# Graph Algorithms using basic python Constructs. # Narayana Chikkam, Dec, 22, 2015. from collections import defaultdict from heapq import * import itertools import copy from lib.unionfind import ( UnionFind ) from lib.prioritydict import ( priorityDictionary ) class Vertex: def __init__(self, id): self.id = id self.neighbours = {} def addNeighbour(self, id, weight): self.neighbours[id] = weight def __str__(self): return str(self.id) + ': ' + str(self.neighbours.keys()) def getNeighbours(self): return self.neighbours #.keys() def getName(self): return self.id def getWeight(self, id): return self.neighbours[id] class Graph: def __init__(self): self.v = {} self.count = 0 def addVertex(self, key): self.count += 1 newV = Vertex(key) self.v[key] = newV def getVertex(self, id): if id in self.v.keys(): return self.v[id] return None def __contains__(self, id): return id in self.v.keys() def addEdge(self, vertexOne, vertexTwo, weight=None): # vertexOne, vertexTwo, cost-of-the-edge if vertexOne not in self.v.keys(): self.addVertex(vertexOne) if vertexTwo not in self.v.keys(): self.addVertex(vertexTwo) self.v[vertexOne].addNeighbour(vertexTwo, weight) def updateEdge(self, vertexOne, vertexTwo, weight=None): # vertexOne, vertexTwo, cost-of-the-edge self.v[vertexOne].addNeighbour(vertexTwo, weight) def getVertices(self): return self.v.keys() def __str__(self): ret = "{ " for v in self.v.keys(): ret += str(self.v[v].__str__()) + ", " return ret + " }" def __iter__(self): return iter(self.v.values()) def getNeighbours(self, vertex): if vertex not in self.v.keys(): raise "Node %s not in graph" % vertex return self.v[vertex].neighbours #.keys() def getEdges(self): edges = [] for node in self.v.keys(): neighbours = self.v[node].getNeighbours() for w in neighbours: edges.append((node, w, neighbours[w])) #tuple, srcVertex, dstVertex, weightBetween return edges def findIsolated(self): isolated = [] for node in self.v: deadNode = False reachable = True # dead node, can't reach any other node from this if len(self.v[node].getNeighbours()) == 0: deadNode = True # reachable from other nodes ? nbrs = [n.neighbours.keys() for n in self.v.values()] # flatten the nested list nbrs = list(itertools.chain(*nbrs)) if node not in nbrs: reachable = False if deadNode == True and reachable == False: isolated.append(node) return isolated def getPath(self, start, end, path=[]): path = path + [start] if start == end: return path if start not in self.v: return None for vertex in self.v[start].getNeighbours(): if vertex not in path: extended_path = self.getPath(vertex, end, path) if extended_path: return extended_path return None def getAllPaths(self, start, end, path=[]): path = path + [start] if start == end: return [path] if start not in self.v: return [] paths = [] for vertex in self.v[start].getNeighbours(): if vertex not in path: extended_paths = self.getAllPaths(vertex, end, path) for p in extended_paths: paths.append(p) return paths def inDegree(self, vertex): """ how many edges coming into this vertex """ nbrs = [n.neighbours.keys() for n in self.v.values()] # flatten the nested list nbrs = list(itertools.chain(*nbrs)) return nbrs.count(vertex) def outDegree(self, vertex): """ how many vertices are neighbours to this vertex """ adj_vertices = self.v[vertex].getNeighbours() return len(adj_vertices) """ The degree of a vertex is the no of edges connecting to it. loop is counted twice for an undirected Graph deg(v) = indegree(v) + outdegree(v) """ def getDegree(self, vertex): return self.inDegree(vertex) + self.outDegree(vertex) def verifyDegreeSumFormula(self): """Handshaking lemma - Vdeg(v) = 2 |E| """ degSum = 0 for v in self.v: degSum += self.getDegree(v) return degSum == (2* len(self.getEdges())) def delta(self): """ the minimum degree of the Graph V """ min = 2**64 for vertex in self.v: vertex_degree = self.getDegree(vertex) if vertex_degree < min: min = vertex_degree return min def Delta(self): """ the maximum degree of the Graph V """ max = -2**64 for vertex in self.v: vertex_degree = self.getDegree(vertex) if vertex_degree > max: max = vertex_degree return max def degreeSequence(self): """ degree sequence is the reverse sorder of the vertices degrees Isomorphic graphs have the same degree sequence. However, two graphs with the same degree sequence are not necessarily isomorphic. More-Info: http://en.wikipedia.org/wiki/Graph_realization_problem """ seq = [] for vertex in self.v: seq.append(self.getDegree(vertex)) seq.sort(reverse=True) return tuple(seq) # helper to check if the given sequence is in non-increasing Order ;) @staticmethod def