Solve the recurrence relations below. Indicate which strategy you are using and show your work. Big-theta or exact answer required? Big-theta. On any problem where it is appropriate to use the Master Theorem, big-theta is sufficient (you can\'t get better from the theorem). But if the master theorem doesn\'t apply for a certain problem you\'ll have to use another technique like telescoping, or filling out a table and doing guess and check. And then you\'ll have an exact answer anyway on your way to getting big-theta. So write the exact answer too as part of your solution. a. T(1) = 1, T(N) = 2 T(N/4) + N^0.5 b. T(1) = 1, T(N) = T(N - 2) + 2, assuming N is odd. C. T(1) = 1, T(N) = 3 T(N/2) + 2N
Solution
g(n) = g(n-1) + 2n - 1 = [g(n-2) + 2(n-1) - 1] + 2n - 1 // because g(n-1) = g(n-2) + 2(n-1) -1 // = g(n-2) + 2(n-1) + 2n - 2 = [g(n-3) + 2(n-2) -1] + 2(n-1) + 2n - 2 // because g(n-2) = g(n-3) + 2(n-2) -1 // = g(n-3) + 2(n-2) + 2(n-1) + 2n - 3 ... = g(n-i) + 2(n-i+1) +...+ 2n - i ... = g(n-n) + 2(n-n+1) +...+ 2n - n = 0 + 2 + 4 +...+ 2n - n // because g(0) = 0 // = 2 + 4 +...+ 2n - n = 2*n*(n+1)/2 - n // using arithmetic progression formula 1+...+n = n(n+1)/2 // = n^2 T(m,n) = 2*T(m/2,n/2) + m*n, m > 1, n > 1 T(m,n) = n, if m = 1 T(m,n) = m, if n = 1 We can solve this recurrence using the iteration method as follows. Assume m <= n. Then T(m,n) = 2*T(m/2,n/2) + m*n = 2^2*T(m/2^2,n/2^2) + 2*(m*n/4) + m*n = 2^2*T(m/2^2,n/2^2) + m*n/2 + m*n = 2^3*T(m/2^3,n/2^3) + m*n/2^2 + m*n/2 + m*n ... = 2^i*T(m/2^i,n/2^i) + m*n/2^(i-1) +...+ m*n/2^2 + m*n/2 + m*n Let k = log_2 m. Then we have T(m,n) = 2^k*T(m/2^k,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*T(m/m,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*T(1,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n/2^k + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n*(2-1/2^k) = Theta(m*n)
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Solve the recurrence relations below- Indicate which strategy you are.docx
1. Solve the recurrence relations below. Indicate which strategy you are using and show your work.
Big-theta or exact answer required? Big-theta. On any problem where it is appropriate to use the
Master Theorem, big-theta is sufficient (you can't get better from the theorem). But if the master
theorem doesn't apply for a certain problem you'll have to use another technique like
telescoping, or filling out a table and doing guess and check. And then you'll have an exact
answer anyway on your way to getting big-theta. So write the exact answer too as part of your
solution. a. T(1) = 1, T(N) = 2 T(N/4) + N^0.5 b. T(1) = 1, T(N) = T(N - 2) + 2, assuming N is
odd. C. T(1) = 1, T(N) = 3 T(N/2) + 2N
Solution
g(n) = g(n-1) + 2n - 1 = [g(n-2) + 2(n-1) - 1] + 2n - 1 // because g(n-1) = g(n-2) + 2(n-1) -1 // =
g(n-2) + 2(n-1) + 2n - 2 = [g(n-3) + 2(n-2) -1] + 2(n-1) + 2n - 2 // because g(n-2) = g(n-3) + 2(n-
2) -1 // = g(n-3) + 2(n-2) + 2(n-1) + 2n - 3 ... = g(n-i) + 2(n-i+1) +...+ 2n - i ... = g(n-n) + 2(n-
n+1) +...+ 2n - n = 0 + 2 + 4 +...+ 2n - n // because g(0) = 0 // = 2 + 4 +...+ 2n - n = 2*n*(n+1)/2
- n // using arithmetic progression formula 1+...+n = n(n+1)/2 // = n^2 T(m,n) = 2*T(m/2,n/2) +
m*n, m > 1, n > 1 T(m,n) = n, if m = 1 T(m,n) = m, if n = 1 We can solve this recurrence using
the iteration method as follows. Assume m <= n. Then T(m,n) = 2*T(m/2,n/2) + m*n =
2^2*T(m/2^2,n/2^2) + 2*(m*n/4) + m*n = 2^2*T(m/2^2,n/2^2) + m*n/2 + m*n =
2^3*T(m/2^3,n/2^3) + m*n/2^2 + m*n/2 + m*n ... = 2^i*T(m/2^i,n/2^i) + m*n/2^(i-1) +...+
m*n/2^2 + m*n/2 + m*n Let k = log_2 m. Then we have T(m,n) = 2^k*T(m/2^k,n/2^k) +
m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*T(m/m,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 +
m*n/2 + m*n = m*T(1,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n/2^k +
m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n*(2-1/2^k) = Theta(m*n)
![Solve the recurrence relations below. Indicate which strategy you are using and show your work.
