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.
Dijkstra's Algorithm
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From the course by Princeton University
Algorithms, Part II
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From the lesson
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.
Meet the Instructors
Robert SedgewickWilliam O. Baker *39 Professor of Computer Science
Computer Science
Kevin WayneSenior Lecturer
Computer Science
Now we'll look for, at Dijkstra's algorithm for finding shortest paths in
graphs with non-negative edge weights. Dijsktra is a famous figure in computer
science. And you can amuse yourself by reading some
of the things that he wrote. He was a very prolific writer, and, kept
notebooks. And then here really some of his, famous
quotes. Now, you need to may know something about,
old languages to appreciate some of these. For example, COBOL was one of the first
business oriented languages, programming languages and it tried to, make it so that
programs kind of read like English language sentences.
But purists in computer science, such as Dijkstra really didn't like that language
as you can tell, The use of COBOL cripples the mind.
It's teaching should therefore be regarded as a criminal offense.
Actually, when I first came to Princeton and was setting up the department here,
One of, there was a big donor, who actually, wanted to have me fired and the
department closed down because we wouldn't teach COBOL.
I didn't know about Dijkstra's quote at that time,
I wish I had. And there's some other opinions that
Dijkstra had. You can amuse yourself on the web, reading
some of his writing, But he was, here's another one that's
actually pretty relevant today. Object-oriented programming is an
exceptionally bad idea that could only have originated in California.
Dijkstra worked in Texas, the University of Texas for a while, and of course, came
from the Netherlands. Okay.
But, Dijkstra, did, invent, a very, very important algorithm, that's very widely
used for solving the shortest paths problem.
It's the one that's in your maps app and in your, car, and many, many other
applications. So let's take a look at how that algorithm
works. It's a very simple algorithm.
We, consider, what we're going to do is consider the vertices
In increasing order of their distance from the source.
So and a way we're going to do that is take the next vertex, add it to the tree
and relax all the edges that point from that vertex.
So let's talk about it in terms of a demo and then we'll look at the code.
So our mandate is to consider vertices in increasing order of distance from the
source. So, our source is zero in this case so
what vertices are closest to the source? Well, we can look at the edges that point
from that vertex. Well, start out with vertex is zero, is
the distance zero from the source, so we pick it.
Then we add that vertex to the shortest paths tree, and relax all its edges.
And relaxing all the edges pointing from zero in this case is just going to uptake
the, this two and edgeTo entries for five, eight, nine.
So in this case, this says that the shortest path from zero to one is to go
along zero, one and that's distance five and from zero to four, it's to go along
zero, four and that's distance nine, And zero to seven, let's go take it zero,
seven that's distance eight. And the edge weights are positive.
We're not going to find a shorter path to anyone, of what,
Sorry. The edge weights are positive and those
are the shortest paths that we know so far to the vertices.
Now, the first key point of the algorithm is take the closest one.
If you take the closest one, in this case, it's one, Then we know we're not going to
find a shorter, shorter path to one. The only other way to go out of zero is to
take one of these other edges, and they're longer.
So, and all the edge weights are positive. So we're not going to find a shorter way
from zero to one, than to take the edge, 0-1.
So that means that edge 0-1 is on the shortest path tree,
That's what the algorithm says. Take the next closest vertex to the
source. in this case, it's one. And then,
We're going to add that edge, that vertex to the tree.
And relax all the edges that point from that.
So now both zero and one are on the tree, so now let's look at the edges pointing to
that and we have to relax each one of those.
So in this case if you go from one to four, Then we look at four, we know a path
of length nine. And one to four doesn't give us a shorter
path, so we leave that edge alone. One to three gives us a shorter path to
three which is going to be twenty, cuz you went from zero to one, and then one to
three is of length twenty. And one to two, also, we didn't know a way
to two, But now we know now we know one, or better one is seventeen.
So that completes the relaxation of all the edges pointing from one.
So now we have zero and one on the tree, And.
We would, consider next, The next closest vertex to the source.
