text: '''
  altgraph.GraphAlgo - Graph algorithms
  =====================================
  '''
  from . import GraphError
  
  def dijkstra(graph, start, end=None):
      """
      Dijkstra's algorithm for shortest paths
  
      `David Eppstein, UC Irvine, 4 April 2002 <http://www.ics.uci.edu/~eppstein/161/python/>`_
  
      `Python Cookbook Recipe <http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466>`_
  
      Find shortest paths from the  start node to all nodes nearer than or equal to the end node.
  
      Dijkstra's algorithm is only guaranteed to work correctly when all edge lengths are positive.
      This code does not verify this property for all edges (only the edges examined until the end
      vertex is reached), but will correctly compute shortest paths even for some graphs with negative
      edges, and will raise an exception if it discovers that a negative edge has caused it to make a mistake.
  
      *Adapted to altgraph by Istvan Albert, Pennsylvania State University - June, 9 2004*
  
      """
      D = {}    # dictionary of final distances
      P = {}    # dictionary of predecessors
      Q = _priorityDictionary()    # estimated distances of non-final vertices
      Q[start] = 0
  
      for v in Q:
          D[v] = Q[v]
          if v == end: break
  
          for w in graph.out_nbrs(v):
              edge_id  = graph.edge_by_node(v,w)
              vwLength = D[v] + graph.edge_data(edge_id)
              if w in D:
                  if vwLength < D[w]:
                      raise GraphError("Dijkstra: found better path to already-final vertex")
              elif w not in Q or vwLength < Q[w]:
                  Q[w] = vwLength
                  P[w] = v
  
      return (D,P)
  
  def shortest_path(graph, start, end):
      """
      Find a single shortest path from the given start node to the given end node.
      The input has the same conventions as dijkstra(). The output is a list of the nodes
      in order along the shortest path.
  
      **Note that the distances must be stored in the edge data as numeric data**
      """
  
      D,P = dijkstra(graph, start, end)
      Path = []
      while 1:
          Path.append(end)
          if end == start: break
          end = P[end]
      Path.reverse()
      return Path
  
  #
  # Utility classes and functions
  #
  class _priorityDictionary(dict):
      '''
      Priority dictionary using binary heaps (internal use only)
  
      David Eppstein, UC Irvine, 8 Mar 2002
  
      Implements a data structure that acts almost like a dictionary, with two modifications:
          1. D.smallest() returns the value x minimizing D[x].  For this to work correctly,
              all values D[x] stored in the dictionary must be comparable.
          2. iterating "for x in D" finds and removes the items from D in sorted order.
              Each item is not removed until the next item is requested, so D[x] will still
              return a useful value until the next iteration of the for-loop.
              Each operation takes logarithmic amortized time.
      '''
      def __init__(self):
          '''
          Initialize priorityDictionary by creating binary heap of pairs (value,key).
          Note that changing or removing a dict entry will not remove the old pair from the heap
          until it is found by smallest() or until the heap is rebuilt.
          '''
          self.__heap = []
          dict.__init__(self)
  
      def smallest(self):
          '''
          Find smallest item after removing deleted items from front of heap.
          '''
          if len(self) == 0:
              raise IndexError("smallest of empty priorityDictionary")
          heap = self.__heap
          while heap[0][1] not in self or self[heap[0][1]] != heap[0][0]:
              lastItem = heap.pop()
              insertionPoint = 0
              while 1:
                  smallChild = 2*insertionPoint+1
                  if smallChild+1 < len(heap) and heap[smallChild] > heap[smallChild+1] :
                      smallChild += 1
                  if smallChild >= len(heap) or lastItem <= heap[smallChild]:
                      heap[insertionPoint] = lastItem
                      break
                  heap[insertionPoint] = heap[smallChild]
                  insertionPoint = smallChild
          return heap[0][1]
  
      def __iter__(self):
          '''
          Create destructive sorted iterator of priorityDictionary.
          '''
          def iterfn():
              while len(self) > 0:
                  x = self.smallest()
                  yield x
                  del self[x]
          return iterfn()
  
      def __setitem__(self,key,val):
          '''
          Change value stored in dictionary and add corresponding pair to heap.
          Rebuilds the heap if the number of deleted items gets large, to avoid memory leakage.
          '''
          dict.__setitem__(self,key,val)
          heap = self.__heap
          if len(heap) > 2 * len(self):
              self.__heap = [(v,k) for k,v in self.iteritems()]
              self.__heap.sort()  # builtin sort probably faster than O(n)-time heapify
          else:
              newPair = (val,key)
              insertionPoint = len(heap)
              heap.append(None)
              while insertionPoint > 0 and newPair < heap[(insertionPoint-1)//2]:
                  heap[insertionPoint] = heap[(insertionPoint-1)//2]
                  insertionPoint = (insertionPoint-1)//2
              heap[insertionPoint] = newPair
  
      def setdefault(self,key,val):
          '''
          Reimplement setdefault to pass through our customized __setitem__.
          '''
          if key not in self:
              self[key] = val
          return self[key]
  