In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets (replacing them with their union), and finding a representative member of a set. The last operation makes it possible to determine efficiently whether any two elements belong to the same set or to different sets.
While there are several ways of implementing disjoint-set data structures, in practice they are often identified with a particular implementation known as a disjoint-set forest. This specialized type of forest performs union and find operations in near-constant amortized time. For a sequence of addition, union, or find operations on a disjoint-set forest with nodes, the total time required is , where is the extremely slow-growing inverse Ackermann function. Although disjoint-set forests do not guarantee this time per operation, each operation rebalances the structure (via tree compression) so that subsequent operations become faster. As a result, disjoint-set forests are both asymptotically optimal and practically efficient.
Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures support a wide variety of algorithms. In addition, these data structures find applications in symbolic computation and in compilers, especially for register allocation problems.
History
Disjoint-set forests were first described by Bernard A. Galler and Michael J. Fischer in 1964.. The paper originating disjoint-set forests. In 1973, their time complexity was bounded to , the iterated logarithm of , by Hopcroft and Ullman. In 1975, Robert Tarjan was the first to prove the (inverse Ackermann function) upper bound on the algorithm's time complexity. He also proved it to be tight. In 1979, he showed that this was the lower bound for a certain class of algorithms, pointer algorithms, that include the Galler-Fischer structure. In 1989, Fredman and Saks showed that (amortized) words of bits must be accessed by any disjoint-set data structure per operation, thereby proving the optimality of the data structure in this model.
In 1991, Galil and Italiano published a survey of data structures for disjoint-sets.
In 1994, Richard J. Anderson and Heather Woll described a parallelized version of Union–Find that never needs to block.
In 2007, Sylvain Conchon and Jean-Christophe Filliâtre developed a semi-persistent version of the disjoint-set forest data structure and formalized its correctness using the proof assistant Rocq (then: Coq). "Semi-persistent" means that previous versions of the structure are efficiently retained, but accessing previous versions of the data structure invalidates later ones. Their fastest implementation achieves performance almost as efficient as the non-persistent algorithm. They do not perform a complexity analysis.
Variants of disjoint-set data structures with better performance on a restricted class of problems have also been considered. Gabow and Tarjan showed that if the possible unions are restricted in certain ways, then a truly linear time algorithm is possible.Harold N. Gabow, Robert Endre Tarjan, "A linear-time algorithm for a special case of disjoint set union," Journal of Computer and System Sciences, Volume 30, Issue 2, 1985, pp. 209–221, ISSN 0022-0000, https://doi.org/10.1016/0022-0000(85)90014-5 In particular, linear time is achievable if a "union tree" is given a priori. This is a tree that includes all elements of the sets. Let p[v] denote the parent in the tree, then the assumption is that union operations must have the form union(v,p[v]) for some v.
Representation
In this and the following section we describe the most common implementation of the disjoint-set data structure, as a forest of parent pointer trees. This representation is known as Galler-Fischer trees.
Each node in a disjoint-set forest consists of a pointer and some auxiliary information, either a size or a rank (but not both). The pointers are used to make parent pointer trees, where each node that is not the root of a tree points to its parent. To distinguish root nodes from others, their parent pointers have invalid values, such as a circular reference to the node or a sentinel value. Each tree represents a set stored in the forest, with the members of the set being the nodes in the tree. Root nodes provide set representatives: Two nodes are in the same set if and only if the roots of the trees containing the nodes are equal.
Nodes in the forest can be stored in any way convenient to the application, but a common technique is to store them in an array. In this case, parents can be indicated by their array index. Every array entry requires bits of storage for the parent pointer. A comparable or lesser amount of storage is required for the rest of the entry, so the number of bits required to store the forest is . If an implementation uses fixed size nodes (thereby limiting the maximum size of the forest that can be stored), then the necessary storage is linear in .
Operations
Disjoint-set data structures support three operations: Making a new set containing a new element; Finding the representative of the set containing a given element; and Merging two sets.
Making new sets
The MakeSet operation adds a new element into a new set containing only the new element, and the new set is added to the data structure. If the data structure is instead viewed as a partition of a set, then the MakeSet operation enlarges the set by adding the new element, and it extends the existing partition by putting the new element into a new subset containing only the new element.
