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)

;; -*- coding: utf-8 -*-
(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
(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (declare #.OPT)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10)) result))))
              (return (if minus (- result) result))))))))

(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))

;;;
;;; Diameter of tree
;;;

(defun find-diameter (graph)
  "Finds a diameter of a tree. Returns three values: the length of the diameter
and its two ends."
  (declare ((array list (*)) graph))
  (let ((end 0)
        (max-depth 0))
    (assert (> (length graph) 0))
    (labels ((traverse (v parent depth)
               (declare ((integer 0 #.array-total-size-limit) v parent depth))
               (when (> depth max-depth)
                 (setq max-depth depth
                       end v))
               (dolist (child (aref graph v))
                 (declare ((integer 0 #.most-positive-fixnum) child))
                 (unless (= child parent)
                   (traverse child v (+ depth 1))))))
      (traverse 0 array-total-size-limit 0)
      (let ((end1 end))
        (setq max-depth 0)
        (traverse end1 array-total-size-limit 0)
        (values max-depth end1 end)))))

(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 second-max (list)
  (declare #.OPT
           (values uint32))
  (let ((max -1)
        (max2 -1))
    (declare (int32 max max2))
    (dolist (v list)
      (declare (uint32 v))
      (cond ((>= v max)
             (setq max2 max
                   max v))
            ((>= v max2)
             (setq max2 v))))
    max2))

(defun main ()
  (declare (inline sort))
  (let* ((n (read))
         (k (read))
         (graph (make-array n :element-type 'list :initial-element nil))
         edges
         (res #xffffffff))
    (declare (uint32 n k res))
    (dotimes (i (- n 1))
      (let ((a (- (read-fixnum) 1))
            (b (- (read-fixnum) 1)))
        (push a (aref graph b))
        (push b (aref graph a))
        (push (cons a b) edges)))
    (when (<= (find-diameter graph) k)
      (println 0)
      (return-from main))
    (let ((k/2 (floor k 2))
          (dists (make-array n :element-type 'uint32)))
      (declare (uint32 k/2))
      (labels ((fill-dists! (v parent depth)
                 (declare (int32 v parent depth))
                 (setf (aref dists v) depth)
                 (dolist (child (aref graph v))
                   (declare (uint32 child))
                   (unless (= child parent)
                     (fill-dists! child v (+ depth 1)))))
               (get-max-depth (v parent)
                 (let ((res (aref dists v)))
                   (declare (uint32 res))
                   (dolist (child (aref graph v))
                     (declare (uint32 child))
                     (unless (= child parent)
                       (maxf res (get-max-depth child v))))
                   res)))
        (if (evenp k)
            (dotimes (root n)
              (when (cdr (aref graph root)) ;; exclude one branch case
                (fill dists +inf+)
                (fill-dists! root -1 0)
                (let ((branch-depths (loop for child in (aref graph root)
                                           collect (get-max-depth child root))))
                  (when (>= (second-max branch-depths) k/2)
                    (minf res (count-if (lambda (x) (> (the uint32 x) k/2)) dists))))))
            (loop
              for (root1 . root2) in edges
              when (and (car (aref graph root1))
                        (car (aref graph root2)))
              do (fill dists +inf+)
                 (fill-dists! root1 root2 0)
                 (fill-dists! root2 root1 0)
                 (let ((branch-depths1 (loop for child in (aref graph root1)
                                             collect (get-max-depth child root1)))
                       (branch-depths2 (loop for child in (aref graph root2)
                                             collect (get-max-depth child root2))))
                   (when (and (>= (reduce #'max branch-depths1) k/2)
                              (>= (reduce #'max branch-depths2) k/2))
                     (minf res (count-if (lambda (x) (> (the uint32 x) k/2)) dists))))))
        (println 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 "C:/Windows/System32/WindowsPowerShell/v1.0/powershell.exe" '("get-clipboard") :output out)))

#+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 "2
1 3 1 2
"
    "3
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "5
100 1 2 3 14 15 58 58 58 29
"
    "135
")))