Bounded functions

Show that 8𝑛^2 βˆ’ 5𝑛 + 7 = 𝑂(𝑛^3)
To show that 8𝑛^2 βˆ’ 5𝑛 + 7 = 𝑂(𝑛^3), we need to find constants C and k such that for all values of n greater than k, the inequality 8𝑛^2 βˆ’ 5𝑛 + 7 ≀ C𝑛^3 holds true.
Let’s simplify the equation:
8𝑛^2 βˆ’ 5𝑛 + 7 ≀ C𝑛^3
Rearranging the terms:
8𝑛^2 βˆ’ 5𝑛 + 7 – C𝑛^3 ≀ 0
Now, let’s take the limit as n approaches infinity:
lim(n->∞) (8𝑛^2 βˆ’ 5𝑛 + 7 – C𝑛^3) ≀ 0
Since the limit is less than or equal to zero, we can conclude that 8𝑛^2 βˆ’ 5𝑛 + 7 = 𝑂(𝑛^3).
Determine whether each of these functions is bounded by O(n). Say yes or no.
a) 𝑓(𝑛) = 3Yes, it is bounded by O(n) because it is a constant function.
b) 𝑓(𝑛) = 𝑛^2 + 𝑛 + 12Yes, it is bounded by O(n) because the highest degree term is n^2, which grows slower than n^3.
c) 𝑓(𝑛) = (𝑛)Yes, it is bounded by O(n) because it is a linear function.
d) 𝑓(𝑛) = 17𝑛 + 7Yes, it is bounded by O(n) because it is a linear function.
e) 𝑓(𝑛) = 15𝑛 log 𝑛Yes, it is bounded by O(n log n) because it grows slower than n^3.
f) 𝑓(𝑛) = (𝑛/2)Yes, it is bounded by O(n) because it is a linear function.
Arrange the following functions in ascending order of growth rate.
𝑓3(𝑛) = (log 𝑛)^3𝑓4(𝑛) = βˆšπ‘› log 𝑛𝑓2(𝑛) = 𝑛^100 + 𝑛^3 log 𝑛𝑓6(𝑛) = 𝑛^99 + 𝑛^98𝑓1(𝑛) = 1.5^𝑛 + 50𝑓7(𝑛) = (𝑛!)^2𝑓5(𝑛) = 10^𝑛
Solve the following recurrences using Master Theorem.
a) 𝑇(𝑛) = 16𝑇 (𝑛/4)+ π‘›βˆšπ‘›The recurrence relation can be written as π‘Ž=16, 𝑏=4, and 𝑐=1/2.Since logβ‚„16 = 2, and 𝑐 = 1/2 < logβ‚„16, we apply case 1 of the Master Theorem.The time complexity of this recurrence relation is Θ(n²√n).
b) 𝑇(𝑛) = 𝑇 (𝑛/5)+ 10The recurrence relation can be written as π‘Ž=1, 𝑏=5, and 𝑐=0.Since logβ‚…1 = 0, and 𝑐 = 0 = logβ‚…1, we apply case 2 of the Master Theorem.The time complexity of this recurrence relation is Θ(log n).
In the divide and conquer closest pair of points, we sorted the coordinates (x or y). Once sorted, are we ready to start computing distances?
No, once the coordinates are sorted, we need to divide the points into smaller subsets and recursively compute the closest pair of points within each subset. Sorting the coordinates is a preprocessing step that allows us to efficiently divide the points and apply the divide and conquer strategy. After dividing the points and finding the closest pairs within each subset, we need to merge the results and compare distances to find the overall closest pair of points. So, sorting the coordinates alone does not complete the computation of distances.


Best Custom Essay Writing Services

Looking for unparalleled custom paper writing services? Our team of experienced professionals at is here to provide you with top-notch assistance that caters to your unique needs.

We understand the importance of producing original, high-quality papers that reflect your personal voice and meet the rigorous standards of academia. That’s why we assure you that our work is completely plagiarism-freeβ€”we craft bespoke solutions tailored exclusively for you.

Why Choose

  • Our papers are 100% original, custom-written from scratch.
  • We’re here to support you around the clock, any day of the year.
  • You’ll find our prices competitive and reasonable.
  • We handle papers across all subjects, regardless of urgency or difficulty.
  • Need a paper urgently? We can deliver within 6 hours!
  • Relax with our on-time delivery commitment.
  • We offer money-back and privacy guarantees to ensure your satisfaction and confidentiality.
  • Benefit from unlimited amendments upon request to get the paper you envisioned.
  • We pledge our dedication to meeting your expectations and achieving the grade you deserve.

Our Process: Getting started with us is as simple as can be. Here’s how to do it: is dedicated to expediting the writing process without compromising on quality. Our roster of writers boasts individuals with advanced degreesβ€”Masters and PhDsβ€”in a myriad of disciplines, ensuring that no matter the complexity or field of your assignment, we have the expertise to tackle it with finesse. Our quick turnover doesn’t mean rushed work; it means efficiency and priority handling, ensuring your deadlines are met with the excellence your academics demand.

ORDER NOW and experience the difference with, where excellence meets timely delivery.