This course covers the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications and scientific performance analysis of Java implementations. Part I covers elementary data structures, sorting, and searching algorithms. Part II focuses on graph- and string-processing algorithms.
Edge-Weighted DAGs
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Algorithms, Part II
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Shortest Paths
In this lecture we study shortest-paths problems. We begin by analyzing some basic properties of shortest paths and a generic algorithm for the problem. We introduce and analyze Dijkstra's algorithm for shortest-paths problems with nonnegative weights. Next, we consider an even faster algorithm for DAGs, which works even if the weights are negative. We conclude with the Bellman−Ford−Moore algorithm for edge-weighted digraphs with no negative cycles. We also consider applications ranging from content-aware fill to arbitrage.
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Robert SedgewickWilliam O. Baker *39 Professor of Computer Science
Computer Science
Kevin WayneSenior Lecturer
Computer Science
Now we're going to look at shortest paths and edge weighted dags.
This is a very common situation and we'll see a couple of important applications.
Suppose that you've got an edge weighted digraph.
That's our input to shortest paths. But you konw there's no directed cycles.
And actually that's true and in many applications we see some.
The question is this fact that it has no directed cycles make it easier to find the
surest path than in a general digraph, And the answer is yes, it certainly does.
So lets take a look at what happens. And the algorithm is very simple.
What we're going to do is, just consider the vertices in topological order.
Since there's no cycles, we know there's a topological ordering where we can lay out
the graph and all the edges point to vertices that we haven't seen yet.
And so we're just relaxed all edges, pointing from each vertex, taking them in
topological order. Again, this'll be easy to see in an
example. Alright, so here's a edge-weighted
directed acyclic graph. And the first thing that we do is sort it
in topological, Sort the vertices in topological order.
And we talked about an algorithm for doing that a couple of lectures ago.
So just using that first search. So we consider vertex zero, and relax all
the edges coming from vertex zero. And that gives us shortest paths to one
four and seven. So next we consider vertex one.
We don't do anything except take the next vertex in topological order.
We could of also taken four in this case. And so we're going to add that.
And then just relax along all the edges coming from that vertex.
In this case that gives us new shortest paths to two and to three.
Next is four relax and that gives us new shortest paths to five and six.
Next in topological left quarter is seven. Relax and that gives us a new shortest
path to two, but not to five. The path that we had to five was better.
So next in the order is five. Relax along its edges.
And it gives us better ways to get to both two and to six.
Then 2's next. Relax along its edges.
And it gives us, new better ways to get both three and six.
Then we do three, and relax. And that doesn't help.
And then we do six. And, when we're done all we did was
consider the edges in topological order and relax.
Then we have a shortest path tree. From the source to every vertex in the
directed acyclic weighed digraph. So that's a demo of a simple algorithm for
that case. Alright, why does that algorithm
considering the vertices and topological work for edge-weighted dags.
Now, one thing that's pretty important about this is that the edge weights could
even be negative doesn't matter to this algorithm.
So let's look again. Every edge is relaxed exactly once.
When, we add a vertex to the tree we, relax each of the edges that go from that
vertex. And, again, just as, with Dykestra, after
we're done with the relaxation, we have this condition holds that, distance to w
is less than or equal to the distance to v plus the edge weight.
Either it was before-hand and we ignored the weight, or we made it equal.
And so. when we relax an edge we have that condition.
And again, inequality holds until the algorithm terminates.
Well, first, again, we only. Decrease values in the distribute array.
We don't change it to increase. So the only thing that can happen to this
2W is that it gets smaller. And that's not going to violate the
inequality. And then the other thing is, this 2V's not
going to change because of the topological order.
When we process v, it's in topological order.
That means we're not going to find an edge that points 2V.
So this two v's not going to change. And edge weight doesn't change, so the
inequality holds. And this is true even if the edge weights
are negative. So shortest path optimality conditions
hold. And so this simple algorithm finds
shortest paths in edge-weighted directed acyclic graphs.
Whether or not the edge weights are positive or negative.
Pretty easy algorithm to implement. So this, acyclic, shortest path.
So, the goal is to compute edge two and disc two.
