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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 high rank matrix completion in general is NP-Hard. However, with certain assumptions, some incomplete high rank matrix or even full rank matrix can be completed. Eriksson, Balzano and Nowak [10] have considered the problem of completing a matrix with the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces.
A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...
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 ...
The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.
A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries, M ∈ C n × m {\displaystyle M\in \mathbb {C} ^{n\times m}} has rank 1 {\displaystyle 1} if ...