Quicksort:
art of analysis

&

Power of Randomization

MIT OpenCourse

Introduction to Algorithms, Lec 4

Tzu-Li Tai (Gordon Tai)

@ NCKU TechOrange

What will be covered

• Learn the Quicksort algorithm

• Today is going to be all about analysis !!!

• Further learn the Randomized Quicksort

Quicksort

DIVIDE: Divide A[p...r] into left/right partition according to a pivot value, so that elements on the left is smaller than elements on the right. Calculate the index of the pivot.

CONQUER: Recursively Quicksort the partitions

The partiTion

Looks hard...right?
Well... it's actually super easy

Try to love math

Where it starts to get fun:

Performance analysis

• Worst case

• Best case

• 亂七八糟 case

亂中有序 -

Randomized quicksort

Randomly select the pivot element (instead of always using the last element as pivot).

Remember to swap the chosen pivot with last element.

What's good about randomized quicksort?

• Independent of input ordering

• You might ask: what if we still always pick a bad pivot?

• That's caused by the random-number generator, not the input.

iT'S rANDOM.

wHAT CAN WE "EXPECT"?

What we did today...

• Learn a new sorting algorithm

• More importantly, we analyzed the algorithm

• We found out that the algorithm performs very bad on worst cases. However, it perform very well on some other cases.

What we did today...

• Used a "randomization" technique to eliminate dependency on algorithm input.

• By doing so, we can expect the performance of the randomized algorithm, no matter what the input is!

• This, is the power of randomized algorithms

EXTRA:

Designing your own algorithm -

Walking a maze