lisp - LISP 广度优先搜索

Question

我有一个使用列表表示的树。例如:

(1 ((2 (3)) (3 (2)))) (2 ((1 (3)) (3 (1)))) (3 ((1 (2)) (2 (1)))))`

现在我需要在维护层次结构树的同时逐层遍历它。例如:

1. 遍历根节点 (1)
2. 遍历深度1 (1 2) (1 3) (2 1) (3 1) (3 1) (3 2)
3. 遍历 深度 2 (1 2 3) (1 3 2) (2 1 3) (2 3 1) (3 1 2) (3 2 1)

我不知道如何在 Lisp 中做到这一点。感谢任何帮助(甚至是伪代码)。我想到了几种方法，但似乎没有一种是合法的。

Answer 1

使用议程的广度优先搜索

进行广度优先搜索的经典方法是维护一个议程:接下来要查看的事情列表。然后，您只需将对象从议程的开头剥离，并将它们的子对象添加到议程的结尾。对于这样的议程，一种非常简单的方法是节点列表:添加到列表的末尾，然后使用 append。

我无法理解您的树结构(请在询问需要数据结构或算法规范的问题时给出该规范:尝试支持是浪费每个人的时间-猜猜)所以我在列表方面做了自己的:一棵树是一个缺点，它的车是它的值(value)，它的 cdr 是 child 的列表。以下是创建和访问此类树结构和示例树的函数。

(defun tree-node-value (n)
  (car n))

(defun tree-node-children (n)
  (cdr n))

(defun make-tree-node (value &optional (children '()))
  (cons value children))

(defparameter *sample-tree*
  (make-tree-node
   1
   (list
    (make-tree-node 2 (list (make-tree-node 3)))
    (make-tree-node 4 (list (make-tree-node 5) (make-tree-node 6)))
    (make-tree-node 7 (list (make-tree-node 8 (list (make-tree-node 9))))))))

现在我再也不用担心树的显式结构了。

现在这是一个使用议程的函数，该议程将在这棵树中搜索给定的节点值:

(defun search-tree/breadth-first (tree predicate)
  ;; search a tree, breadth first, until predicate matches on a node's
  ;; value.  Return the node that matches.
  (labels ((walk (agenda)
             (if (null agenda)
                 ;; we're done: nothing matched
                 (return-from search-tree/breadth-first nil)
               (destructuring-bind (this . next) agenda
                 (if (funcall predicate (tree-node-value this))
                     ;; found it, return the node
                     (return-from search-tree/breadth-first this)
                   ;; missed, add our children to the agenda and
                   ;; carry on
                   (walk (append next (tree-node-children this))))))))
    (walk (list tree))))

为了比较这里是深度优先搜索:

(defun search-tree/depth-first (tree predicate)
  ;; search a tree, depth first, until predicate matches on a node's
  ;; value
  (labels ((walk (node)
             (if (funcall predicate (tree-node-value node))
                 (return-from search-tree/depth-first node)
               (dolist (child (tree-node-children node) nil)
                 (walk child)))))
    (walk tree)))

您现在可以通过打印其参数但始终失败的谓词来比较这些实现，从而导致遍历整个树:

> (search-tree/breadth-first *sample-tree*
                             (lambda (v)
                               (print v)
                               nil))

1 
2 
4 
7 
3 
5 
6 
8 
9 
nil

> (search-tree/depth-first *sample-tree*
                           (lambda (v)
                             (print v)
                              nil))

1 
2 
3 
4 
5 
6 
7 
8 
9 
nil

---

附录 1:更好的议程实现

这种天真的议程实现的一个问题是我们最终总是调用 append。更聪明的实现允许将项目有效地附加到末尾。这是这样一个实现:

(defun make-empty-agenda ()
  ;; an agenda is a cons whose car is the list of items in the agenda
  ;; and whose cdr is the last cons in that list, or nil is the list
  ;; is empty.  An empty agenda is therefore (nil . nil)
  (cons nil nil))