And then we use those to respond to client queries of the length of the shortest path
and the path itself. This is the constructor.
We build the arrays. We initialize this distances to infinity.
Distance the source to zero. Then we use our topological sort algorithm
on di-graphs to compute an iterable that gives us the vertices in topological
order. And that's the order method in this
topological class. So that's using the API that we set up for
topological sorting, Which has to be adapted to weighted
di-graphs. But that's just adding some nomenclature.
So we take the vertices in topological order.
We take for every vertex in topological order, taken in topological order.
We take every edge that emanates from that vertex and relax it.
It couldn't be simpler. And then we're done we have the other
shortest pass. And that works even for negative edge
weights. Pretty simple.
Now, as an example of a, familiar application of, shortest paths in dags and
weighted dags we'll look at, the idea of scene carving.
And this is a relatively recent algorithm that, was developed that has, important
application that you'll see, in this, YouTube film clip.
Scene carving for content-aware image resizing.
Digital media today has the ability to support dynamic page layouts.
By changing the window or display size, we can change the layout of a document.
However, images imbedded in a document typically remain rigid in size and shape.
Resizing is also important to put images into different displays.
Current techniques, including cropping or scaling, are limited in their abilities to
capture the image content. We present a method for content aware
resizing of images. Our technique enables resizing while
adapting the image content and layout. This is sometimes called re-targeting.
We also define a flexible, multi-size representation for images that supports
continuous resizing. An image can be shrunk in a non-uniform
manner while preserving its features, but it can also be extended beyond its
original size. Instead of scrolling through images that
do not fit in a given display, we gracefully re-size them to fit inside the
window. For example, assume that we need to change
the width of an image. The simplest way to do this is to remove
columns of pixels from the image. The best column to remove would be the
least noticeable, or least important column.
We can search for this column by defining the importance or energy function on the
image. In this example, we use the gradient
magnitude of the image. Next, we look for the column which
contains the least energy and remove it. However, using such an approach quickly
leads to serious artifacts. Therefore, instead of using rigid columns,
we search for connected paths of pixels, or seams, from one side of the image to
the other, that contain the least energy. This can be done using a simple dynamic
programming algorithm as described in the paper in both vertical and horizontal
directions. Here's another example of an image, its
energy function, and the least noticeable horizontal and vertical seams.
By successively removing horizontal and vertical seams, the image can be resized
in a non-uniform manner. The order of seam removal in an image
defines an order on the pixels on the image by storing the simple indexing
information we create content to where multisize images.
So that's a amazingly powerful image processing process.
And you'll find it all over image processing applications, from Photoshop to
GIMP to Image Magic. And actually nowadays almost any image
that you see on your cell phone or your Tablet.
And what really, does that have to do with shortest paths?
Well, it's actually a direct application of shortest paths on a directed acyclic
graph. So what we do is, build a huge directed
acyclic graph. And this is a really huge graph.
Because images now, at high resolution, can have millions or billions of pixels.
And so. Every pixel corresponds to a vertex in
this graph. And the edges are gonna be just directed
edges from every pixel to its three downward neighbors.
And then there's, what weight do we assign to the pixels?
Well, there's an energy function that has to do with the image processing, that is,
a function of the eight neighboring pixels.
Every, every pixel, has a relationship with all its neighboring pixels.
And, the energy function, has it, is, is a image processing, concept that, is.
Perfect for, assigning weights, in, in this instance.
And so, that gives, the weights. And then, a seam is just the shortest
path, that goes from the top to the bottom.
So you can, imagine an imaginary source that goes all to the top ones, and, just,
run the shortest pass algorithm that we just gave, and, it'll find a seam.
So that's the lowest energy, one pixel cut, through the image, and then, the
resizing algorithm, just deletes, one, one pixel on each row, along that seam, and
that's the algorithm. Which.
The shortest pass algorithm, I'im I'm sorry, allows and enables the
re-sizing for a really huge graph. And, and it's really important to realize
that you have to have an efficient implementation of the shortest pass
algorithm. Because there's so many pixels.
And certainly the topological sort algorithm that we gave is extremely
efficient linear time algorithm. And you can see the effects of that
efficient algorithm all around you. Here's another completely different
application of shortest paths in directed acyclic graphs.
