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Do curso por University of California San Diego

Advanced Algorithms and Complexity

248 classificações

University of California San Diego

Advanced Algorithms and Complexity

248 classificações

Curso 5 de 6 em Specialization Data Structures and Algorithms

You've learned the basic algorithms now and are ready to step into the area of more complex problems and algorithms to solve them. Advanced algorithms build upon basic ones and use new ideas. We will start with networks flows which are used in more typical applications such as optimal matchings, finding disjoint paths and flight scheduling as well as more surprising ones like image segmentation in computer vision. We then proceed to linear programming with applications in optimizing budget allocation, portfolio optimization, finding the cheapest diet satisfying all requirements and many others. Next we discuss inherently hard problems for which no exact good solutions are known (and not likely to be found) and how to solve them in practice. We finish with a soft introduction to streaming algorithms that are heavily used in Big Data processing. Such algorithms are usually designed to be able to process huge datasets without being able even to store a dataset.

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Coping with NP-completeness

After the previous module you might be sad: you've just went through 5 courses in Algorithms only to learn that they are not suitable for most real-world problems. However, don't give up yet! People are creative, and they need to solve these problems anyway, so in practice there are often ways to cope with an NP-complete problem at hand. We first show that some special cases on NP-complete problems can, in fact, be solved in polynomial time. We then consider exact algorithms that find a solution much faster than the brute force algorithm. We conclude with approximation algorithms that work in polynomial time and find a solution that is close to being optimal.

Vertex Cover9:13

Metric TSP12:50

TSP: Local Search6:18

Conheça os instrutores

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

We now describe a local search heuristic for the traveling salesman problem.

We've already seen the main idea of the local search algorithm,

but let me describe it once again.

So to solve an optimization problem with a local search heuristic

means the following, we usually start with some initial solution s,

then we'll repeat the following procedure.

For a current solution s, we look into some neighborhood of this solution and

we try to find a solution in this neighborhood,

if the solution [INAUDIBLE] which is better than the current solution s.

If there is some solution such a solution s', and we replace s with s' and

repeat this and we stop when s is the best one in its neighborhood.

So first of all, the first thing to note is that this algorithm might

return a sub optimal solution because the only thing which we can guarantee about

the return solution it is that it is the local optimum, not the global optimum.

So namely it is optimum in its neighborhood.

But also, the quality and the running time of this

algorithm depends on how we define the neighborhood actually.

For example, we might define the neighborhood of some solution

as the search of all possible solutions.

Then this algorithm in one iteration will need to

go through all possible candidate solutions and to find the best one.

So, this is algorithm will be actually the same as the brute force algorithm.

How do we define the neighborhood in case of the traveling salesman problem?

Well, for this we first need to define a distance between two solutions.

We call that, in case of traveling salesman,

a candidate solution is a cycle that visits each vertex exactly once.

So assume that we have two stage cycles, s and s'.

We say that the distance between them is at most d, if we can get s'

from s by first deleting the edges, at most, the edges from s, and

then adding probably some other d edges to the resulting structure, okay?

Then we can define a neighborhood, again, as in the case with algorithm,

namely, we define a neighborhood of s with radius r as all the cycles

that we can obtain from s by changing it's most r edges.

To give you a specific example, consider the following graph

containing of eight vertices, and, again, we assume that the vertices here

are just points on a plane and the distance between any two vertices

is just equal to the distance between the corresponding points on a plane.

Assume that our initial solution looks as follows, It is clearly suboptimal but

just changing the two edges in this solution makes it optimal.

So instead of using these three edges, we use the following two edges.

This gives us an optimal solution.

Sometimes, however,

just changing two edges is not enough to get to improve a solution.

For example, consider the following situation.

Again, it is not difficult to see that this solution is suboptimal.

In particular, it is clear that instead of this vertex from this one,

instead of using this path, it would be much better to use this part.

However, changing just two edges,

by changing just two edges we cannot improve this.

So, in this case, to improve the solution to find a better solution in its

neighborhood, we need to allow changing three edges in particular instead of

this three edges, we want to use the following three edges.

Right?

So what we can see, there is a trade off between the quality of solution and

the running time of a single iteration.

Namely, when we increase the size of the bowl, the size of the neighborhood,

we increase our chances to find a better solution in a current bowl,

but we also need to go over all this solution inside this ball.

Right?

In any case, it is not excluded, the total no

of iteration of your algorithm will be exponentially increase in this case.

Even we use it is not excluded that from this solution we go to that,

to that, to that, to that, and so on and we always improve it a little bit.

And the total number of steps of iterations is exponential.

And at the same time,

it is not excluded that the quality of the final solution will be poor.

But on the other hand, this heuristic usually works well in practice, but

it probably with some additional tricks.

For example, an obvious additional trick that is reasonable to use

is to allow your algorithm to restart.

Namely, not to use just one initial solution but

to use one solution then do some local search and

then to restart from some completely different point and probably to select

many such points, probably a trend among other trend, and then return and

then the solution which is the best one that you've seen in this process.

So to conclude, the role module, let me repeat the main message.

When you face an NP-complete problem in practice, don't give up.

There are several possibilities.

First of all, it is not excluded that all the instances of your NP-complete

problem are some special cases for each, there exist an efficient algorithm.

Also, it might be the case that you can implement an algorithm

which have no theoretical upper bound on its running time.

But still, it works well in practice because it intelligently searches for

the space of all candidate solutions.

And finally, you might want to implement an algorithm, which returns not an optimal

solution, but something which is close to an optimal solution.

For some applications, it might be good enough.

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