/usr/share/common-lisp/source/spatial-trees/r-trees.lisp is in cl-spatial-trees 0.2-4.
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;;;; STRUCTURE FOR SPATIAL SEARCHING", Antonin Guttman, Proc. ACM
;;;; SIGMOD Int. Conf. on Management of Data, 1984.
;;;;
;;;; Differences with the algorithms described in that paper:
;;;;
;;;; * we keep the minimum bounding rectangle of a node's children in
;;;; the node itself and not in an index pointing to the node, except
;;;; for leaves where there is one extra level of indirection (to
;;;; assist in implementation of R+-trees).
;;;;
;;;; * we implement (in SPLIT-NODE) a guarantee of the stated
;;;; invariant that each node contains at least m elements. This is
;;;; referred to in Section "4. Performance Tests", "... decreased
;;;; cost of insertion with a stricter node balance reflects the fact
;;;; that when one group becomes too full, all split algorithms simply
;;;; put the remaining elements in the other group without further
;;;; comparisons."
(in-package "SPATIAL-TREES-IMPL")
(defgeneric mbr (thing tree))
(defmethod mbr ((n spatial-tree-node) (tree spatial-tree))
(declare (ignore tree))
#+nil
(check (not (eq n (root-node tree)))
"Root of ~S asked for its MBR" tree)
(slot-value n 'mbr))
(defun minimum-bound-of (objects tree)
(reduce #'minimum-bound objects :key (lambda (x) (mbr x tree))))
(defstruct leaf-node-entry rectangle datum)
(defmethod mbr ((o leaf-node-entry) (tree spatial-tree))
(declare (ignore tree))
(leaf-node-entry-rectangle o))
(defclass r-tree (spatial-tree)
())
(defmethod make-spatial-tree ((kind (eql :r)) &rest initargs)
(apply #'make-instance 'r-tree
:root-node (make-instance 'spatial-tree-leaf-node :records nil)
initargs))
;;; 3.1. Searching
(defmethod search ((o t) (tree r-tree))
(search (funcall (rectfun tree) o) tree))
(defmethod search ((r rectangle) (tree r-tree))
(labels ((%search (r node)
(cond
((typep node 'spatial-tree-leaf-node)
(let (result)
(dolist (entry (records node) (nreverse result))
(when (intersectp r (leaf-node-entry-rectangle entry))
(push (leaf-node-entry-datum entry) result)))))
(t
(let (result)
(dolist (child (children node) result)
(when (intersectp r (mbr child tree))
(setq result (append (%search r child) result)))))))))
(let ((root (root-node tree)))
(%search r root))))
;;; 3.2. Insertion
(defmethod choose-leaf (r (tree r-tree))
(labels ((%choose-leaf (r node)
(cond
((typep node 'spatial-tree-leaf-node) node)
(t (do* ((children (children node) (cdr children))
(child (car children) (car children))
(candidate child)
(min-extension (- (area (minimum-bound (mbr child tree) (mbr r tree)))
(area (mbr child tree)))))
((null children) (%choose-leaf r candidate))
(let* ((new-area (area (minimum-bound (mbr child tree)
(mbr r tree))))
(old-area (area (mbr child tree)))
(extension (- new-area old-area)))
(when (or (< extension min-extension)
(and (= extension min-extension)
(< old-area (area (mbr candidate tree)))))
(setf min-extension extension
candidate child))))))))
(let ((n (root-node tree)))
(%choose-leaf r n))))
(defmethod insert ((r t) (tree r-tree))
(let* ((r (make-leaf-node-entry :datum r
:rectangle (funcall (rectfun tree) r)))
(leaf-node (choose-leaf r tree)))
(cond
((< (length (records leaf-node)) (max-per-node tree))
(push r (records leaf-node))
(adjust-tree tree leaf-node))
(t
(let ((new-node (split-node tree r leaf-node)))
(check (<= (min-per-node tree)
(length (records new-node))
(max-per-node tree))
"invariant (1) violated for the new node ~S in ~S"
new-node tree)
(check (<= (min-per-node tree)
(length (records leaf-node))
(max-per-node tree))
"invariant (1) violated for the old node ~S in ~S"
leaf-node tree)
(let ((new (adjust-tree tree leaf-node new-node)))
(when new
(let ((new-root
(make-instance 'spatial-tree-node
:children (list (root-node tree) new))))
(setf (parent (root-node tree)) new-root
(root-node tree) new-root
(parent new) new-root))))))))
tree)
(defgeneric adjust-tree (tree node &optional new))
(defmethod adjust-tree ((tree r-tree) node &optional new)
(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)
(adjust-tree tree parent))
(t
(let ((new-parent (split-node tree new parent)))
(check (<= (min-per-node tree)
(length (children new-parent))
(max-per-node tree))
"invariant (3) violated for the new parent node ~S in ~S"
new-parent tree)
(check (<= (min-per-node tree)
(length (children parent))
(max-per-node tree))
"invariant (3) violated for the old parent node ~S in ~S"
parent tree)
(dolist (child (children parent))