And the idea is that actually since negative weights are allowed, we can find
longest paths in edge-weighted DAGs, just by negating all the weights.
So what I want is I have edge. Again I have an edge weighted dag.
And what I want is, this is with, I guess five as the source.
I, I don't want the shortest path from five to two.
I want the longest path from five to two. We'll see an application why that's
important in a minute. And in order to get that, all I do is just
negate all the weights, and the run shortest paths.
And if you add up negative weights to get the smallest negative number, then to
negate the answer that's the longest path, and it works because the algorithm, top
logical sort algorithm, doesn't care whether the weights are positive or
negative. In general graphs it does matter as we'll
see in a minute, but for a cyclic graph it doesn't matter, we can find longest path
in a cyclic graph just by negating all the weights, therefore we can solve the
longest paths problem. So that's a key point.
And now, now you can look at applications that had problems.
And there's a really important application for longest paths in edge weighted
directed A-cyclic graphs. And that's called the Job Scheduling
Problem. And this is just another example to show
the importance of the shortest paths problem as a problem solving model.
This particular problem doesn't seem related to shortest paths at all.
But we'll show how to formulate it as a shortest paths problem.
And that's great, because we have an efficient solution to the shortest paths
problems. Or actually, it's a longest paths problem
in this case. So this is just an example that arises
say, in man, in manufacturing. Or other applica, applications we have a
complex set of interacting processes. So this we call, job scheduling.
Parallel job scheduling. So we've got a bunch of let's say a
factory manager has a bunch of jobs to manage, And they have.
Durations and precedents constraints. So durations just means the job takes a
certain amount of time. And precedents constraints means that you
can't start job one until after job zero is done, or seven, or nine.
One, seven and nine have to start sometime after jo-, job zero.
And similarly three and eight have to start after six, and so forth.
So maybe this is manufacturing a car. And, you know you can't, paint the car
until after you put the doors on. Or whatever else you can imagine.
Easily, setting up, precedence constraints, that are natural, for trying
to complete this whole, set of jobs. And so what we want to do is, find a start
time for each job. That minimizes the completion time.
While still respecting all the precedence constraints.
So when do we start each job? That's the parallel job scheduling
problem. We, we assume that we got enough workers
that we can have a bunch of jobs going on at the same time, same time.
So, This graph model shows that we can change the job scheduling problem into a
graph processing problem. So, what we're going to do is, create an
edge weighted directed acyclic graph the following way.
We're going to have a source and sync vortex that the source is, begin
everything and the sync is, end everything.
And then, well at the zero weightage from the.
For every job will have a start and a finish vertex for that job, and then we'll
have an edge, whose weight is the duration.
And from the finished vertex of every job, we'll have edges to the start vertex of
the jobs that it has to complete before. That's, those are the precedence
constraints. We have a zero weight edge from, the,
overall source to every job start, and a zero weight edge from the overall, from
every job finish to, the, sink vertex. And so, in summary, there's three edges
for every job. There's from the begin to the end, the
start to the finish, weighted by the duration.
From the source to the beginning, zero weight, and from the end to the sync, zero
weight, in the edges for the precedence constraints, also have zero weight.
So that's a graph model, we took our scheduling problem and now we have a
graph. And what relates this to what we've been
talking about is the longest path from the source to each job.
It turns out to give a schedule. So the job scheduling problem corresponds
to a solution to the longest path problem in directed acyclic graph by the way this
graph doesn't have any cycles because that would mean we have to do a job before
another job insist that one be done after zero and that two be done after one.
We can't also insist that zero be done after two because that would be a present
cycle that couldn't be satisfied at all. And so the, the n-, now you have to think
of this quite a while to understand why the longest path from the source.
We'll schedule each job, but, that's a fact in that it means, then, we have, a
fast, linear time algorithm for solving this problem.
Negate all the weights, run shortest paths with topological sort, and negate the
answer, and you have the start times for every job.
In fact, in the book and the book site, you'll find code that not solves, this,
schedule, parallel job scheduling problem using the critical path method, Again,
showing how important it is to have, a fast and efficient solution to the
shortest paths problem.
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