrich Gauss (1777ñ1855) once
noticed
that
although
the
product
of two
complex
numbers
(
a
+
bi
)(
c
+
di
) =
ac
bd
+ (
bc
+
ad
)
i
seems
to
involve
four
real-number
multiplications
, it can
in
fact
be
done
with
just
three
:
ac
,
bd
, and
(
a
+
b
)(
c
+
d
)
, since
bc
+
ad
=
(
a
+
b
)(
c
+
d
)
ac
bd:
In
our
big-
O
way of thinking
, reducing
the
number
of multiplications
from
four
to
three
seems
wasted
ingenuity
. But
this
modest
improvement
becomes
very
signicant
when
applied
recur
-
sivel
y
.
55
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Question 1 - Buzz Matrix For any number n, such that n is a power of...
We know that binary search on a sorted array of size n takes O(log n) time....
We know that binary search on a sorted array of size n takes O(log n) time. Design a similar divide-and-conquer algorithm for searching in a sorted singly linked list of size n. Describe the steps of your algorithm in plain English. Write a recurrence equation for the runtime complexity. Solve the equation by the master theorem.
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int...
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int recurseFunc(int n) If n 0, return 1. If n 1, return 1 while i< n do while j <n do print("hi") j 1 end while i i 1 end while int a recurse Func(n/9); int b recurse Func (n/9) int c recurse Func (n/9) return a b c (1) Set up a runtime recurrence for the runtime T n) of this algorithm. (2) Solve...
3. Determine the asymptotic complexity of the function defined by the recurrence relation. Justify your solution...
3. Determine the asymptotic complexity of the function defined by the recurrence relation. Justify your solution using expansion/substitution and upper and/or lower bounds, when necessary. You may not use the Master Theorem as justification of your answer. Simplify and express your answer as O(n*) or O(nk log2 n) whenever possible. If the algorithm is exponential just give exponential lower bounds c) T(n) T(n-4) cn, T(0) c' d) T(n) 3T(n/3) c, T() c' e) T(n) T(n-1)T(n-4)clog2n, T(0) c'
3. Determine the...
*algorithm analysis and design* Solve the following recurrence relation T(n) = Tỉn/2) + 1 Using: 1-Recurrence...
*algorithm analysis and design*
Solve the following recurrence relation T(n) = Tỉn/2) + 1 Using: 1-Recurrence Tree. 2-Master Therom.
Question #4 (15 points) In class, we discussed a divide-and-conquer algorithm for matrix multiplication that involved...
Question #4 (15 points) In class, we discussed a divide-and-conquer algorithm for matrix multiplication that involved solving eight subproblems, each half the size of the original, and performing a constant number of e(n) addition operations. Strassen's Algo- rithm, which we did not cover, reduces the number of (half-sized) subproblems to seven, with a constant number of e(n) addition and subtraction operations. Provide clear, concise answers to each of the following related questions. • (7 points). Express the runtime of Strassen's...
1. Solve the recurrence relation T(n) = 2T(n/2) + n, T(1) = 1 and prove your...
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...
Subject: Algorithm solve only part 4 and 5 please. need urgent. 1 Part I Mathematical Tools and Definitions- 20 points, 4 points each 1. Compare f(n) 4n log n + n and g(n)-n-n. Is f E Ω(g),fe 0(g)...
Subject: Algorithm
solve only part 4 and 5 please.
need urgent.
1 Part I Mathematical Tools and Definitions- 20 points, 4 points each 1. Compare f(n) 4n log n + n and g(n)-n-n. Is f E Ω(g),fe 0(g), or f E (9)? Prove your answer. 2. Draw the first 3 levels of a recursion tree for the recurrence T(n) 4T(+ n. How many levels does it have? Find a summation for the running time. (Extra Credit: Solve it) 3. Use...
Weird recursion tree analysis. Suppose we have an algorithm that on problems of size n, recursively...
Weird recursion tree analysis. Suppose we have an algorithm that on problems of size n, recursively solves two problems of size n/2, with a “local running time” bounded by t(n) for some function t(n). That is, the algorithm’s total running time T(n) satisfies the recurrence relation T(n) ≤ 2T(n/2) + t(n). For simplicity, assume that n is a power of 2. Prove the following using a recursion tree analysis (a) If t(n) = O(n log n), then T(n) = O(n(log...