So what we have in this two is the shortest paths to all the vertex, vertices
that we've seen so far and this one says that the we've been to the zero and one so
the next closest one is seven, which is distance eight.
So we are going to choose vertex seven. And again, that's the shortest path we've
seen so far, we are not going to find a shorter one cuz to get to every other
vertex is longer. and so, we know it's on the shortest paths tree, and now we're
going to relax all the edges pointing from seven.
In this case, both of them, the one from seven to two, gives us a shorter way to
two than what we knew before, and the one from seven to five gives us a shorter way
to five than we knew before. Well, we hadn't been to five before.
So that relaxes seven, so now seven is on the shortest paths tree.
So now 4's the next closest path vertex to the source.
From the edge zero, four which is of length nine.
So that's the one that we're going to pick next for the tree.
We're not going to find a shorter path there,
And we relax, relax along all its edges. In this case, we find a shorter path to
five than we knew before and a shorter path to six, well, we visit six for the
first time. Okay, so that's four, so now we just have
to worry about two, three, five, and six, and five is the one over there, and so we
select vertex five, Relax along its edges. In this case those edges both give us
better paths than we knew before. So now we're left with three vertices, and
two is the winner. This is the two from the sources fourteen,
To three, it's twenty and to six, it's 26. Relax it's edges, and it gives us again,
new shorter paths to both three and six. And then the last, this next to last step
is to pick three and that does not give us a new way to six, and then finally we pick
six. And then we now know,
The shortest path to all the vertices from vertex zero.
If we just take the edgeTo edges, That's from one, to one you take zero, to
five you take two, And so forth you get the shortest paths tree and that gives the
shortest way to get from the source to every vertex.
That's a simulation of Dijkstra's algorithm.
Here's a demo of Dijsktra's algorithm running in a large digraph.
So the source is the protects the center with the open circle and you can see it
considers the vertices in the graph in order of their distance from the source.
Now you can see this is maybe like a lake in the middle of the graph,
And so it's not going to move on from those vertices.
It's going to take a little while to get around the lake.
And again, if, this visualization is produced by our code,
Then it, and it gives a very, natural feeling for the way that the algorithm
works, This, in principle. And I think, in fact,
in many cases, is what your, car does when you turn the map system on,
It computes the shortest path to everywhere,
And then it's all ready when you ask for a certain place, how to get there.
So here's just, starting from another point in the graph.
You have, if you happen to live at a corner of the Earth, then it's going to be
slightly longer to get to the other corner.
And, a nice feeling for how the algorithm gets its job done.
Again, when it gets into those blank areas, it takes a little while to get over
to the other side. And of course if we had islands, if we had
little roads in the middle there that were not connected,
Then we know where to get to them from the source and we wouldn't see them.
And that's fine for the way our algorithm works.
We just leave that out of the demo and the proof to avoid adding an extra comment
about that for every algorithm. That's a visualization of Dijkstra's
algorithm on a large graph. So, so how do we prove it's correct?
Well, essentially we've prove that it's an instance of the generic algorithm.
So, first thing is that every edge is relaxed exactly once.
Every time we, everytime we put a vertex onto the tree, we relax all the edges that
come from that vertex and we never reconsider the vertex.
And what does relaxation do? Relaxation ensures that after the
relaxation, either way there was before, afterwards, you have the distance to w is
less than or equal the distance to v plus e to the, e plus the way to the edge.
Either it's equal cuz we just made it that way, or it's less than it was before and
the edge was not relevant. And this inequality is going to hold for
every an, entry for every edge corresponding to every edge, for just two
entries corresponding to every edge because number one, the, these two values
are always increasing. We never I'm sorry, we're always
decreasing. The only thing we ever change, only reason
we ever change distTo w is to make it smaller.
If we, if we found an edge that would make it bigger we ignor that edge,
So it's always decreasing. So when we change distTo w, we're not
going to make that inequality holds. We're just going to make it better.
And this distTo v is not going to change at all.
Once we relax an edge from a vertex, we're done with that vertex.