In a disjoint-set forest, MakeSet initializes the node's parent pointer and the node's size or rank. If a root is represented by a node that points to itself, then adding an element can be described using the following pseudocode:
function MakeSet(x) is if x is not already in the forest then x.parent := x x.size := 1 // if nodes store size x.rank := 0 // if nodes store rank end if end function
This operation has linear time complexity. In particular, initializing a disjoint-set forest with nodes requires time.
Lack of a parent assigned to the node implies that the node is not present in the forest.
In practice, MakeSet must be preceded by an operation that allocates memory to hold . As long as memory allocation is an amortized constant-time operation, as it is for a good dynamic array implementation, it does not change the asymptotic performance of the random-set forest.
Finding set representatives
The Find operation follows the chain of parent pointers from a specified query node until it reaches a root element. This root element represents the set to which belongs and may be itself. Find returns the root element it reaches.
Performing a Find operation presents an important opportunity for improving the forest. The time in a Find operation is spent chasing parent pointers, so a flatter tree leads to faster Find operations. When a Find is executed, there is no faster way to reach the root than by following each parent pointer in succession. However, the parent pointers visited during this search can be updated to point closer to the root. Because every element visited on the way to a root is part of the same set, this does not change the sets stored in the forest. But it makes future Find operations faster, not only for the nodes between the query node and the root, but also for their descendants. This updating is an important part of the disjoint-set forest's amortized performance guarantee.
There are several algorithms for Find that achieve the asymptotically optimal time complexity. One family of algorithms, known as path compression, makes every node between the query node and the root point to the root. Path compression can be implemented using a simple recursion as follows:
function Find(x) is if x.parent ≠ x then x.parent := Find(x.parent) return x.parent else return x end if end function
This implementation makes two passes, one up the tree and one back down. It requires enough scratch memory to store the path from the query node to the root (in the above pseudocode, the path is implicitly represented using the call stack). This can be decreased to a constant amount of memory by performing both passes in the same direction. The constant memory implementation walks from the query node to the root twice, once to find the root and once to update pointers:
function Find(x) is root := x while root.parent ≠ root do root := root.parent end while
while x.parent ≠ root do parent := x.parent x.parent := root x := parent end while
return root end function
Tarjan and Van Leeuwen also developed one-pass Find algorithms that retain the same worst-case complexity but are more efficient in practice. These are called path splitting and path halving. Both of these update the parent pointers of nodes on the path between the query node and the root. Path splitting replaces every parent pointer on that path by a pointer to the node's grandparent:
function Find(x) is while x.parent ≠ x do (x, x.parent) := (x.parent, x.parent.parent) end while return x end function
Path halving works similarly but replaces only every other parent pointer:
function Find(x) is while x.parent ≠ x do x.parent := x.parent.parent x := x.parent end while return x end function
Merging two sets
The operation Union(x, y) replaces the set containing and the set containing with their union. Union first uses Find to determine the roots of the trees containing and . If the roots are the same, there is nothing more to do. Otherwise, the two trees must be merged. This is done by either setting the parent pointer of 's root to 's, or setting the parent pointer of 's root to 's.
The choice of which node becomes the parent has consequences for the complexity of future operations on the tree. If it is done carelessly, trees can become excessively tall. For example, suppose that Union always made the tree containing a subtree of the tree containing . Begin with a forest that has just been initialized with elements and execute Union(1, 2), Union(2, 3), ..., Union(n - 1, n). The resulting forest contains a single tree whose root is , and the path from 1 to passes through every node in the tree. For this forest, the time to run Find(1) is .
In an efficient implementation, tree height is controlled using union by size or union by rank. Both of these require a node to store information besides just its parent pointer. This information is used to decide which root becomes the new parent. Both strategies ensure that trees do not become too deep.
Union by size
In the case of union by size, a node stores its size, which is simply its number of descendants (including the node itself). When the trees with roots and are merged, the node with more descendants becomes the parent. If the two nodes have the same number of descendants, then either one can become the parent. In both cases, the size of the new parent node is set to its new total number of descendants.
function Union(x, y) is // Replace nodes by roots x := Find(x) y := Find(y)
if x = y then return // x and y are already in the same set end if
// If necessary, swap variables to ensure that // x has at least as many descendants as y if x.size < y.size then (x, y) := (y, x) end if
// Make x the new root y.parent := x // Update the size of x x.size := x.size + y.size end function
The number of bits necessary to store the size is clearly the number of bits necessary to store . This adds a constant factor to the forest's required storage.
Union by rank