sortedInDescendingOrder(seq): return all (x>=y for x,y in zip(seq, seq[1:])) @staticmethod def isGraphicSequence(seq): """ Assumes that the degreeSequence is a list of non negative integers http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Gallai_theorem """ # Check to ensure there are an even number of odd degrees if sum(seq)%2 != 0: return False # Erdos-Gallai theorem for k in range(1, len(seq)+1): leftSum = sum(seq[:(k)]) rightSum = k * (k-1) + sum([min(x, k) for x in seq[k:]]) if leftSum > rightSum: return False return True @staticmethod def isGraphicSequenceIterative(s): # successively reduce degree sequence by removing node of maximum degree # as in Havel-Hakimi algorithm while s: s.sort() # sort in increasing order if s[0]<0: return False # check if removed too many from some node d=s.pop() # pop largest degree if d==0: return True # done! rest must be zero due to ordering # degree must be <= number of available nodes if d>len(s): return False # remove edges to nodes of next higher degrees #s.reverse() # to make it easy to get at higher degree nodes. for i in range(len(s)-1,len(s)-(d+1),-1): s[i]-=1 # should never get here b/c either d==0, d>len(s) or d<0 before s=[] return False def density(self): """ In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph. The distinction between sparse and dense graphs is rather vague, and depends on the context. For undirected simple graphs, the graph density is defined as: D = (2*No-Of-Edges)/((v*(v-1))/2) For a complete Graph, the Density D is 1 """ """ method to calculate the density of a graph """ V = len(self.v.keys()) E = len(self.getEdges()) return 2.0 * E / (V *(V - 1)) """ Choose an arbitrary node x of the graph G as the starting point Determine the set A of all the nodes which can be reached from x. If A is equal to the set of nodes of G, the graph is connected; otherwise it is disconnected. """ def isConnected(self, start=None): if start == None: start = self.v.keys()[0] reachables = self.dfs(start, []) return len(reachables) == len(self.v.keys()) """ ToDo: USE CLR Approach for this Later """ def dfs(self, start, path = []): path = path + [start] for v in self.v[start].getNeighbours().keys(): if v not in path: path = self.dfs(v, path) return path """ CLR Sytle """ def CLR_Dfs(self): paths = [] for v in self.v.keys(): explored = self.dfs(v, []) if len(explored) == len(self.v.keys()): paths.append(explored) return paths def BFS(self, start): # initialize lists maxV = len(self.v.keys()) processed = [False] * (maxV) # which vertices have been processed discovered = [False] * (maxV) # which vertices have been found parent= [-1] * (maxV) # discovery relation q = [] # queue of vertices to visit */ # enqueue(&q,start); q.append(start) discovered[start] = True while (len(q) != 0): v = q.pop(0) processed[v] = True nbrs = self.v[v].getNeighbours().keys() # print nbrs for n in nbrs: # if processed[n] == False if discovered[n] == False: q.append(n) discovered[n] = True parent[n] = v return (discovered, parent) def findPath(self, start, end, parents, path): if ((start == end) or (end == -1)): path.append(start) else: self.findPath(start, parents[end], parents, path) path.append(end) """ Find path between two given nodes """ def find_path(self, start, end, path=[]): path = path + [start] if start == end: return path if not self.v.has_key(start): return None for node in self.v[start].getNeighbours().keys(): if node not in path: newpath = self.find_path(node, end, path) if newpath: return newpath return None """ Find all paths """ def find_all_paths(self, start, end, path=[]): path = path + [start] if start == end: return [path] if not self.v.has_key(start): return [] paths = [] for node in self.v[start].getNeighbours().keys(): if node not in path: newpaths = self.find_all_paths(node, end, path) for newpath in newpaths: paths.append(newpath) return paths """ Find shorted path w.r.t no of vertices on the path """ def find_shortest_path(self, start, end, path=[]): path = path + [start] if start == end: return path if not self.v.has_key(start): return None shortest = None for node in self.v[start].getNeighbours().keys(): if node not in path: newpath = self.find_shortest_path(node, end, path) if newpath: if not shortest or len(newpath) < len(shortest): shortest = newpath return shortest """ prim's algorithm - properties: tree could be not connected during the finding process as it finds edges with min cost - greedy strategy Prims always stays as a tree If you don't know all the weight on edges use Prim's algorithm f you only need partial solution on the graph use Prim's algorithm """ def mspPrims(self): nodes = self.v.keys() edges = [(u, v, c) for u in self.v.keys() for v, c in self.v[u].getNeighbours().items()] return self.prim(nodes, edges) def prim(self, nodes, edges): conn = defaultdict( list ) for n1,n2,c in edges: # makes graph undirected conn[ n1 ].append( (c, n1, n2) ) conn[ n2 ].append( (c, n2, n1) ) mst = [] used = set() used.add( nodes[0] ) usable_edges = conn[ nodes[0] ][:] heapify( usable_edges ) while usable_edges: cost, n1, n2 = heappop( usable_edges ) if n2 not in used: used.add( n2 ) mst.append( ( n1, n2, cost ) ) for e in conn[ n2 ]: if e[ 2 ] not in used: heappush( usable_edges, e ) return mst """ Kruskals begins with forest and merge into a tree """ def mspKrushkals(self): nodes = self.v.keys() edges = [(c, u, v) for u in self.v.keys() for v, c in self.v[u].getNeighbours().items()] return self.krushkal(edges) def pprint(self): print ("{ ", end=" ") for u in self.v.keys(): print (u, end=" ") print (": { ", end=" ") for v in self.v[u].getNeighbours().keys(): print (v, ":", self.v[u].getNeighbours()[v], end=" ") print(" }", end= " ") print (" }\n") def krushkal(self, edges): """ Return the minimum spanning tree of an undirected graph G. G should be represented in such a way that iter(G) lists its vertices, iter(G[u]) lists the neighbors of u, G[u][v] gives the length of edge u,v, and G[u][v] should always equal G[v][u]. The tree is returned as a list of edges. """ # Kruskal's algorithm: sort edges by weight, and add them one at a time. # We use Kruskal's algorithm, first because it is very simple to # implement once UnionFind exists, and second, because the only slow # part (the sort) is sped up by being built in to Python. subtrees = UnionFind() tree = [] for c,u,v in sorted(edges): # take from small weight to large in order if subtrees[u] != subtrees[v]: tree.append((u,v, c)) subtrees.union(u,v) return tree def adj(self, missing=float('inf')): # makes the adj dict will all possible cells, similar to matrix """ G= { 0 : { 1 : 6, 2 : 4 } 1 : { 2 : 3, 5 : 7 } 2 : { 3 : 9, 4 : 1 } 3 : { 4 : 1 } 4 : { 5 : 5, 6 : 2 } 5 : { } 6 : { } } adj(G) >> { 0: {0: 0, 1: 6, 2: 4, 3: inf, 4: inf, 5: inf, 6: inf}, 1: {0: inf, 1: 0, 2: 3, 3: inf, 4: inf, 5: 7, 6: inf}, 2: {0: inf, 1: inf, 2: 0, 3: 9, 4: 1, 5: inf, 6: inf}, 3: {0: inf, 1: inf, 2: inf, 3: 0, 4: 1, 5: inf, 6: inf}, 4: {0: inf, 1: inf, 2: inf, 3: inf, 4: 0, 5: 5, 6: 2}, 5: {0: inf, 1: inf, 2: inf, 3: inf, 4: inf, 5: 0, 6: inf}, 6: {0: inf, 1: inf, 2: inf, 3: inf, 4: inf, 5: inf, 6: 0} } """ vertices = self.v.keys() return {v1: {v2: 0 if v1 == v2 else self.v[v1].getNeighbours().get(v2, missing) for v2 in vertices } for v1 in vertices } def floyds(self): """ All pair shortest Path Idea: for k in (0, n): for i in (0, n): for j in (0, n): g[i][j] = min(graph[i][j], graph[i][k]+graph[k][j]) Find the shortest distance between every pair of vertices in the weighted Graph G """ d = self.adj() # prepare the adjacency list representation for the algorithm vertices = self.v.keys() for v2 in vertices: d = {v1: {v3: min(d[v1][v3], d[v1][v2] + d[v2][v3]) for v3 in vertices} for v1 in vertices} return d def reachability(self): """ Idea: graph reachability floyd-warshall for k in (0, n): for i in (0, n): for j in (0, n): g[i][j] = graph[i][j] || (graph[i][k]&&graph[k][j])) """ vertices = self.v.keys() d = self.adj(float('0')) for u in vertices: for v in vertices: if u ==v or d[u][v]: d[u][v] = True else: d[u][v] = False for v2 in vertices: d = {v1: {v3: d[v1][v3] or (d[v1][v2] and d[v2][v3]) # path for v1->v3 or v1->v2, v2-?v3 for v3 in vertices} for v1 in vertices} return d def pathRecoveryFloydWarshall(self): d = self.adj() # missing edges will have -1.0 value vertices = self.v.keys() parentMap = copy.deepcopy(d) for v1 in vertices: for v2 in vertices: if (v1 == v2) or d[v1][v2] == float('inf'): parentMap[v1][v2] = -1 else: parentMap[v1][v2] = v1 for i in vertices: for j in vertices: for k in vertices: temp = d[i][k] + d[k][j] if temp < d[i][j]: d[i][j] = temp parentMap[i][j] = parentMap[k][j] return parentMap def getFloydPath(self, parentMap, u, v, path=[]): """ recursive procedure to get the path from parentMap matrix """ path.append(v) if u != v and v != -1: self.getFloydPath(parentMap, u, parentMap[u][v], path) # from active recipes - handy thoughts to think about heap for this algorithm def dijkstra(self, 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> 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. Introduction to Algorithms, 1st edition), page 528: G = { 's':{'u':10, 'x':5}, ' u':{'v':1, 'x':2}, 'v':{'y':4}, 'x':{'u':3, 