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;;; Modifications to the R*-tree as in "The X-tree: An Index Structure
;;; for High-Dimensional Data", Berchtold, Keim and Kriegel,
;;; Proc. 22th Int. Conf. on Very Large Databases, 1996
(in-package "SPATIAL-TREES-IMPL")
(defclass x-tree (r*-tree)
((max-overlap :initarg :max-overlap :reader max-overlap)))
(defmethod initialize-instance :after ((tree x-tree) &rest args)
(declare (ignore args))
(unless (slot-boundp tree 'max-overlap)
(setf (slot-value tree 'max-overlap) 1/5)))
(defmethod make-spatial-tree ((kind (eql :x)) &rest initargs)
(apply #'make-instance 'x-tree
:root-node (make-instance 'spatial-tree-leaf-node :records nil)
initargs))
(defclass x-tree-node (spatial-tree-node)
((split-tree :initarg :split-tree :accessor split-tree)))
;;; Do we actually need to keep track of what is and what isn't a
;;; supernode? I'm not sure that we do...
(defclass x-tree-supernode (x-tree-node)
())
;;; FIXME: leaf supernodes (and leaf nodes) don't need a split tree.
(defclass x-tree-leaf-supernode (x-tree-supernode spatial-tree-leaf-node)
())
;;; Split trees.
;;;
;;; In the X-tree algorithms, we are required to store the 'split
;;; history' of a node. In typical academic fashion, this is
;;; extremely badly explained in the paper itself; I think I've
;;; reconstructed what's necessary, but it's not obvious that it's a
;;; huge win.
;;;
;;; In clearer terms, then, when we split a node, we record in the
;;; node's parent (which is possibly the new root node of the tree)
;;; the split in a 'split tree', which we represent using
;;; non-NULL-terminated conses. When a node overflows, we replace it
;;; in its parent's split tree with a cons of the original node and
;;; the new one; when the root node overflows, the new root node
;;; acquires a split tree of (old . new).
;;;
;;; A new non-leaf node, meanwhile, must result from a split of a
;;; previous such non-leaf node with its own split-tree information.
;;; We construct two new split trees from the original node's split
;;; tree, such that the tree structure is preserved as much as
;;; possible while retaining only those children contained in the
;;; redistributed nodes.
;;;
;;; When a node comes to be split, one potentially good split of its
;;; children is into the two sets defined by its left- and right-split
;;; trees; this is exploited, unless the split is too unbalanced, if
;;; the ordinary topological split fails to find a sufficiently good
;;; partition.
(defun find-cons-with-leaf (object conses)
(cond
((atom conses) nil)
((eq (car conses) object) conses)
((eq (cdr conses) object) conses)
(t (or (find-cons-with-leaf object (car conses))
(find-cons-with-leaf object (cdr conses))))))
(defun leaves-of-split-tree (split-tree)
(cond
((atom split-tree) (list split-tree))
(t (append (leaves-of-split-tree (car split-tree))
(leaves-of-split-tree (cdr split-tree))))))
(defun split-tree-from-set (set split-tree)
(cond
((atom split-tree) (find split-tree set))
(t (let ((car (split-tree-from-set set (car split-tree)))
(cdr (split-tree-from-set set (cdr split-tree))))
(cond
((null car) cdr)
((null cdr) car)
(t (cons car cdr)))))))
(defvar *split*)
;;; Figure 7: X-tree Insertion Algorithm for Directory Nodes
(defmethod adjust-tree ((tree x-tree) node &optional new)
(check (or (null new)
(or (and (typep node 'spatial-tree-leaf-node)
(typep new 'spatial-tree-leaf-node))
(and (not (typep node 'spatial-tree-leaf-node))
(not (typep new 'spatial-tree-leaf-node)))))
"oh dear")
(cond
((eq node (root-node tree)) new)
(t
(setf (slot-value node 'mbr) (minimum-bound-of (children node) tree))
(let ((parent (parent node)))
(if new
(cond
((< (length (children parent)) (max-per-node tree))
(push new (children parent))
(setf (parent new) parent)
(let ((cons (find-cons-with-leaf node (split-tree parent))))
(if (eq node (car cons))
(rplaca cons (cons node new))
(progn
(check (eq node (cdr cons)) "Aargh1")
(rplacd cons (cons node new)))))
(adjust-tree tree parent))
(t
(let ((cons (find-cons-with-leaf node (split-tree parent))))