(defun agenda-empty-p (agenda)
  ;; an agenda is empty if it has no entries in its list.
  (null (car agenda)))

(defun agenda-next-item (agenda)
  ;; Return the next entry from the agenda, removing it
  (when (agenda-empty-p agenda)
    (error "empty agenda"))
  (let ((item (pop (car agenda))))
    (when (null (car agenda))
      (setf (cdr agenda) nil))
    item))

(defun agenda-add-item (agenda item)
  ;; add an item to the end of the agenda, returning it
  (let ((item-holder (list item)))
    (if (agenda-empty-p agenda)
        (setf (car agenda) item-holder
              (cdr agenda) item-holder)
      (setf (cdr (cdr agenda)) item-holder
            (cdr agenda) item-holder))
    item))

请注意，无法复制所提供的这些议程之一。

这是一个明确的迭代函数，它使用了这个“聪明”的议程:

(defun search-tree/breadth-first/iterative (tree predicate)
  (loop with agenda = (make-empty-agenda)
        initially (agenda-add-item agenda tree)
        while (not (agenda-empty-p agenda))
        for node = (agenda-next-item agenda)
        when (funcall predicate (tree-node-value node))
        do (return-from search-tree/breadth-first/iterative node)
        else do (loop for c in (tree-node-children node)
                      do (agenda-add-item agenda c))
        finally (return nil)))

最后，任何基于议程的搜索都可以轻松修改为可重新启动:它只需要在匹配点返回当前议程，并允许传递议程。这是支持重新开始搜索的上述功能的变体:

(defun search-tree/breadth-first/iterative (tree predicate 
                                                 &optional (agenda
                                                            (make-empty-agenda)))
  ;; search TREE using PREDICATE.  if AGENDA is given and is not empty
  ;; instead restart using it (TREE is ignored in this case).  Return
  ;; the node found, or nil, and the remaining agenda
  (loop initially (unless (not (agenda-empty-p agenda))
                    (agenda-add-item agenda tree))
        while (not (agenda-empty-p agenda))
        for node = (agenda-next-item agenda)
        when (funcall predicate (tree-node-value node))
        do (return-from search-tree/breadth-first/iterative
             (values node agenda))
        else do (loop for c in (tree-node-children node)
                      do (agenda-add-item agenda c))
        finally (return (values nil agenda))))

附录 2:带有议程的一般搜索

事实上，可以进一步推广基于议程的方法来搜索树。特别是:

• 如果议程是一个队列 (FIFO)，那么您将进行广度优先搜索；
• 如果议程是堆栈式 (LIFO)，那么您将进行深度优先搜索。

对于这两种情况，实际的搜索实现可以是相同的，这很简洁。

下面是一些演示这一点的代码。这定义了树访问的通用函数(使用基于 cons 的树的方法)，因此无需关心它，并进一步定义了具有两个具体类的议程协议(protocol)，queue 和 stack 有适当的方法。然后，搜索功能完全不知道它是进行深度优先搜索还是广度优先搜索，并且在任何一种情况下都可以重新启动。

这是相当大的一段代码:我把它留在这里以防万一它对任何人都有用。

;;;; Trees
;;;

(defgeneric tree-node-value (n)
  (:documentation "The value of a tree node"))

(defgeneric tree-node-children (n)
  (:documentation "The children of a tree"))

;;;; Consy trees
;;;

(defmethod tree-node-value ((n cons))
  (car n))

(defmethod tree-node-children ((n cons))
  (cdr n))

(defun make-cons-tree-node (value &optional (children '()))
  ;; consy trees: I could do some clever EQL method thing perhaps to
  ;; abstract this?
  (cons value children))

(defun form->tree (form &key (node-maker #'make-cons-tree-node))
  (labels ((walk-form (f)
             (destructuring-bind (value . child-forms) f
               (funcall node-maker
                        value
                        (mapcar #'walk-form child-forms)))))
    (walk-form form)))

(defparameter *sample-tree*
  (form->tree '(1 (2 (3))
                  (4 (5) (6))
                   (7 (8 (9))))))


