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testing subscripting

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Jidong Xiao
2023-09-19 23:32:54 -04:00
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10
lectures/07_order_notation_recursion/README.md
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@@ -47,14 +47,10 @@ Which company has the “best” algorithm?
In this course we will focus on the intuition of order notation. This topic will be covered again, in more depth, in In this course we will focus on the intuition of order notation. This topic will be covered again, in more depth, in
later computer science courses. later computer science courses.
- Definition: Algorithm A is order f(n) — denoted O(f(n)) — if constants k and n0 exist such that A requires - Definition: Algorithm A is order f(n) — denoted O(f(n)) — if constants k and n<sup>0</sup> exist such that A requires no more than k * f(n) time units (operations) to solve a problem of size n ≥ n<sup>0</sup>.
no more than k ∗ f(n) time units (operations) to solve a problem of size n ≥ n0.
- For example, algorithms requiring 3n + 2, 5n − 3, and 14 + 17n operations are all O(n). - For example, algorithms requiring 3n + 2, 5n − 3, and 14 + 17n operations are all O(n).
This is because we can select values for k and n0 such that the definition above holds. (What values?) This is because we can select values for k and n0 such that the definition above holds. (What values?)
Likewise, algorithms requiring n 2/10 + 15n − 3 and 10000 + 35n Likewise, algorithms requiring n<sup>2</sup>/10 + 15n − 3 and 10000 + 35n<sup>2</sup> are all O(n<sup>2</sup>).
2 are all O(n
2
).
- Intuitively, we determine the order by finding the asymptotically dominant term (function of n) and throwing - Intuitively, we determine the order by finding the asymptotically dominant term (function of n) and throwing
out the leading constant. This term could involve logarithmic or exponential functions of n. Implications for out the leading constant. This term could involve logarithmic or exponential functions of n. Implications for
analysis: analysis:
@@ -71,7 +67,7 @@ quadratic root.
- O(n), a.k.a. LINEAR. e.g., sum up a list. - O(n), a.k.a. LINEAR. e.g., sum up a list.
- O(n log n), e.g., sorting. - O(n log n), e.g., sorting.
- O(n<sup>2</sup>), O(n<sup>3</sup>), O(n<sup>k</sup>), a.k.a. POLYNOMIAL. e.g., find closest pair of points. - O(n<sup>2</sup>), O(n<sup>3</sup>), O(n<sup>k</sup>), a.k.a. POLYNOMIAL. e.g., find closest pair of points.
- O(2n), O(kn), a.k.a. EXPONENTIAL. e.g., Fibonacci, playing chess. - O(2<sup>n</sup>), O(k<sup>n</sup>), a.k.a. EXPONENTIAL. e.g., Fibonacci, playing chess.
## 7.6 Exercise: A Slightly Harder Example ## 7.6 Exercise: A Slightly Harder Example