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def min_set_size(arr)
  min_size = (arr.length + 1) / 2

  q = RubyPriorityQueue.new
  arr.each do |num|
    if q.has_key?(num)
      q.push(num, q[num] - 1)
    else
      q.push(num, -1)
    end
  end

  count = 0
  size = 0
  while count < min_size
    if q.empty?
      return size
    else
      count += -(q.delete_min[1])
      size += 1
    end
  end

  size
end

# Pure ruby Priority Queue
class RubyPriorityQueue

  include Enumerable

  private

  #
  def link_nodes(b1, b2)
    return link_nodes(b2, b1) if b2.priority < b1.priority

    b2.parent = b1    
    child = b1.child
    b1.child = b2
    if child 
      b2.left  = child.left
      b2.left.right = b2
      b2.right = child
      child.left = b2
    else 
      b2.left = b2
      b2.right = b2
    end
    b1.degree += 1
    b2.mark = false # TODO: Check if this is correct, or if b1 should be marked as false
    return b1
  end

  # Does not change length
  def delete_first
    return nil unless @rootlist

    result = @rootlist
    if result == result.right
      @min = @rootlist = nil
    else 
      @rootlist = result.right
      @rootlist.left = result.left
      @rootlist.left.right = @rootlist
      
      result.right = result.left = result      
    end
    return result;
  end

  def cut_node(n)
    return self unless n.parent
    n.parent.degree -= 1
    if n.parent.child == n 
      if n.right == n
	n.parent.child = nil
      else
	n.parent.child = n.right;  
      end
    end
    n.parent = nil
    n.right.left = n.left
    n.left.right = n.right

    n.right = @rootlist
    n.left  = @rootlist.left
    @rootlist.left.right = n
    @rootlist.left = n

    n.mark = false

    return self
  end
  
  # Does not change length
  def insert_tree(tree)
    if @rootlist == nil
      @rootlist = @min = tree
    else
      l = @rootlist.left
      l.right = tree
      @rootlist.left = tree
      tree.left = l
      tree.right = @rootlist
      @min = tree if tree.priority < @min.priority
    end
    self
  end
  
  def consolidate
    return self if self.empty?
    array_size = (2.0 * Math.log(self.length) / Math.log(2) + 1.0).ceil
    tree_by_degree = Array.new(array_size)

    while n = delete_first      
      while n1 = tree_by_degree[n.degree]	
	tree_by_degree[n.degree] = nil;
	n = link_nodes(n, n1);
      end
      tree_by_degree[n.degree] = n;
    end

    @rootlist = @min = nil;
    tree_by_degree.each do | tree | 
      next unless tree
      insert_tree(tree)
    end
    self
  end

  # Node class used internally
  class Node # :nodoc:
    attr_accessor :parent, :left, :right, :key, :priority, :degree, :mark
    attr_reader :child

    def child=(c)
      raise "Circular Child" if c == self
      raise "Child is neighbour" if c == self.right
      raise "Child is neighbour" if c == self.left
      @child = c
    end
    
    def to_dot(only_down = false, known_nodes = [])
      p known_nodes.map { | n | n.dot_id }
      p self.dot_id
      result = []
      if only_down
	raise "Circular #{caller.inspect}" if known_nodes.include?(self)
	known_nodes << self

	result << "#{dot_id} [label=\"#{@key}: #{@priority}\"];"
	l = "  "
	#l << "#{@left.dot_id} <- #{dot_id}; " if @left 
	l << "#{dot_id} -> #{@left.dot_id} [constraint=false]; " if @left and @left.dot_id < self.dot_id
	l << "#{dot_id} -> #{@right.dot_id} [constraint=false];\t\t\t\t/*neighbours*/" if @right and @right.dot_id <= self.dot_id
	result << l
	result << "  #{dot_id} -> #{@child.dot_id}; //child" if @child
	result << @child.to_dot(false, known_nodes) if @child
      else
	n = self
	begin
	  result.concat(n.to_dot(true, known_nodes))
	  n = n.right
	end while n != self
      end
      result.flatten.map{|r| "  " << r}
    end

    def dot_id
      "N#{@key}"
    end

    def initialize(key, priority)
      @key = key; @priority = priority; @degree = 0
    end
  end

  public

  # Returns the number of elements of the queue.
  # 
  #    q = PriorityQueue.new
  #    q.length #=> 0
  #    q[0] = 1
  #    q.length #=> 1
  attr_reader :length

  # Create a new, empty PriorityQueue
  def initialize
    @nodes = Hash.new
    @rootlist = nil
    @min = nil
    @length = 0
  end
  
