Disjoint-set data structure

Given a set of elements, it is often useful to break them up or partition them into a number of separate, nonoverlapping sets. A disjoint-set data structure is a data structure that keeps track of such a partitioning. A union-find algorithm is an algorithm that performs two useful operations on such a data structure:

Find: Determine which set a particular element is in. Also useful for determining if two elements are in the same set.
Union: Combine or merge two sets into a single set.

Because it supports these two operations, a disjoint-set data structure is sometimes called a merge-find set. The other important operation, MakeSet, which makes a set containing only a given element (a singleton), is generally trivial. With these three operations, many practical partitioning problems can be solved (see the Applications section).

In order to define these operations more precisely, we need some way of representing the sets. One common approach is to select a fixed element of each set, called its representative, to represent the set as a whole. Then, Find(x) returns the representative of the set that x belongs to, and Union takes two set representatives as its arguments.

Disjoint-set linked lists

Perhaps the simplest approach to creating a disjoint-set data structure is to create a linked list for each set. We choose the element at the head of the list as the representative.

MakeSet is obvious, creating a list of one element. Union simply appends the two lists, a constant-time operation. Unfortunately, with this implementation Find requires Ω(n) or linear time with this approach.

We can avoid this by including in each linked list node a pointer to the head of the list; then Find takes constant time. However, we've now ruined the time of Union, which has to go through the elements of the list being appended to make them point to the head of the new combined list, requiring Ω(n) time.

We can ameliorate this by always appending the smaller list to the longer, called the weighted union heuristic. This also requires keeping track of the length of each list as we perform operations to be efficient. Using this, a sequence of m MakeSet, Union, and Find operations on n elements requires O(m + nlog n) time. To make any further progress, we need to start over with a different data structure.

Disjoint-set forests

We now turn to disjoint-set forests, a data structure where each set is represented by a tree data structure where each node holds a reference to its parent node. Disjoint-set forests were first described by Bernard A. Galler and Michael J. Fisher in 1964,^[1] although their precise analysis took years.

In a disjoint-set forest, the representative of each set is the root of that set's tree. Find simply follows parent nodes until it reaches the root. Union combines two trees into one by attaching the root of one to the root of the other. One way of implementing these might be:

 function MakeSet(x)
     x.parent := null
 
 function Find(x)
     if x.parent == null
        return x
        
     return Find(x.parent)
 
 function Union(x, y)
     xRoot := Find(x)
     yRoot := Find(y)
     xRoot.parent := yRoot

In this naive form, this approach is no better than the linked-list approach, because the tree it creates can be highly unbalanced, but it can be enhanced in two ways.

The first way, called union by rank, is to always attach the smaller tree to the root of the larger tree, rather than vice versa. To evaluate which tree is larger, we use a simple heuristic called rank: one-element trees have a rank of zero, and whenever two trees of the same rank are unioned together, the result has one greater rank. Just applying this technique alone yields an amortized running-time of $O(\log n)$ per MakeSet, Union, or Find operation. Here are the improved MakeSet and Union:

 function MakeSet(x)
     x.parent := null
     x.rank   := 0
 
 function Union(x, y)
     xRoot = Find(x)
     yRoot = Find(y)
     if xRoot.rank > yRoot.rank
         yRoot.parent := xRoot
     else if xRoot.rank < yRoot.rank
         xRoot.parent := yRoot
     else if xRoot != yRoot
         yRoot.parent := xRoot
         xRoot.rank := xRoot.rank + 1

The second improvement, called path compression, is a way of flattening the structure of the tree whenever we use Find on it. The idea is that each node we visit on our way to a root node may as well be attached directly to the root node; they all share the same representative. To effect this, we make one traversal up to the root node, to find out what it is, and then make another traversal, making this root node the immediate parent of all nodes along the path. The resulting tree is much flatter, speeding up future operations not only on these elements but on those referencing them, directly or indirectly. Here is the improved Find:

 function Find(x)
     if x.parent == null
        return x
  
     x.parent = Find(x.parent)
     return x.parent

These two techniques complement each other; applied together, the amortized time per operation is only $O(\alpha (n))$ , where $\alpha (n)$ is the inverse of the function $f(n)=A(n,n)$ , and $A$ is the extremely quickly-growing Ackermann function. Since $\alpha (n)$ is its inverse, it's less than 5 for all remotely practical values of $n$ ^[2]. Thus, the amortized running time per operation is effectively a small constant.