(setf (parent child) parent))
(dolist (child (children new-parent))
(setf (parent child) new-parent))
(adjust-tree tree parent new-parent))))
(adjust-tree tree parent))))))
;;; 3.3. Deletion
(defmethod delete ((r t) (tree r-tree))
(let ((leaf-node (find-leaf r (root-node tree) tree)))
(when leaf-node
(setf (records leaf-node) (remove r (records leaf-node) :key #'leaf-node-entry-datum))
(condense-tree leaf-node tree)
(unless (typep (root-node tree) 'spatial-tree-leaf-node)
(when (null (cdr (children (root-node tree))))
(check (car (children (root-node tree)))
"non-leaf root node with no children")
(setf (root-node tree) (car (children (root-node tree))))
(slot-makunbound (root-node tree) 'parent)
(slot-makunbound (root-node tree) 'mbr)))
tree)))
(defun find-leaf (obj node tree)
(labels ((%find-leaf (leaf-entry node tree)
(if (typep node 'spatial-tree-leaf-node)
(when (member (leaf-node-entry-datum leaf-entry)
(records node)
:key #'leaf-node-entry-datum)
(return-from find-leaf node))
(dolist (entry (children node))
(when (intersectp (mbr leaf-entry tree) (mbr entry tree))
(%find-leaf leaf-entry entry tree))))))
(%find-leaf (make-leaf-node-entry :datum obj
:rectangle (funcall (rectfun tree) obj))
node
tree)))
(defun condense-tree (node tree)
(labels ((all-leaves-below (node)
(if (typep node 'spatial-tree-leaf-node)
(records node)
(apply #'append
(mapcar #'all-leaves-below (children node))))))
(do ((node node (parent node))
(q nil))
((eq node (root-node tree))
(dolist (orphan q)
;; NOTE: this interpretation (reinsert every leaf)
;; disagrees with BKSS (R*-trees), section 4.3, "... is
;; based on the ability of the insert routine to insert
;; entries on every level of the tree as already required
;; by the deletion algorithm [Gut 84]."
(dolist (oleaf (all-leaves-below orphan))
(insert (leaf-node-entry-datum oleaf) tree))))
(cond
((< (length (children node)) (min-per-node tree))
(setf (children (parent node)) (remove node (children (parent node))))
(push node q))
(t
(setf (slot-value node 'mbr) (minimum-bound-of (children node) tree)))))))
;;; 3.5. Node Splitting
(defun d (r1 r2 tree)
(- (area (minimum-bound (mbr r1 tree) (mbr r2 tree)))
(area (mbr r1 tree))
(area (mbr r2 tree))))
(defun pick-seeds (entries tree)
(do* ((entry1 (car entries) (car entries))
(entries (cdr entries) (cdr entries))
(maxentry1 entry1)
(maxentry2 (car entries))
(maxd (d maxentry1 maxentry2 tree)))
((null entries) (values maxentry1 maxentry2))
(dolist (entry2 entries)
(when (> (d entry1 entry2 tree) maxd)
(setf maxd (d entry1 entry2 tree)
maxentry1 entry1
maxentry2 entry2)))))
(defun pick-next (entries node new-node tree)
(let* ((maxentry (car entries))
(maxdelta (- (d maxentry new-node tree) (d maxentry node tree))))
(dolist (entry (cdr entries) (values maxentry maxdelta))
(let ((delta (- (d entry new-node tree) (d entry node tree))))
(when (> (abs delta) (abs maxdelta))
(setf maxentry entry
maxdelta delta))))))
(defun make-node-like (node)
(make-instance (class-of node)))
(defmethod split-node ((tree r-tree) new node)
(let ((new-node (make-node-like node))
(entries (cons new (children node))))
(multiple-value-bind (s1 s2)
(pick-seeds entries tree)
(setf (children node) (list s1)
(slot-value node 'mbr) (mbr s1 tree))
(setf (children new-node) (list s2)
(slot-value new-node 'mbr) (mbr s2 tree))
(do ((entries (remove s1 (remove s2 entries))))
((null entries) new-node)
(multiple-value-bind (maxentry maxdelta)
(pick-next entries node new-node tree)
(cond
((or (plusp maxdelta)
(and (zerop maxdelta)
(or (< (area (mbr node tree)) (area (mbr new-node tree)))
(and (= (area (mbr node tree)) (area (mbr new-node tree)))
(> (length (children new-node))
(length (children node)))))))
(push maxentry (children node))
(setf (slot-value node 'mbr) (minimum-bound-of (children node) tree)))
(t
(push maxentry (children new-node))
(setf (slot-value new-node 'mbr)
(minimum-bound-of (children new-node) tree))))
(setf entries (remove maxentry entries))
;; the extra bit (see notes at head of file)
(when (= (length entries)
(- (min-per-node tree) (length (children node))))
(setf (children node) (append entries (children node))
(slot-value node 'mbr) (minimum-bound-of (children node) tree))
(return new-node))
(when (= (length entries)
(- (min-per-node tree) (length (children new-node))))
(setf (children new-node) (append entries (children new-node))
(slot-value new-node 'mbr) (minimum-bound-of (children new-node) tree))
(return new-node)))))))
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