We're not going to process that at all. So, then when we're done we have the
optimality conditions hold that exactly is the optimality condition.
And, not only that, we have a path from the source to every vertex.
So that's the correctness proof for Dijkstra's algorithm based on the
optimality conditions. Here's the code.
It's similar to code that we've seen before.
We're going to use the indexed priority queue that allows us to implement the
decreased key operation. And we have our edgeTo and distTo arrays
that are part of the shortest paths computation and the goal of the shortest
paths computation. So we initialized the constructor,
initializes those rays and including the Index
Minimum PQ. and we start out with all the distances infinity except for the distance
of the source is zero. We put the source on the priority Q and
then what we're going to do is take the, Edge that closest to the source off the
priority. That's on the priority queue off and then
relax all the edges that are adjacent to that, so I'm using our standard iterator
to get all the edges that emanate from that vertex and relax them. And then the
relax code is just like the code that was showed when describing relaxation, except
that it also updates the priority queue. If the, vertex, that, that edge goes to is
on the priority queue, it gives a new shorter way to get to that,
So we have to, decrease the key on the priority queue,
Cuz if it's not on the priority queue, we insert it,
And that's it. That's a complete implementation, of
Dijsktra's algorithm using modern data structures.
Now, this algorithm might seem very familiar than, if you've been paying
attention. It's essentially the same algorithm as
Prim's algorithm. The difference is that, in both cases,
we're building what's called a spanning tree of the graph.
But in Prim's algorithm, we take a vertex that, that's not on the tree using the
rule of let's take the vertex that's closest to the tree, anywhere on the tree,
Closest to some vertex on the tree. For Dijsktra's algorithm, we take next,
the vertex that's closest to the source through a path, they go through a tree and
then into a non-tree vertex. That's the difference and the differences
in the code have to do with the fact that Prim's algorithm is for an undirected
graph, Dijsktra's algorithm is for a directed graph,
But essentially, they're the same algorithm.
And actually, several of the algorithms that we've talked about are in this same
family. They compute a spanning tree.
I have, you, you have a tree that takes care of where that records where you've
been in the graph, From every vertex, back to where you
started. Then you used different rules for choosing
which vertex to add next. For breadth-first search, you use a Q, for
depth-first search, you use something like a stack and then you have to decide what
to do, if you encounter a vertex that you've been to before.
But many graph algorithms use this same basic idea.
So in particular, when we are talking about what the running time of Dijsktra's
algorithm, it depends on what priority Q the implementation we use,
And it's the same considerations as for Prim's algorithm.
We have V insert operations, every vertex goes on to the priority queue. V
delete-min, every vertex comes off the priority queue. and for every edge, in the
worst case, we could compute decrease-key operation.
So, the original implementations of Dijsktra's algorithm used an unordered
array, Which would mean that it would take time
proportional to V to find the minimum, To find the vertex closest to the source.
So the total running time would be proportional to V squared.
That's not adequate for the huge sparse graph that we see in practice today like
the map in your car. So, the binary heap data structure, makes
the,, makes it feasible to run this algorithm.
And that's where all the operations take time proportional to log V.
We have to use the indexing trick that we talked about last time to support
decrease-key. But still, we get a total running time of
E log V, Which makes it feasible.
And just as with Prim's algorithm, by using a implementation of the priority
queue that can do a faster decrease-key, you can get a faster algorithm.
And in, in practice something like a 4-way hit is going to give quite a fast
algorithm. And so, more expensive to insert and t-,
to insert but much faster to delete min in decrease key.
And again, there's a theoretical data structure that's not useful in practice.
It gets the running time down to E + V log V.
Of course, if your graph, graph is dense, And again, the examples I used, they're
not. The array implementation is optimal, we
can't do any better. We have to go through all the edges and
might as well find the minimum, at the same time.
But in, in, in practice, people use binary heaps, for sparse graphs.
I maybe going to four way, If, the performance is really critical.
The bottom line is, we have, extremely efficient implementations for the huge
graphs that arise in practice nowadays.
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