  # Print a priority queue as a dot-graph. The output can be fed to dot from the
  # vizgraph suite to create a tree depicting the internal datastructure.
  def to_dot
    r = ["digraph fibheap {"]
    #r << @rootlist.to_dot.join("\n") if @rootlist
      r << "ROOT -> #{@rootlist.dot_id};" if @rootlist
    @nodes.to_a.sort.each do | (_, n) |
      r << "  #{n.dot_id} [label=\"#{n.key}: #{n.priority}\"];"
      r << "  #{n.dot_id} -> #{n.right.dot_id} [constraint=false];" if n.right# and n.dot_id < n.right.dot_id
      r << "  #{n.dot_id} -> #{n.left.dot_id} [constraint=false];" if n.left #and n.dot_id < n.left.dot_id
      r << "  #{n.dot_id} -> #{n.child.dot_id}" if n.child
    end
    r << "}"
    r.join("\n")
    r
  end

  # Call dot and gv displaying the datstructure
  def display_dot
    puts to_dot
    system "echo '#{to_dot}' | twopi -Tps -Groot=ROOT -Goverlap=false> /tmp/dotfile.ps; gv /tmp/dotfile.ps"
  end
  
  # call-seq:
  #     [key] = priority
  #     change_priority(key, priority)
  #     push(key, priority)
  # 
  # Set the priority of a key.
  #
  #     q = PriorityQueue.new
  #     q["car"] = 50
  #     q["train"] = 50
  #     q["bike"] = 10
  #     q.min #=> ["bike", 10]
  #     q["car"] = 0
  #     q.min #=> ["car", 0]
  def change_priority(key, priority)
    return push(key, priority) unless @nodes[key]

    n = @nodes[key]
    if n.priority < priority # Priority was increased. Remove the node and reinsert.
      self.delete(key)
      self.push(key, priority);
      return self
    end
    n.priority = priority;
    @min = n if n.priority < @min.priority

    return self if !n.parent or n.parent.priority <= n.priority # Already in rootlist or bigger than parent
    begin # Cascading Cuts
      p = n.parent
      cut_node(n)
      n = p
    end while n.mark and n.parent
    n.mark = true if n.parent

    self
  end
  
  # Add an object to the queue.
  def push(key, priority)    
    return change_priority(key, priority) if @nodes[key]
    @nodes[key] = node = Node.new(key, priority)
    @min = node if !@min or priority < @min.priority
    if not @rootlist
      @rootlist = node
      node.left = node.right = node
    else
      node.left = @rootlist.left
      node.right = @rootlist
      @rootlist.left.right = node
      @rootlist.left = node
    end
    @length += 1
    self
  end
  
  # Returns true if the array is empty, false otherwise.
  def empty?
    @rootlist.nil?
  end

  # call-seq:
  #     [key] -> priority
  #
  # Return the priority of a key or nil if the key is not in the queue.
  #
  #     q = PriorityQueue.new
  #     (0..10).each do | i | q[i.to_s] = i end
  #     q["5"] #=> 5
  #     q[5] #=> nil
  def [](key)
    @nodes[key] and @nodes[key].priority
  end

  # call-seq:
  #     has_key? key -> boolean
  #     
  # Return false if the key is not in the queue, true otherwise.
  #
  #     q = PriorityQueue.new
  #     (0..10).each do | i | q[i.to_s] = i end
  #     q.has_key("5") #=> true 
  #     q.has_key(5)   #=> false
  def has_key?(key)
    @nodes.has_key?(key)
  end

  alias :[]= :push

  # Call the given block with each [key, priority] pair in the queue
  #
  # Beware: Changing the queue in the block may lead to unwanted behaviour and
  # even infinite loops.
  def each
    @nodes.each do | key, node |
      yield(key, node.priority)
    end
  end

  # call-seq:
  #     min -> [object, priority]
  #    
  # Return the pair [object, priority] with minimal priority or nil when the
  # queue is empty.
  #
  #     q = PriorityQueue.new
  #     q["a"] = 10
  #     q["b"] = 20
  #     q.min          #=> ["a", 10]
  #     q.delete_min   #=> ["a", 10]
  #     q.min          #=> ["b", 20]
  #     q.delete_min   #=> ["b", 20]
  #     q.min          #=> nil
  def min
    [@min.key, @min.priority] rescue nil
  end
  
  # call-seq:
  #     min_key -> object
  #    
  # Return the key that has the minimal priority or nil when the queue is empty.
  #
  #     q = PriorityQueue.new
  #     q["a"] = 10
  #     q["b"] = 20
  #     q.min_key      #=> "a"
  #     q.delete_min   #=> ["a", 10]
  #     q.min_key      #=> "b"
  #     q.delete_min   #=> ["b", 20]
  #     q.min_key      #=> nil
  def min_key    
    @min.key rescue nil
  end
  
