Princeton Univerisy Lectures: Algorithms Part I – Union-Find(1) Dynamic Connectivity

by Kevin Wayne, Robert Sedgewick

Welcome back to algorithms. Today, we’re going to talk about the union find problem. A set of algorithms for solving the so-called dynamic connectivity problem. We’ll look at two classic algorithms. Quick Find and Quick Union, and some applications and improvements of those algorithms. The subtext of today’s lecture really is to go through the steps that we’ll follow over and over again to develop a useful algorithm. The first step is to model the problem. Try to understand, basically, what are the main elements of the problem that need to be solved. Then we’ll find some algorithm to solve the problem.

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In many cases, the first algorithm we come up with would be fast enough and maybe it fits in memory and, we’ll go ahead and use it, and be off and running. But in many other cases maybe it’s not fast enough, or there’s not enough memory. So, what we do is try to figure out why, find a way to address whatever’s causing that problem, find a new algorithm and iterate until we’re satisfied.

This is the scientific approach to designing and analyzing algorithms, where we build mathematical models to try and understand what’s going on, and then we do experiments to validate those models and help us improve things. So, first we’ll talk about the dynamic connectivity problem, the model of the problem for union find. So, here’s the idea. They’re going to have a set of N objects. Doesn’t really matter what they are. We’re going to use the numbers, zero through N to model our objects. And then, we have the idea of a connection between two objects. And, we’ll, postulate that there’s going to be a command that says, connect two objects. Given two objects, provide a connection between them. And then key part of the problem is find query or the connected query, which just asks, is there a path connecting the two objects.

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So for example, in this set of ten objects, we performed already, a bunch of union commands, connecting four and three, three and eight, six and five, nine and four, two and one. And now we might have a connected query that says, is zero connected to seven? Well, in this case, there is no connection, so we say no. But if we ask is eight connected to nine? We are going to say yes, even no we don’t have a direct connection between eight and nine. There is a path from eight to three to four to nine. So, that’s our problem, to be able to officially support these two commands for given set of objects.

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Now, let’s say we add a union five, zero. So, that creates a connection between five and zero.  Seven and two creates a connection between seven and two. And six and one,  between six and one. So, now if we ask our zero connected to seven, well one and zero we can do that too. And that’s a redundant  connection. And now, if we ask is zero connected to seven we’re going to answer yes. So that’s our problem, intermix union, commands and connected queries and we need to be able to officially support those commands for a large number of objects. So, here’s a much bigger example.

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And you can see that we’re going to need efficient algorithms for this. First of all, you can see we’re going to need a computer for this. It would take quite, quite some time for a human to figure out whether there’s a connection. In this case there is a connection. Now, the algorithms that we’re looking at today are not going to actually give the path connecting the two objects. It’s just going to be able to answer the question, is there a path? In part two of the course, we’ll consider algorithms that explicitly find paths. They’re not as efficient as union find because they have more work to do.

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Now, applications of these, these algorithms involve objects of all types. These are used for digital photos, where the objects are pixels they’re used for networks, where the objects are computers, social networks, where it’s people, or computer chips, where it’s circuit elements or abstract things like variable names in a program, or elements in a mathematical set, or physical things like metallic sites in a composite system. So, all different types of objects for, but for programming we’re going to associate each object with a name and we’ll just name the objects with a number, integers from zero to N-1.

That’s a very convenient initial starting point for our programs because we can use integers as an index into an array then, and then quickly access information relevant to each object. And it also just supresses a lot of details that are not relevant to union find. In fact, to make this mapping from an object name to the integer zero through N – one is to find application of a symbol table or a searching algorithm, which is one of the things that we’ll be studying later in this course algorithms and data structures for solving that problem.

Now, the connections, well, we need, a few abstract properties that these connections have to satisfy. And they’re all quite natural and intuitive. So we assume that is connected to is an equivalence elation. That is, every object’s connected to itself, it’s symmetric. If P’s connected to Q, then Q’s connected to P, and it’s transitive. If P’s connected to Q, and Q’s connected to R, then P’s connected to R.

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Now these properties are very intuitive. But it’s worthwhile to state them explicitly and make sure that our algorithms maintain them. When we have an equivalence relation a set of objects and connections divide into subsets called connected components. A connected component is a maximal set of objects that’s mutually connected. For example in this small example here, here’s three connected components. One consisting of just object zero, second one objects one, four and five. And third one the other four objects. And these components have the property that if any two objects in them are connected and there is no object outside that is connected to those objects, that’s connected components. Our algorithms will gain efficiency by maintaining connected components and using that knowledge to efficiently answer the query that’s, that they’re presented with.

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Okay, so to implement the operations, we have to find query and the union command. And so we’re going to maintain the connected components. The find is going to have to check if two objects are in the same component and the union command is going to have to replace components containing two objects with their union.

So, for example, if we have these components, and we get the command to union connect, two and five. Essentially, we need to merge the connected components containing the one containing two or the one containing five to get a big connected components and now we have only two connected components.

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All of that leads up to, in a programming world to specifying, a data type which is simply specification of the methods that we are want to going to implement in order to solve this problem. So you know, typical Java model, what we will do is create a class called UF that contains two methods, one to implement union, the other one to implement connected, which returns a boolean. The constructor, takes SR unit, the number of objects, so that it can build data structure based on the number of objects. So, and we have to, bear in mind, as we’re building our logarithms, that both the number of objects can be huge, but also, the number of operations.

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We can have a, a very large number, of  union and connected, operations and our algorithms are going to have to be efficient, under those conditions.

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One of the practices that will follow often in this course is to check our API design before getting too far into dealing with the problem, by building a client that is going to use the data type that we develop. So, for this example, we’ve got a client that, Will read information from standard input.

First, an integer which is the number of objects that are going to be processed. And then a series of pairs of object names. And what the client does is it, it’ll, first it’ll read the integer from standard input, and create a, a UF object. And then as long as standard input is not empty, it’s going to read two integers from the input. And if they’re not connected, then it’ll connect them and print them out. If they are connected it’ll ignore. So, that’s our test client and that’s a fine test client to make sure that any implementation does what we expect that it will. So, that’s the setup. We’ve described the operations we want to implement all the way down to code and we have client code that we’re going to have to be able to service with our applications.

 

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