#### Transcript COMP 320: Introduction to Computer Organization

**Asymptotes: Why?**

How to describe an algorithm’s running time?

(or space, …) How does the running time depend on the input?

T(x) = running time for instance x

**Problem:**

Impractical to use, e.g., “15 steps to sort [3 9 1 7], 13 steps to sort [1 2 0 3 9], …” Need to abstract away from the individual instances.

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**Asymptotes: Why?**

**Standard solution:**

Abstract based on

**size**

of input.

How does the running time depend on the input?

T(n) = running time for instances of size n

**Problem:**

Time also depends on other factors.

E.g., on sortedness of array.

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**Asymptotes: Why?**

**Solution:**

Provide a bound over these instances.

Most common. Default.

Worst case Best case Average case T(n) = max{T(x) | x is an instance of size n} T(n) = min{T(x) | x is an instance of size n} T(n) = |x|=n Pr{x} T(x) Determining the input probability distribution can be difficult.

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**Asymptotes: Why?**

What’s confusing about this notation?

Worst case Best case Average case T(n) = max{T(x) | x is an instance of size n} T(n) = min{T(x) | x is an instance of size n} T(n) = |x|=n Pr{x} T(x) Two different kinds of functions: T(instance) T(size of instance) Won’t use T(instance) notation again, so can ignore.

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**Asymptotes: Why?**

**Problem:**

T(n) = 3n 2 + 14n + 27 Too much detail: constants may reflect implementation details & lower terms are insignificant.

**Solution:**

Ignore the constants & low-order terms.

(Omitted details still important pragmatically.) n 1 10 3n 2 3 300 14n+17 31 157 100 1000 30,000 3,000,000 1,417 14,017 10000 300,000,000 140,017 3n 2 > 14n+17 “large enough” n 5

**Upper Bounds**

Creating an algorithm proves we can solve the problem within a given bound.

But another algorithm might be faster.

E.g., sorting an array.

Insertion sort O(n 2 ) What are example algorithms for O(1), O(log n), O(n), O(n log n), O(n 2 ), O(n 3 ), O(2 n )?

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**Lower Bounds**

Sometimes can prove that we cannot compute something without a sufficient amount of time.

That doesn't necessarily mean we know how to compute it in this lower bound.

E.g., sorting an array.

# comparisons needed in worst case (n log n) Shown in COMP 482.

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**Definitions: O, **

T(n) O(g(n)) constants C,k > 0 such that n k, T(n) C g(n) C g(n) T(n) (g’(n)) constants C’,k’ > 0 such that n k’, T(n) C’ g’(n) T(n) C’ g’(n) k k’ 8

**Examples: O, **

2n+13 2n+13 O( ?

( ?

) ) O(n) Also, O(n 2 ), O(5n), … Can always weaken the bound.

(n), also (log n), (1), … 2 n O(n) ?

(n) ?

Given a C, 2 n (n), not O(n).

C n, for all but small n.

n log n O(n 5 ) ?

No. Given a C, log n C 5, for all large enough n. Thus, (n 5 ).

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**Definitions: **

T(n) T(n) (g(n)) O(g(n)) and T(n) (g(n)) Ideally, find algorithms that are asymptotically as good as possible.

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**Notation**

O(), (), () are sets of functions.

But common to abuse notation, writing T(n) = O(…) instead of T(n) O(…) as well as T(n) = f(n) + O(…) 11