Question 1. Solving Recursive Relations [3 mark]. A naive multiplication of two matrices of order n...
Question 1. Solving Recursive Relations [3 mark]. A naive multiplication of two matrices of order n requires O(nᵒ) additions. By using a divide and conquer approach, Strassen devised another algorithm that requires T(n) additions where T(n) = 7T(n/2)+cna, where c is a constant independent of n and T(1) = 0 (as multiplying two numbers re- quires no additions). Use the method of backward substitution (introduced in Week 2's lecture) to show that Strassen’s algorithm requires O(nlog27) = O(n2.81) additions, which...
Discrete Math Give a big-Theta estimate for the number of additions in the following algorithm a)...
Discrete Math
Give a big-Theta estimate for the number of additions in the following algorithm a) procedure f (n: integer) bar = 0; for i = 1 to n^3 for j = 1 to n^2 bar = bar + i + j return bar b) Consider the procedure T given below. procedure T (n: positive integer) if n = 1 return 2 for i = 1 to n^3 x = x + x + x return T(/4) + T(/4) +...
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int recurseFunc(int n) If n 0, return 1. If n 1, return 1 while i< n do while j <n do print("hi") j 1 end while i i 1 end while int a recurse Func(n/9); int b recurse Func (n/9) int c recurse Func (n/9) return a b c (1) Set up a runtime recurrence for the runtime T n) of this algorithm. (2) Solve...
3. Determine the asymptotic complexity of the function defined by the recurrence relation. Justify your solution using expansion/substitution and upper and/or lower bounds, when necessary. You may not use the Master Theorem as justification of your answer. Simplify and express your answer as O(n*) or O(nk log2 n) whenever possible. If the algorithm is exponential just give exponential lower bounds c) T(n) T(n-4) cn, T(0) c' d) T(n) 3T(n/3) c, T() c' e) T(n) T(n-1)T(n-4)clog2n, T(0) c'
3. Determine the...
*algorithm analysis and design*
Solve the following recurrence relation T(n) = Tỉn/2) + 1 Using: 1-Recurrence Tree. 2-Master Therom.
Question #4 (15 points) In class, we discussed a divide-and-conquer algorithm for matrix multiplication that involved solving eight subproblems, each half the size of the original, and performing a constant number of e(n) addition operations. Strassen's Algo- rithm, which we did not cover, reduces the number of (half-sized) subproblems to seven, with a constant number of e(n) addition and subtraction operations. Provide clear, concise answers to each of the following related questions. • (7 points). Express the runtime of Strassen's...
Subject: Algorithm
solve only part 4 and 5 please.
need urgent.
1 Part I Mathematical Tools and Definitions- 20 points, 4 points each 1. Compare f(n) 4n log n + n and g(n)-n-n. Is f E Ω(g),fe 0(g), or f E (9)? Prove your answer. 2. Draw the first 3 levels of a recursion tree for the recurrence T(n) 4T(+ n. How many levels does it have? Find a summation for the running time. (Extra Credit: Solve it) 3. Use...
Question 1. Solving Recursive Relations [3 mark]. A naive multiplication of two matrices of order n requires O(nᵒ) additions. By using a divide and conquer approach, Strassen devised another algorithm that requires T(n) additions where T(n) = 7T(n/2)+cna, where c is a constant independent of n and T(1) = 0 (as multiplying two numbers re- quires no additions). Use the method of backward substitution (introduced in Week 2's lecture) to show that Strassen’s algorithm requires O(nlog27) = O(n2.81) additions, which...
Discrete Math
Give a big-Theta estimate for the number of additions in the following algorithm a) procedure f (n: integer) bar = 0; for i = 1 to n^3 for j = 1 to n^2 bar = bar + i + j return bar b) Consider the procedure T given below. procedure T (n: positive integer) if n = 1 return 2 for i = 1 to n^3 x = x + x + x return T(/4) + T(/4) +...
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