《算法图解》读书笔记7-狄克斯特拉算法

概念

狄克斯特拉算法是从一个顶点到其余各顶点的最短路径算法,解决的是有向图中最短路径问题 (该算法不能处理包含负边的图)。主要特点是以起始点为中心向外层层扩展,直到扩展到终点为止。

Python实现(收藏)

#coding:utf8

graph = {
'N1': {'N2': 1},
'N12':{'N13':1},
'N2': {'N4': 1, 'N13':1},
'N4': {'N5': 1, 'N12': 1},
'N5': {'N6': 1},
'N6': {'N7': 1},
'N7': {'N8': 1},
'N8': {'N9':1},
'N9': {'N10': 1, 'N11': 1, 'N12':1},
'N10': {'N11':1},
'N11': {'N12':1},
'N13':{'N12':1}
}

# Dijkstra's algorithm for shortest paths
# David Eppstein, UC Irvine, 4 April 2002

# http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/117228
from priodict import priorityDictionary

def Dijkstra(G,start,end=None):
    """
    Find shortest paths from the start vertex to all
    vertices nearer than or equal to the end.
    The input graph G is assumed to have the following
    representation: A vertex can be any object that can
    be used as an index into a dictionary.  G is a
    dictionary, indexed by vertices.  For any vertex v,
    G[v] is itself a dictionary, indexed by the neighbors
    of v.  For any edge v->w, G[v][w] is the length of
    the edge.  This is related to the representation in
    <http://www.python.org/doc/essays/graphs.html>
    where Guido van Rossum suggests representing graphs
    as dictionaries mapping vertices to lists of neighbors,
    however dictionaries of edges have many advantages
    over lists: they can store extra information (here,
    the lengths), they support fast existence tests,
    and they allow easy modification of the graph by edge
    insertion and removal.  Such modifications are not
    needed here but are important in other graph algorithms.
    Since dictionaries obey iterator protocol, a graph
    represented as described here could be handed without
    modification to an algorithm using Guido's representation.
    Of course, G and G[v] need not be Python dict objects;
    they can be any other object that obeys dict protocol,
    for instance a wrapper in which vertices are URLs
    and a call to G[v] loads the web page and finds its links.

    The output is a pair (D,P) where D[v] is the distance
    from start to v and P[v] is the predecessor of v along
    the shortest path from s to v.

    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 seen
    before 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.
    """

    D = {}  # dictionary of final distances
    P = {}  # dictionary of predecessors
    Q = priorityDictionary()   # est.dist. of non-final vert.
    Q[start] = 0

    for v in Q:
        D[v] = Q[v]
        if v == end: break

        for w in G[v]:
            vwLength = D[v] + G[v][w]
            if w in D:
                if vwLength < D[w]:
                    raise ValueError, \
  "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 shortestPath(G,start,end):
    """
    Find a single shortest path from the given start vertex
    to the given end vertex.
    The input has the same conventions as Dijkstra().
    The output is a list of the vertices in order along
    the shortest path.
    """

    D,P = Dijkstra(G,start,end)
    Path = []
    while 1:
        Path.append(end)
        if end == start: break
        end = P[end]
    Path.reverse()
    return Path


print shortestPath(graph, 'N1','N13')
print shortestPath(graph, 'N1','N12')

priodict.py

# Priority dictionary using binary heaps
# 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.

from __future__ import generators

class priorityDictionary(dict):
    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]

gist地址:

https://gist.github.com/theonewolf/6175427

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