In fact, we can't get better than this: Fredman and Saks showed in 1989 that $\Omega (\alpha (n))$ words must be accessed by any disjoint-set data structure per operation on average.

Applications

Disjoint-set data structures arise naturally in many applications, particularly where some kind of partitioning or equivalence relation is involved, and this section discusses some of them.

Tracking the connected components of an undirected graph

Suppose we have an undirected graph and we want to efficiently make queries regarding the connected components of that graph, such as:

Are two vertices of the graph in the same connected component?
List all vertices of the graph in a particular component.
How many connected components are there?

If the graph is static (not changing), we can simply use breadth-first search to associate a component with each vertex. However, if we want to keep track of these components while adding additional vertices and edges to the graph, a disjoint-set data structure is much more efficient.

We assume the graph is empty initially. Each time we add a vertex, we use MakeSet to make a set containing only that vertex. Each time we add an edge, we use Union to union the sets of the two vertices incident to that edge. Now, each set will contain the vertices of a single connected component, and we can use Find to determine which connected component a particular vertex is in, or whether two vertices are in the same connected component.

This technique is used by the Boost Graph Library to implement its Incremental Connected Components functionality.

Note that this scheme doesn't allow deletion of edges — even without path compression or the rank heuristic, this is not as easy, although more complex schemes have been designed that can deal with this type of incremental update.

Computing shorelines of a terrain

When computing the contours of a 3D surface, one of the first steps is to compute the "shorelines," which surround local minima or "lake bottoms." We imagine we are sweeping a plane, which we refer to as the "water level," from below the surface upwards. We will form a series of contour lines as we move upwards, categorized by which local minima they contain. In the end, we will have a single contour containing all local minima.

Whenever the water level rises just above a new local minimum, it creates a small "lake," a new contour line that surrounds the local minimum; this is done with the MakeSet operation.

As the water level continues to rise, it may touch a saddle point, or "pass." When we reach such a pass, we follow the steepest downhill route from it on each side until we arrive a local minimum. We use Find to determine which contours surround these two local minima, then use Union to combine them. Eventually, all contours will be combined into one, and we are done.

Classifying a set of atoms into molecules or fragments

In computational chemistry, collisions involving the fragmentation of large molecules can be simulated using molecular dynamics. The result is a list of atoms and their positions. In the analysis, the union-find algorithm can be used to classify these atoms into fragments. Each atom is initially considered to be part of its own fragment. The Find step usually consists of testing the distance between pairs of atoms, though other criterion like the electronic charge between the atoms could be used. The Union merges two fragments together. In the end, the sizes and characteristics of each fragment can be analyzed. [3]

History

While the ideas used in disjoint-set forests have long been familiar, Robert Tarjan was the first to prove the upper bound (and a restricted version of the lower bound) in terms of the inverse Ackermann function. Until this time the best bound on the time per operation, proven by Hopcroft and Ullman, was O(log^* n), the iterated logarithm of n, another slowly-growing function (but not quite as slow as the inverse Ackermann function). Tarjan and van Leeuwen also developed one-pass Find algorithms that are more efficient in practice. The algorithm was made well-known by the popular textbook Introduction to Algorithms.^[4]

External links

Compaq Research: Zeus: Animation of Union-Find Algorithms
Java applet: A Graphical Union-Find Implementation, by Rory L. P. McGuire
The abstract data type Union-Find, a simple C implementation by Vašek Chvátal
Wait-free Parallel Algorithms for the Union-Find Problem, a 1994 paper by Richard J. Anderson and Heather Woll describing a parallelized version of Union-Find that never needs to block

Notes

^ The first n such that α(n) = 5 has a base-2 logarithm whose own base-2 logarithm is greater than the number of particles in the universe raised to the power 200.

References

^ Bernard A. Galler and Michael J. Fisher. An improved equivalence algorithm. Communications of the ACM, Volume 7, Issue 5 (May 1964), pages 301-303. The paper originating disjoint-set forests. ACM Digital Library

^ Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 21: Data structures for Disjoint Sets, pp.498–524.

Zvi Galil and Giuseppe F. Italiano. Data structures and algorithms for disjoint set union problems, ACM Computing Surveys, Volume 23, Issue 3 (September 1991), pages 319-344. ACM Digital Library

J. A. Spirko and A. P. Hickman, Molecular-dynamics simulations of collisions of Ne with La@C₈₂, Phys. Rev. A 57, 3674–3682 (1998).