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This specialized type of forest performs union and find operations in near-constant amortized time. For a sequence of m addition, union, or find operations on a disjoint-set forest with n nodes, the total time required is O(mα(n)), where α(n) is the extremely slow-growing inverse Ackermann function. Although disjoint-set forests do not ...
The algorithm begins with scanning the grid cell by cell and checking whether the cell is occupied or not. If the cell is occupied, then it must be labeled with a cluster label. This cluster label is assigned based on the neighbors of that cell. (For this we are going to use Union-Find Algorithm which is explained in the next section.) If the ...
It uses the MakeSet, Find, and Union functions of a disjoint-set data structure. MakeSet(u) removes u to a singleton set, Find(u) returns the standard representative of the set containing u, and Union(u,v) merges the set containing u with the set containing v. TarjanOLCA(r) is first called on the root r.
The final iteration through all edges performs two find operations and possibly one union operation per edge. These operations take amortized time O(α(V)) time per operation, giving worst-case total time O(E α(V)) for this loop, where α is the extremely slowly growing inverse Ackermann function. This part of the time bound is much smaller ...
Union-find essentially stores labels which correspond to the same blob in a disjoint-set data structure, making it easy to remember the equivalence of two labels by the use of an interface method E.g.: findSet(l). findSet(l) returns the minimum label value that is equivalent to the function argument 'l'.
The primary advantage of a tagged union over an untagged union is that all accesses are safe, and the compiler can even check that all cases are handled. Untagged unions depend on program logic to correctly identify the currently active field, which may result in strange behavior and hard-to-find bugs if that logic fails.
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Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. He also obtained a complexity O ( n log n ) {\displaystyle O(n\log n)} algorithm, now known as Bentley's algorithm , based on reducing the problem to n 1 -dimensional problems: this is done by ...