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CMSC 132 Exam Review With 100% Correct And Verified Answers 2024
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CMSC 132 Exam Review With 100% Correct And Verified Answers 2024
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CMSC 132 Exam Review With 100% Correct And Verified Answers 2024Analyzing runtime Insert element into position 0 of an array of size n (in java) - Correct Answer-Runs in linear time, (if the array is size n and you double it, time will be doubles as well, this is a good way to think about linear time).
Analyzing runtime Retrieving an element from an array of size n (at a particular index, in java). - Correct Answer-When the CPU has to do the arithmetic to get the element from the specified index, it's a constant time operation. The arithmetic barley takes any time regardless of what index it is asking for.
(note: the size of arrays in java are bounded to 2^32)
Analyzing runtime Program that prints all of the n-digit numbers - Correct Answer-It is an exponential function multiplied by a linear function.
Think about 2 parabolas where one is shallow and the other is steep.
Can we make the shallow one worse than the steep one by multiplying it by a large constant? - Correct Answer-- Yes, all parabolas are in the same "ballpark."
- It doesn't matter if one parabola starts better by the other, if you multiply it by a big enough number, it is the same thing.
Imagine a parabola and a shallow line where the line is clearly faster, what if we start multiplying it? will the line still be better? - Correct Answer-- Multiplying does slow it down but even when we multiply by a large number, after the crossover we still see that the red line is faster. - As n goes to infinity, there is no way u can multiply the line by a number and make it worse.
- Conclusion: lines are better/faster than parabolas.
f(n) as O(g(n)) means: - Correct Answer-As n goes toward infinity, f is either better, (faster/smaller) than g, or "in the same ballpark."
What does "In the same ballpark" mean? - Correct Answer-- Even though g might seem
better (smaller), we can multiply it by a big number and make it worse (bigger) than f(for
big values of n). - We can find a big enough value m, so that f(n) < mg(n) for sufficiently large n. On a test Fawzi may ask: show 3n^2 + 15n + 20 is O(n^2)
How would you show that? - Correct Answer-- Determine what number to multiply by n^2 to show that it can be worse than the first function.
- answer: "I choose the multiplier n = 4, now 3n^2 + 15n + 20 < m(n^2) as long as n > 20.
Show 100n + 150 is O(n) - Correct Answer-- Keep plugging numbers into n until you could prove the answer is true.
- I choose m = 101
- Now 100n + 150 < m(n) as long as n > 1000
What is the big O of the binary search algorithm? - Correct Answer-O(log n)
If f is linear and g is quadratic, - Correct Answer-then f is O(g) but g is NOT O(f)
If f is linear and g is logarithmic, - Correct Answer-then f is NOT O(g) but g IS O(f)
if f(n) = 2^n and g(n) = 3^n, - Correct Answer-then f is O(g) but g is not O(f)
f is O(g) meaning - Correct Answer-- f is either better (smaller) than g or at least not dramatically worse.
- this is big O notation
f is o(g) meaning - Correct Answer-f is dramatically better (smaller) than g.
Suppose we have two functions, f(n) and g(n), and we want to know: Are they "in the same ballpark"? if not, which one is better/worse?
evaluate this limit: lim (f(n)/g(n))
n --> infinity
What does it mean if this limit comes out as 0? - Correct Answer-If this limit comes out as zero that means whatever is in the denominator is getting bigger dramatically faster than whatever is on top, so f is faster. We can write this in little o notation:
f(n) is o(g(n))
Suppose we have two functions, f(n) and g(n), and we want to know: Are they "in the same ballpark"? if not, which one is better/worse?
evaluate this limit: lim (f(n)/g(n))
n --> infinity What does it mean if this limit diverges? (goes to infinity) - Correct Answer-If the limit diverges (goes to infinity) f is growing much faster so in little o notation we would write:
g(n) is o(f(n))
Suppose we have two functions, f(n) and g(n), and we want to know: Are they "in the same ballpark"? if not, which one is better/worse?
evaluate this limit: lim (f(n)/g(n))
n --> infinity
What does it mean if this limit equals a non zero constant? - Correct Answer-f(n) is theta(g(n))
(check lecture 40 for theta notation)
Analyzing code fragments
for (int i= 0; i< n; i++) {
System.out.println("Hi");
} - Correct Answer-O(n)
Analyzing code fragments
for (int i= 0; i< 100 * n; i++) {
System.out.println("HI");
} - Correct Answer-O(n)
Analyzing code fragments
for (int i= 0; i< n; i++) { for (int j = 0; j < n; j++) { System.out.println("HI"); } } - Correct Answer-O(n^2)
Analyzing code fragments
for (int i= 0; i< n; i++) { for (int j = i; j < n; j++) { System.out.println("HI"); } } - Correct Answer-O(n^2) - represents gauss' formula
Analyzing code fragments
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