Big-theta or exact answer required? Big-theta. On any problem where it is appropriate to use the
Master Theorem, big-theta is sufficient (you can't get better from the theorem). But if the master
theorem doesn't apply for a certain problem you'll have to use another technique like
telescoping, or filling out a table and doing guess and check. And then you'll have an exact
answer anyway on your way to getting big-theta. So write the exact answer too as part of your
solution. a. T(1) = 1, T(N) = 2 T(N/4) + N^0.5 b. T(1) = 1, T(N) = T(N - 2) + 2, assuming N is
odd. C. T(1) = 1, T(N) = 3 T(N/2) + 2N
Solution
g(n) = g(n-1) + 2n - 1 = [g(n-2) + 2(n-1) - 1] + 2n - 1 // because g(n-1) = g(n-2) + 2(n-1) -1 // =
g(n-2) + 2(n-1) + 2n - 2 = [g(n-3) + 2(n-2) -1] + 2(n-1) + 2n - 2 // because g(n-2) = g(n-3) + 2(n-
2) -1 // = g(n-3) + 2(n-2) + 2(n-1) + 2n - 3 ... = g(n-i) + 2(n-i+1) +...+ 2n - i ... = g(n-n) + 2(n-
n+1) +...+ 2n - n = 0 + 2 + 4 +...+ 2n - n // because g(0) = 0 // = 2 + 4 +...+ 2n - n = 2*n*(n+1)/2
- n // using arithmetic progression formula 1+...+n = n(n+1)/2 // = n^2 T(m,n) = 2*T(m/2,n/2) +
m*n, m > 1, n > 1 T(m,n) = n, if m = 1 T(m,n) = m, if n = 1 We can solve this recurrence using
the iteration method as follows. Assume m <= n. Then T(m,n) = 2*T(m/2,n/2) + m*n =
2^2*T(m/2^2,n/2^2) + 2*(m*n/4) + m*n = 2^2*T(m/2^2,n/2^2) + m*n/2 + m*n =
2^3*T(m/2^3,n/2^3) + m*n/2^2 + m*n/2 + m*n ... = 2^i*T(m/2^i,n/2^i) + m*n/2^(i-1) +...+
m*n/2^2 + m*n/2 + m*n Let k = log_2 m. Then we have T(m,n) = 2^k*T(m/2^k,n/2^k) +
m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*T(m/m,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 +
m*n/2 + m*n = m*T(1,n/2^k) + m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n/2^k +
m*n/2^(k-1) +...+ m*n/2^2 + m*n/2 + m*n = m*n*(2-1/2^k) = Theta(m*n)](https://image.slidesharecdn.com/solvetherecurrencerelationsbelow-indicatewhichstrategyyouare-230209002420-4067a280/85/Solve-the-recurrence-relations-below-Indicate-which-strategy-you-are-docx-1-320.jpg)