;;;; Agendas
;;;

(defclass agenda ()
  ())

(defgeneric agenda-empty-p (agenda)
  (:documentation "Return true if AGENDA is empty"))

(defgeneric agenda-next-item (agenda)
  (:documentation "Return the next item from AGENDA.
If there is no next item, signal an error: there is a before method which does this.")
  (:method :before ((agenda agenda))
   (when (agenda-empty-p agenda)
     (error "empty agenda"))))

(defmethod initialize-instance :after ((agenda agenda) &key
                                       (item nil itemp)
                                       (items (if itemp (list item) '()))
                                       (ordered nil))
  (agenda-add-items agenda items :ordered ordered))

(defgeneric agenda-add-item (agenda item)
  (:documentation "Add ITEM to AGENDA, returning ITEM.
There is an around method which arranges for ITEM to be returned.")
  (:method :around ((agenda agenda) item)
   (call-next-method)
   item))

(defgeneric agenda-add-items (agenda items &key ordered)
  (:documentation "Add ITEMS to AGENDA.
If ORDERED is true do so in a way that AGENDA-NEXT-ITEM will pull them
off in the same order.  Return AGENDA (there is an around method which
arranges for this).  The default method just adds the items in the
order given.")
  (:method :around ((agenda agenda) items &key ordered)
   (declare (ignorable ordered))
   (call-next-method)
   agenda)
  (:method ((agenda agenda) items &key ordered)
   (declare (ignorable ordered))
   (loop for item in items
         do (agenda-add-item agenda item))))

;;;; Queues are FIFO agendas
;;;

(defclass queue (agenda)
  ((q :initform (cons nil nil)))
  (:documentation "A queue"))

(defmethod agenda-empty-p ((queue queue))
  (null (car (slot-value queue 'q))))

(defmethod agenda-next-item ((queue queue))
  (let* ((q (slot-value queue 'q))
         (item (pop (car q))))
    (when (null (car q))
      (setf (cdr q) nil))
    item))

(defmethod agenda-add-item ((queue queue) item)
  (let ((q (slot-value queue 'q))
        (item-holder (list item)))
    (if (null (car q))
        (setf (car q) item-holder
              (cdr q) item-holder)
      (setf (cdr (cdr q)) item-holder
            (cdr q) item-holder))))

;;;; Stacks are LIFO agendas
;;;

(defclass stack (agenda)
  ((s :initform '()))
  (:documentation "A stack"))

(defmethod agenda-empty-p ((stack stack))
  (null (slot-value stack 's)))

(defmethod agenda-next-item ((stack stack))
  (pop (slot-value stack 's)))

(defmethod agenda-add-item ((stack stack) item)
  (push item (slot-value stack 's)))

(defmethod agenda-add-items ((stack stack) items &key ordered)
  (loop for item in (if ordered (reverse items) items)
        do (agenda-add-item stack item)))


;;;; Searching with agendas
;;;

(defun tree-search (tree predicate &key (agenda-class 'stack))
  ;; search TREE using PREDICATE.  AGENDA-CLASS (default STACK)
  ;; defines the type of search: a STACK will result in a depth-first
  ;; search while a QUEUE will result in a breadth-first search.  This
  ;; is a wrapper around AGENDA-SEARCH.
  (agenda-search (make-instance agenda-class :item tree) predicate))

(defun agenda-search (agenda predicate)
  ;; Search using an agenda.  PREDICATE is compared against the value
  ;; of a tree node.  On success return the node matched and the
  ;; agenda, on failure return NIL and NIL.  If the returned agenda is
  ;; not empty it can be used to restart the search.
  (loop while (not (agenda-empty-p agenda))
        for node = (agenda-next-item agenda)
        when (funcall predicate (tree-node-value node))
        do (return-from agenda-search
             (values node agenda))
        else do (agenda-add-items agenda (tree-node-children node)
                                  :ordered t)
        finally (return (values nil nil))))