  # call-seq:
  #     min_priority -> priority
  #
  # Return the minimal priority or nil when the queue is empty.
  #
  #     q = PriorityQueue.new
  #     q["a"] = 10
  #     q["b"] = 20
  #     q.min_priority #=> 10
  #     q.delete_min   #=> ["a", 10]
  #     q.min_priority #=> 20
  #     q.delete_min   #=> ["b", 20]
  #     q.min_priority #=> nil
  def min_priority
    @min.priority rescue nil
  end
  
  # call-seq:
  #     delete(key) -> [key, priority]
  #     delete(key) -> nil
  #
  # Delete a key from the priority queue. Returns nil when the key was not in
  # the queue and [key, priority] otherwise.
  #
  #     q = PriorityQueue.new
  #     (0..10).each do | i | q[i.to_s] = i end
  #     q.delete(5)                              #=> ["5", 5]
  #     q.delete(5)                              #=> nil
  def delete(key)
    return nil unless n = @nodes.delete(key)

    if n.child 
      c = n.child
      e = n.child
      begin
	r = c.right
	cut_node(c)
	c = r
      end while c != e
    end
    cut_node(n) if n.parent

    if n == n.right 
      @min = nil;
      @rootlist = nil;
    else 
      @rootlist = n.right if @rootlist == n
      if @min == n
	n1 = n.right
	@min = n1
	begin
	  @min = n1 if n1.priority < @min.priority
	  n1 = n1.right
	end while(n1 != n);
      end
      n.right.left = n.left
      n.left.right = n.right
      n.left = n
      n.right = n
    end
    @length -= 1
    return [n.key, n.priority]
  end

  # call-seq:
  #    delete_min_return_key -> key
  #    
  # Delete key with minimal priority and return the key
  #
  #    q = PriorityQueue.new
  #    q["a"] = 1
  #    q["b"] = 0
  #    q.delete_min_return_key #=> "b"
  #    q.delete_min_return_key #=> "a"
  #    q.delete_min_return_key #=> nil
  def delete_min_return_key
    delete_min[0] rescue nil
  end
  
  # call-seq:
  #    delete_min_return_priority -> priority
  #    
  # Delete key with minimal priority and return the priority value
  #
  #    q = PriorityQueue.new
  #    q["a"] = 1
  #    q["b"] = 0
  #    q.delete_min_return_priority #=> 0
  #    q.delete_min_return_priority #=> 1
  #    q.delete_min_return_priority #=> nil
  def delete_min_return_priority
    delete_min[1] rescue nil
  end

  # call-seq:
  #    delete_min -> [key, priority]
  #    
  # Delete key with minimal priority and return [key, priority]
  #
  #    q = PriorityQueue.new
  #    q["a"] = 1
  #    q["b"] = 0
  #    q.delete_min #=> ["b", 0]
  #    q.delete_min #=> ["a", 1]
  #    q.delete_min #=> nil
  def delete_min
    return nil if self.empty?
    result = self.min

    @nodes.delete(@min.key)

    if @length == 1
      @rootlist = @min = nil
      @length = 0
    else 
      min = @min
      if @min == @rootlist # If the rootlist is anchored at the minimum, shift to the right
	if @rootlist == @rootlist.right 
	  @rootlist = @min = nil
	else
	  @rootlist = @min = @min.right
	end 
      end
      min.left.right = min.right;
      min.right.left = min.left;
      min.left = min.right = min;
      if min.child
	# Kinder und Eltern trennen, Markierung aufheben
	n = min.child;
	begin
	  n.parent = nil;
	  n.mark = false;
	  n = n.right;
	end while n != min.child

	# Kinder einfügen
	if @rootlist
	  l1 = @rootlist.left
	  l2 = n.left

	  l1.right = n
	  n.left = l1
	  l2.right = @rootlist
	  @rootlist.left = l2
	else
	  @rootlist = n
	end
      end

      # Größe anpassen
      @length -= 1

      # Wieder aufhübschen
      consolidate
    end

    result
  end
  
  # Returns a string representation of the priority queue.
  def inspect
    "<PriorityQueue: #{@nodes.map{|(_, n)| [n.key, n.priority]}.sort_by{|(_,p)|p}.inspect}>"
  end

  def initialize_copy(copy)
    copy_nodes = @nodes
    @nodes = {}

    copy_nodes.each do | (_, cn) |
      n = @nodes[cn.key] = Node.new(cn.key, cn.priority)
      n.mark = cn.mark
      n.degree = cn.degree
    end
    
    copy_nodes.each do | (_, cn) |
      n = @nodes[cn.key]
      n.left   = @nodes[cn.left.key]   if cn.left
      n.right  = @nodes[cn.right.key]  if cn.right
      n.parent = @nodes[cn.parent.key] if cn.parent
      n.child  = @nodes[cn.child.key]  if cn.child
    end
    @rootlist = @nodes[@rootlist.key] if @rootlist
    @min = @nodes[@min.key] if @min
    self
  end
end