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Relaxed K-d Tree
A relaxed ''K''-d tree or relaxed ''K''-dimensional tree is a data structure which is a variant of K-d trees. Like K-dimensional trees, a relaxed K-dimensional tree stores a set of n-multidimensional records, each one having a unique K-dimensional key ''x=(x0,... ,xK−1)''. Unlike K-d trees, in a relaxed K-d tree, the discriminants in each node are arbitrary. Relaxed K-d trees were introduced in 1998. Definitions A relaxed K-d tree for a set of K-dimensional keys is a binary tree in which: # Each node contains a K-dimensional record and has associated an arbitrary discriminant ''j ∈ ''. # For every node with key ''x'' and discriminant ''j'', the following invariant is true: any record in the right subtree with key y satisfies ''yj < xj'' and any record in the left subtree with key y satisfies ''yj ≥ xj.'' If ''K = 1'', a relaxed K-d tree is a binary search tree. As in a K-d tree, a relaxed K-d tree of size ''n'' induces a partition of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree. Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-d Tree
In computer science, a ''k''-d tree (short for ''k-dimensional tree'') is a space-partitioning data structure for organizing points in a ''k''-dimensional space. ''k''-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches) and creating point clouds. ''k''-d trees are a special case of binary space partitioning trees. Description The ''k''-d tree is a binary tree in which ''every'' node is a ''k''-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree. The hyperplane direction is chosen in the following way: every node in the tree is associated with one of the ''k'' dimensions, with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree. Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relaxed K-d Tree Example
Relaxation stands quite generally for a release of tension, a return to equilibrium. In the sciences, the term is used in the following ways: * Relaxation (physics), and more in particular: ** Relaxation (NMR), processes by which nuclear magnetization returns to the equilibrium distribution ** Dielectric relaxation, the delay in the dielectric constant of a material ** Vibrational energy relaxation, the process by which molecules in high energy quantum states return to the Maxwell-Boltzmann distribution ** Chemical relaxation methods, related to temperature jump ** Relaxation oscillator, a type of electronic oscillator In mathematics: :* Relaxation (approximation), a technique for transforming hard constraints into easier ones :* Relaxation (iterative method), a technique for the numerical solution of equations :* Relaxation (extension method), a technique for a natural extension in mathematical optimization or variational problems In computer science: :* Relaxation (computing) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-d Tree
In computer science, a ''k''-d tree (short for ''k-dimensional tree'') is a space-partitioning data structure for organizing points in a ''k''-dimensional space. ''k''-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches) and creating point clouds. ''k''-d trees are a special case of binary space partitioning trees. Description The ''k''-d tree is a binary tree in which ''every'' node is a ''k''-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree. The hyperplane direction is chosen in the following way: every node in the tree is associated with one of the ''k'' dimensions, with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Kd-tree
An implicit ''k''-d tree is a ''k''-d tree defined implicitly above a rectilinear grid. Its split planes' positions and orientations are not given explicitly but implicitly by some recursive splitting-function defined on the hyperrectangles belonging to the tree's nodes. Each inner node's split plane is positioned on a grid plane of the underlying grid, partitioning the node's grid into two subgrids. Nomenclature and references The terms " min/max ''k''-d tree" and "implicit ''k''-d tree" are sometimes mixed up. This is because the first publication using the term "implicit ''k''-d tree" did actually use explicit min/max ''k''-d trees but referred to them as "implicit ''k''-d trees" to indicate that they may be used to ray trace implicitly given iso surfaces. Nevertheless, this publication used also slim ''k''-d trees which are a subset of the implicit ''k''-d trees with the restriction that they can only be built over integer hyperrectangles with sidelengths that are powers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Min/max Kd-tree
A min/max ''k''d-tree is a ''k''-d tree with two scalar values - a minimum and a maximum - assigned to its nodes. The minimum/maximum of an inner node is equal to the minimum/maximum of its children's minima/maxima. Construction Min/max ''k''d-trees may be constructed recursively. Starting with the root node, the splitting plane orientation and position is evaluated. Then the children's splitting planes and min/max values are evaluated recursively. The min/max value of the current node is simply the minimum/maximum of its children's minima/maxima. Properties The min/max ''k''dtree has - besides the properties of an ''k''d-tree - the special property that an inner node's min/max values coincide each with a min/max value of either one child. This allows to discard the storage of min/max values at the leaf nodes by storing two bits at inner nodes, assigning min/max values to the children: Each inner node's min/max values will be known in advance, where the root node's min/max values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
