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This inverse Ackermann function f −1 is usually denoted by α. In fact, α ( n ) is less than 5 for any practical input size n , since A (4, 4) is on the order of 2 2 2 2 16 {\displaystyle 2^{2^{2^{2^{16}}}}} .
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 guarantee this time per operation, each operation rebalances the structure (via tree compression) so that ...
The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term; [15] so a is the base, b is the exponent (or hyperexponent), [12] and n is the rank (or grade), [6] and moreover, (,) is read as "the bth n-ation of a", e.g. (,) is read as "the 9th tetration of 7", and (,) is read as "the 789th 123 ...
The subtrees of are stored in union-find data structures, which is why checking whether or not two vertices are in the same subtree is possible in amortised ((,)) where (,) is the inverse Ackermann function.
Created Date: 8/30/2012 4:52:52 PM
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The Ackermann hierarchy is clearly distinct from the hyperoperators, because in his original paper in 1928, Ackermann defines the operators (,,) which are equivalent to the Knuth arrow only for n less than 3.
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