Min/max kd-tree
k-d tree with two scalar values
A min/max kd-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 kd-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 kd-tree has—besides the properties of an kd-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 are stored separately. Each inner node has besides two min/max values also two bits given, defining to which child those min/max values are assigned (0: to the left child 1: to the right child). The non-assigned min/max values of the children are the from the current node already known min/max values.
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