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Algorithmic Toolbox

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University of California San Diego

Algorithmic Toolbox

4233 notes

Cours 1 sur 6 dans Specialization Data Structures and Algorithms

The course covers basic algorithmic techniques and ideas for computational problems arising frequently in practical applications: sorting and searching, divide and conquer, greedy algorithms, dynamic programming. We will learn a lot of theory: how to sort data and how it helps for searching; how to break a large problem into pieces and solve them recursively; when it makes sense to proceed greedily; how dynamic programming is used in genomic studies. You will practice solving computational problems, designing new algorithms, and implementing solutions efficiently (so that they run in less than a second).

À partir de la leçon

Divide-and-Conquer

In this module you will learn about a powerful algorithmic technique called Divide and Conquer. Based on this technique, you will see how to search huge databases millions of times faster than using naïve linear search. You will even learn that the standard way to multiply numbers (that you learned in the grade school) is far from the being the fastest! We will then apply the divide-and-conquer technique to design two efficient algorithms (merge sort and quick sort) for sorting huge lists, a problem that finds many applications in practice. Finally, we will show that these two algorithms are optimal, that is, no algorithm can sort faster!

Random Pivot13:04

Running Time Analysis (optional)15:32

Equal Elements6:31

Final Remarks8:12

Rencontrer les enseignants

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering
Pavel Pevzner
Professor
Department of Computer Science and Engineering
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics

In this video, we'll provide the full outline of the Greek word algorithm.

So as you remember that algorithm is recursive, for

this reason we pass through this procedure [INAUDIBLE] a and

also doing this is l and r in this array for left and right and

this procedure saw the sub array inside r within this is form l to r.

Well we first check whether l is at least r.

And if yes then this means that they can respond in sub array contains at

most one element.

And this in turn means that nothing needs to be done so we just return.

Otherwise we call the partition procedure with the same parameters.

It returns an index m between l and r.

So it rearranges all the elements inside

this sub array with the following property.

After the call to this procedure, A of m stays in its final position,

meaning that all the elements to the left of an element A of m are at most A of m.

And all the elements to the right are greater than A of m.

Well once again, after the call to the partition procedure,

A of m stays in its final position.

So what remains to be done is to sort all the elements that are at most A to m.

They stay to the left of A of m, and all the elements that stay to the right.

So we do this just by making two recursive calls.

So this is how the wall algorithm looks pictorially.

Again, we are given an array A, with two indices l and r,

and we are going to sort the sub array inside from indices L to R.

So with first call the participation procedure which parameter A, l and r.

And it gives us an index m between l and r was the following property.

All elements to the left of them are at most the element A of m.

All the elements to the right are great as an A of m.

Then we make two recursive calls to sort the left part within this is from

l to m- 1 and to solve the right part within this is from m + 1 to r.

And immediately after these two recursive call, we have a sorted array.

So before showing the actual of the partition procedure,

we explain it's main ideas again, on a toy example.

So first of all, we will take is the element A[l] and denoted by x.

This will be called our pivot element.

So what pivot is exactly is the element with respect to which

we're going to partition our sub array.

So x will be placed in its final position.

So our goal now is to rearrange all the elements inside our current sub array so

that x stays in its final position and all the elements to the left of x.

At most x and all the elements to the right of x are greater than x.

So we will do this gradually increasing the region of already discovered elements.

So for this we will use a counter i, and we will maintain the following invariant.

So I will go from l + 1 to r, and

at each point of time when we have already have the i element

we will keep to region in sizes these region from l + 1 to i.

In the first region from l + y to j, we will keep all elements that are at most x.

In the second adjacent region within this is from j +

1 to i we will have all elements that are greater than x.

Let's see it for example.

So I assume that we are somewhere in the middle of this process.

In this case, x is equal to 6, and

we need to partition all the elements with respect to x.

We already have two sub regions so in the red region,

we keep all elements that are at most x.

There are at most 6 in the blue region we have holds elements that

are greater than 6.

Okay, now we move i to the next position and we discover the element 9.

So this element is greater than 6, so

we just need to extend the second region, the blue region.

The region of elements is at greater than 6.

So in this case we just do nothing.

Well the next case is more interesting, we move i to the next position, and

we discover the element 4.

In this case, we need to somehow move this element to the red region,

to the region of elements which at most 6.

So to do this we just swoop it to currently

first element of the blue region, in this case was 9.

So if we do this 4 will be the last element of currently red region and

9 will go to the blue region.

So we do this and now, we increase also the just

to reflect the fact that our red region had just been extended.

Then we will find to the next element so we discover element 7 which is

greater than 6 which means that we can just extend the blue region,

then we discover another element which is 6.

6 is at most 6 and it is actually equal to 6, so

we need to move it to the red region.

Again, we swap it with the first element of the blue region and

then we extend the red region.

We increase g to reflect the fact that the red region has just been extended,

then we discover another element, which is at most 6.

We move it to the end of the red region.

And finally, what we also need to do in the very end is to move the pivot

element which is 6 in this case to its final position.

And its final position actually can easily be found in this case.

So we have red region and we have blue region.

In red region, all the elements are at most 6, and in blue region,

all the elements are greater than 6.

So we can just swap 6 with the last element of the red region.

In this case it is 1, so if we swap these two elements then you

can see that all the elements in the blue region are indeed greater than 6.

All the elements in the red region are smaller than 6.

So we are done with this partition procedure.

Where now ready to present the Soutacot of the petition procedure.

We called it what we're going to do is to place some element x,

which is called the pivot, into it's final place so that all the elements before

x are at most x and all the elements after x is greater than x.

So as the pivot element in this procedure, we are going to use just the first

element of the correspondence of rate, so x is assigned A of l.

We're also going to remain the following subregions.

So first of all, we will readily increase the region of discovered elements.

So i goes from l +1 to r and inside this region of

[INAUDIBLE] elements, we will maintain two sub regions.

In the first region with indices from l +1 to j,

we will keep all of the elements at most x.

In the second region with indices from j+1 to i, we will keep all of the elements

that are greater than x and we will gradually and freeze the value of i.

So when i is increased, so I assumed that i has just been increase so

we discovered a new element of A of i.

So if A of i is greater than x then just the second

of region of elements that are greater than x,

is extended automatically and we do not need to do anything in this case.

However, if the newly discovered element is at most x,

then we need to move it to the first region.

So we do this as follows.

So we just increase the value of j to indicate the fact that

the first region has just been increased, and then swap the elements.

A[j] and A[i], so this way,

we just maintain our invariant each time when i is increased.

So in the very end, when i reaches the value of r,

we also need to place our initial element that states that at

the beginning our pivot between our two regions.

So for this we just swap elements A[l], so this is our pivot with element A[j].

And we then return the value j as an index of our pivot element.

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