(if (eq node (car cons))
(rplaca cons (cons node new))
(progn
(check (eq node (cdr cons)) "Aargh2")
(rplacd cons (cons node new)))))
(let ((new-parent (let ((*split* node))
(split-node tree new parent))))
(dolist (child (children parent))
(setf (parent child) parent))
(when new-parent
(dolist (child (children new-parent))
(setf (parent child) new-parent)))
(adjust-tree tree parent new-parent))))
(adjust-tree tree parent))))))
(defmethod insert ((o t) (tree x-tree))
(let* ((entry (make-leaf-node-entry :datum o
:rectangle (funcall (rectfun tree) o)))
(node (choose-subtree entry (root-node tree) (height tree) tree)))
(cond
((< (length (children node)) (max-per-node tree))
(push entry (children node))
(adjust-tree tree node))
(t
(let ((new-node (split-node tree entry node)))
(let ((new (adjust-tree tree node new-node)))
(when new
(let ((new-root
(make-instance 'x-tree-node
:children (list (root-node tree) new))))
(setf (parent (root-node tree)) new-root
(split-tree new-root) (cons (root-node tree) new)
(root-node tree) new-root
(parent new) new-root)))))))
tree))
;;; Figure 8: X-tree Split Algorithm for Directory Nodes
(defmethod split-node ((tree x-tree) new node)
(let ((new-node (make-node-like node))
(entries (cons new (children node))))
(let ((axis (choose-split-axis entries tree)))
(multiple-value-bind (one two)
(choose-split-index entries axis tree)
(let* ((bone (minimum-bound-of one tree))
(btwo (minimum-bound-of two tree))
(intersection (intersection bone btwo))
(mb (minimum-bound bone btwo)))
(cond
((or (null intersection)
(< (area intersection)
(* (area mb) (max-overlap tree))))
(setf (children node) one
(slot-value node 'mbr) bone)
(when (> (length one) (max-per-node tree))
(check (typep node 'x-tree-supernode) "AARGH"))
(setf (children new-node) two
(slot-value new-node 'mbr) btwo)
(when (> (length two) (max-per-node tree))
(change-class new-node
(if (typep node 'spatial-tree-leaf-node)
'x-tree-leaf-supernode
'x-tree-supernode)))
(when (< (length two) (max-per-node tree))
(change-class new-node
(if (typep node 'spatial-tree-leaf-node)
'spatial-tree-leaf-node
'x-tree-node)))
(unless (typep node 'spatial-tree-leaf-node)
(let ((split-tree (split-tree node)))
(setf (split-tree node) (split-tree-from-set one split-tree)
(split-tree new-node) (split-tree-from-set two split-tree))))
new-node)
((and (not (typep node 'spatial-tree-leaf-node))
(let ((split-tree (split-tree node)))
(destructuring-bind (one . two) split-tree
(let ((l1 (leaves-of-split-tree one))
(l2 (leaves-of-split-tree two)))
(if (find *split* l1)
(push new l1)
(progn
(check (find *split* l2) "Missing node!!")
(push new l2)))
(and (>= (min-per-node tree) (length l1))
(>= (min-per-node tree) (length l2))
(progn
(setf (children node) l1
(slot-value node 'mbr) (minimum-bound-of l1 tree)
(split-tree node) (if (find new l1)
(let ((cons (find-cons-with-leaf *split* one)))
(if (eq (car cons) *split*)
(rplaca cons (cons *split* new))
(rplacd cons (cons *split* new)))
one)
one))
(setf (children new-node) l2
(slot-value new-node 'mbr) (minimum-bound-of l2 tree)
(split-tree node) (if (find new l2)
(let ((cons (find-cons-with-leaf *split* two)))
(if (eq (car cons) *split*)
(rplaca cons (cons *split* new))
(rplacd cons (cons *split* new)))
two)
two))
new-node)))))))
(t (change-class node
(etypecase node
(spatial-tree-leaf-node 'x-tree-leaf-supernode)
(spatial-tree-node 'x-tree-supernode)))
(push new (children node))
(when (not (typep node 'spatial-tree-leaf-node))
(let ((cons (find-cons-with-leaf *split* (split-tree node))))
(if (eq *split* (car cons))
(rplaca cons (cons *split* new))
(progn
(check (eq *split* (cdr cons)) "Aargh2")
(rplacd cons (cons *split* new))))))
nil)))))))
(defmethod check-consistency progn ((tree x-tree))
(labels ((%check (node)
(assert
(or (typep node 'spatial-tree-leaf-node)
(null (set-difference (children node)
(leaves-of-split-tree
(split-tree node))))))
(unless (typep node 'spatial-tree-leaf-node)
(dolist (child (children node))
(%check child)))))
(%check (root-node tree))))
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