Problem:
Score : 600 points

Problem Statement

Given an undirected tree, let the distance between vertices u and v be the
number of edges on the simple path from u to v.
The diameter of a tree is the maximum among the distances between any two vertices.
We will call a tree good if and only if its diameter is at most K.

You are given an undirected tree with N vertices numbered 1 through N.
For each i (1≦i≦N-1), there is an edge connecting vertices A_i and B_i.

You want to remove zero or more vertices from the tree, so that the resulting tree is good.
When a vertex is removed, all incident edges will also be removed.
The resulting graph must be connected.

Find the minimum number of vertices that you need to remove in order to produce a good tree.

Constraints

2≦N≦2000

1≦K≦N-1

1≦A_i≦N, 1≦B_i≦N

The graph defined by A_i and B_i is a tree.

Input

The input is given from Standard Input in the following format:

N K
A_1 B_1
A_2 B_2
:
A_{N-1} B_{N-1}

Output

Print the minimum number of vertices that you need to remove in order to produce a good tree.

Sample Input 1

6 2
1 2
3 2
4 2
1 6
5 6

Sample Output 1

2

The tree is shown below. Removing vertices 5 and 6 will result in a good tree with the diameter of 2.

Sample Input 2

6 5
1 2
3 2
4 2
1 6
5 6

Sample Output 2

0

Since the given tree is already good, you do not need to remove any vertex.

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(macrolet ((def (name fname)
             `(define-modify-macro ,name (new-value) ,fname)))
  (def minf min)
  (def maxf max)
  (def mulf *)
  (def divf /)
  (def iorf logior)
  (def xorf logxor)
  (def andf logand))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;
(defconstant +inf+ #xffffffff)
(defun max2 (list)
  (let ((max -1)
        (max2 -1))
    (assert (cdr list))
    (dolist (x list)
      (cond ((>= x max)
             (setq max2 max max x))
            ((>= x max2)
             (setq max2 x))))
    (values max max2)))

(defun main ()
  (let* ((n (read))
         (k (read))
         (edges (make-array (- n 1) :element-type 'list))
         (graph (make-array n :element-type 'list :initial-element nil)))
    (dotimes (i (- n 1))
      (let ((a (- (read) 1))
            (b (- (read) 1)))
        (setf (aref edges i) (cons a b))
        (push a (aref graph b))
        (push b (aref graph a))))
    (let ((k/2 (floor k 2))
          (res +inf+)
          (dists (make-array n :element-type 'uint32)))
      (labels ((fill-dists (v parent depth)
                 (setf (aref dists v) depth)
                 (dolist (child (aref graph v))
                   (unless (= child parent)
                     (fill-dists child v (+ depth 1)))))
               (get-max-depth (v parent)
                 (let ((res (aref dists v)))
                   (dolist (child (aref graph v))
                     (unless (= child parent)
                       (maxf res (get-max-depth child v))))
                   res)))
        (if (evenp k)
            (dotimes (root n)
              (when (cdr (aref graph root))
                (fill dists +inf+)
                (fill-dists root -1 0)
                (let* ((branch-depths (loop for child in (aref graph root)
                                            collect (get-max-depth child root))))
                  (dbg root dists branch-depths)
                  (multiple-value-bind (max max2) (max2 branch-depths)
                    (dbg max max2)
                    (when (>= max2 k/2)
                      (minf res (count-if (lambda (d) (> d k/2)) dists)))))))
            (loop
              for (r1 . r2) across edges
              when (and (car (aref graph r1))
                        (car (aref graph r2)))
              do (fill dists +inf+)
                 (fill-dists r1 r2 0)
                 (fill-dists r2 r1 0)
                 #>dists
                 (let* ((branch-depths1 (cons 0 (loop for child in (aref graph r1)
                                                      unless (= child r2)
                                                      collect (get-max-depth child r1))))
                        (branch-depths2 (cons 0 (loop for child in (aref graph r2)
                                                      unless (= child r1)
                                                      collect (get-max-depth child r2)))))
                   (dbg branch-depths1 branch-depths2)
                   (let ((max-depth1 (reduce #'max branch-depths1))
                         (max-depth2 (reduce #'max branch-depths2)))
                     (when (>= (min max-depth1 max-depth2) k/2)
                       (minf res (count-if (lambda (x) (> x k/2)) dists))))))))
      (println
       (if (= res +inf+)
           0
           res)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "6 2
1 2
3 2
4 2
1 6
5 6
"
    "2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "6 5
1 2
3 2
4 2
1 6
5 6
"
    "0
")))