by Kevin Wayne, Robert Sedgewick
Okay. So, we’ve looked at the quick union and quick find algorithms. Both of which are easy to implement. But simply can’t support a huge dynamic connectivity problems. So, how are we going to do better? That’s what we’ll look at next. A very effective improvement, it’s called weighting.
And it might have occurred to you while we are looking at these algorithms. The idea is to when implementing the quick union algorithm take steps to avoid having tall trees. If you’ve got a large tree and a small tree to combine together what you want to try to do is avoid putting the large tree lower, that’s going to lead to long tall trees. And there’s a relatively easy way to do that. What we’ll do is we’ll keep track of the number of objects in each tree and then, we’ll maintain balance by always making sure that we link the root of the smaller tree to the root of the larger tree. So, we, we avoid this first situation here where we put the larger tree lower. In the weighted algorithm, we always put the smaller tree lower. How we, let’s see how we implement that. Let’s see a demo first.
Okay, so again start out in our normal starting position, where everybody’s in their own tree. And for when there’s only two items to link it, it works, works the same way as before.
But now, when we have eight to merge with four and three, we put the eight as the child, no matter which order their arguments came, because it’s the smaller tree.
So, six and five doesn’t matter, whichever one goes down doesn’t matter.
Nine and four, so now, nine is the small one four is the big one. So, nine is going to be the one that goes down below.
Two and one, five and zero. So now, five and zero five is in the bigger tree so zero goes below.
Seven and two, two is in the bigger tree so seven goes below.
Six and one they’re in equal size trees. And seven and three, three is in the smaller tree so it goes below. So, the weighted algorithm always makes sure that the smaller tree goes below.
And again, we wind up with a single tree representing all the objects. But this time, we h ave some guarantee that no item is too far from the root and we’ll talk about that explicitly in a second.
So, here’s an example that shows the effect of doing the weighted quick union where we always put the smaller tree down below for the same set of union commands. This is with a hundred sites and 88 union operations. You can see in the top the big tree has some trees, some nodes, a fair distance from the root. In the bottom, for the weighted algorithm all the nodes are within distance four from the root. The average distance to the root is much, much lower.
Let’s look at the Java implementation and then we’ll look in more detail at, at that quantitative information. So, we used the same data structure except, now we need an extra array, that for each item, gives the number of objects in the tree routed at that item. That will maintain in the union operation. Find implementation is identical to for quick union, you’re just checking whether the roots are equal. For the union implementation, we’re going to modify the code to check the sizes. And link the root of the smaller tree to the root of the larger tree in each case. And then after changing the id link, we also change the size array. If we make id, i a child of j, then we have to increment the size of j’s tree by the size of i’s tree. Or if we do the other way around, then we have to increment the size of i’s tree by the size of j’s tree. So, that’s the full code in white for implementing quick union. So, not very much code but much, much better performance.
In fact we can analyze the running time mathematically and show that defined operation, it takes time proportional to how far down the trees are in the node in the tree, the nodes are in the tree, but we can show that it’s guaranteed that the depth of any node in the tree is at most the logarithm to the base two of N. We use the notation Lg always for logarithm to the base two. And, and, so for, if N is a thousand, that’s going to be ten, if N is a million that’s twenty, if N is a billion that’s 30. It’s a very small number compared to N. So, let’s look at the proof of that.
We do some mathematical proofs in, in this course when they’re critical such as this one. And why is it true that the depth of any node x is, at most, log base two of N? Well, the key to understanding that is to, take a look at exactly when does the depth of any node increase? When does it go down further in the tree? Well. The x’s depth will increase by one, when its tree, T1 in this diagram, is merged into some other tree, T2 in this diagram. Well, at that point we said we only do that if the size of T2 was bigger than the or equal to size of T1. So, when the depth of x increases, the size of its tree at least doubles. So, that’s the key because that means that the size of the tree containing x can double at most log N times because if you start with one and double log N times, you get N and there’s only N nodes in the tree. So, that’s a sketch of a proof that the depth of any node x is at most log base two of N. And that has profound impact on the performance of this algorithm.
Now instead of the initialization always takes time proportional to N. But now, both the union and the connected or find operation takes time proportional to log base two of N. And that is an algorithm that scales. If N grows from a million to a billion, that cost goes from twenty to 30, which is quite not acceptable. Now, this was very easy to implement and, and we could stop but usually, what happens in the design of algorithms is now that we understand what it is that gains performance, we take a look and see, well, could we improve it even further.
And in this case, it’s very easy to improve it much, much more. And that’s the idea of path compression. And this idea is that, well, when we’re trying to find the root of the tree containing a, a given node. We’re touching all the nodes on the path from that node to the root. While we’re doi ng that we might as well make each one of those just point to the root. There’s no reason not to.
So when we’re looking, we’re trying to find the root of, of P. After we find it, we might as well just go back and make every node on that path just point to the root.
That’s going to be a constant extra cost. We went up the path once to find the root. Now, we’ll go up again to just flatten the tree out. And the reason would be, no reason not to do that. We had one line of code to flatten the tree, amazingly.
Actually to make a one liner code, we use a, a simple variant where we make every other node in the path point to its grandparent on the way up the tree. Now, that’s not quite as good as totally flattening actually in practice that it actually is just about as good. So, with one line of code, we can keep the trees almost completely flat.
Now, this algorithm people discovered rather early on after figuring out the weighting and it turns out to be fascinating to analyze quite beyond our scope. But we mentioned this example to illustrate how even a simple algorithmah, can have interesting and complex analysis. And what was proved by Hopcroft Ulman and Tarjan was that if you have N objects, any sequence of M union and find operations will touch the array at most a c (N + M lg star N) times. And now, lg N is kind of a funny function. It’s the number of times you have to take the log of N to get one. And the way to think, it’s called the iterated log function. And in the real world, it’s best to think of that as a number less than five because lg two^ 65536 is five.
So, that means that the running time of weighted quick union with path compression is going be linear in the real world and actually could be improved to even a more interesting function called the Ackermann function, which is even more slowly growing than lg.
And another point about this is it seems that this is so close to being linear that is time proportional to N instead of time proportional to N times the slowly growing function in N. Is there a simple algorithm that is linear? And people, looked for a long time for that, and actually it works out to be the case that we can prove that there is no such algorithm. So, there’s a lot of theory that goes behind the algorithms that we use.
And it’s important for us to know that theory and that will help us decide how to choose which algorithms we’re going to use in practice, and where to concentrate our effort in trying to find better algorithms. It’s amazing fact that was eventually proved by Friedman and Sachs, that there is no linear time algorithm for the union find problem. But weighted quick union with path compression in practice is, is close enough that it’s going to enable the solution of huge problems.
So, that’s our summary for algorithms for solving the dynamic connectivity problem. With using weighted quick union and with path compression, we can solve problems that could not otherwise be addressed. For example, if you have a billion operations and a billion objects I said before it might take thirty years. We can do it in six seconds. Now, and what’s most important to recognize about this is that its the algorithm design that enables the solution to the problem. A faster computer wouldn’t help much. You could spend millions on a super computer, and maybe you could get it done in six years instead of 30, or in two months but with a fast logarithm, you can do it in seconds, in seconds on your own PC.