'v':9, 'y':2}, 'y':{'s':7, 'v':6} } """ G = self.adj() 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 shortestPathDijkstra(self, 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 = self.dijkstra(start, end) Path = [] while 1: Path.append(end) if end == start: break end = P[end] Path.reverse() return Path """ smart snippet on the dijkstra alg: def shortestPath(graph, start, end): queue = [(0, start, [])] seen = set() while True: (cost, v, path) = heapq.heappop(queue) if v not in seen: path = path + [v] seen.add(v) if v == end: return cost, path for (next, c) in graph[v].iteritems(): heapq.heappush(queue, (cost + c, next, path)) """ def strongly_connected_components(self): """ Tarjan's Algorithm (named for its discoverer, Robert Tarjan) is a graph theory algorithm for finding the strongly connected components of a graph. Based on: http://en.wikipedia.org/wiki/Tarjan's_strongly_connected_components_algorithm """ index_counter = [0] stack = [] lowlinks = {} index = {} result = [] def strongconnect(node): # set the depth index for this node to the smallest unused index index[node] = index_counter[0] lowlinks[node] = index_counter[0] index_counter[0] += 1 stack.append(node) # Consider successors of `node` try: successors = self.v[node].getNeighbours().keys() print(node, successors) except: successors = [] for successor in successors: if successor not in lowlinks: # Successor has not yet been visited; recurse on it strongconnect(successor) lowlinks[node] = min(lowlinks[node],lowlinks[successor]) elif successor in stack: # the successor is in the stack and hence in the current strongly connected component (SCC) lowlinks[node] = min(lowlinks[node],index[successor]) # If `node` is a root node, pop the stack and generate an SCC if lowlinks[node] == index[node]: connected_component = [] while True: successor = stack.pop() connected_component.append(successor) if successor == node: break component = tuple(connected_component) # storing the result print(component) result.append(component) for node in self.v.keys(): if node not in lowlinks: strongconnect(node) return result def computeFirstUsingSCC(self, initFirst): index_counter = [0] stack = [] lowlinks = {} index = {} result = [] first = {} def computeFirst(node): # set the depth index for this node to the smallest unused index index[node] = index_counter[0] lowlinks[node] = index_counter[0] index_counter[0] += 1 stack.append(node) # Consider successors of `node` try: successors = self.v[node].getNeighbours().keys() except: successors = [] for successor in successors: if successor not in lowlinks: # Successor has not yet been visited; recurse on it computeFirst(successor) lowlinks[node] = min(lowlinks[node],lowlinks[successor]) elif successor in stack: # the successor is in the stack and hence in the current strongly connected component (SCC) lowlinks[node] = min(lowlinks[node],index[successor]) first[node] |= set(first[successor] - set(['epsilon'])).union(set(initFirst[node])) #(*union!*) # If `node` is a root node, pop the stack and generate an SCC if lowlinks[node] == index[node]: connected_component = [] while True: successor = stack.pop() #FIRST[w] := FIRST[v]; (*distribute!*) first[successor] = set(first[node] - set(['epsilon'])).union(set(initFirst[successor]) )#(*distribute!*) connected_component.append(successor) if successor == node: break component = tuple(connected_component) # storing the result result.append(component) for v in initFirst: first[v] = initFirst[v] #(*init!*) #print "init First assignment: ", first for node in self.v.keys(): if node not in lowlinks: computeFirst(node) return first def computeFollowUsingSCC(self, FIRST, initFollow): index_counter = [0] stack = [] lowlinks = {} index = {} result = [] follow = {} def computeFollow(node): # set the depth index for this node to the smallest unused index index[node] = index_counter[0] lowlinks[node] = index_counter[0] index_counter[0] += 1 stack.append(node) # Consider successors of `node` try: successors = self.v[node].getNeighbours().keys() except: successors = [] for successor in successors: if successor not in lowlinks: # Successor has not yet been visited; recurse on it computeFollow(successor) lowlinks[node] = min(lowlinks[node],lowlinks[successor]) elif successor in stack: # the successor is in the stack and hence in the current strongly connected component (SCC) lowlinks[node] = min(lowlinks[node],index[successor]) follow[node] |= follow[successor] #(*union!*) # If `node` is a root node, pop the stack and generate an SCC if lowlinks[node] == index[node]: connected_component = [] while True: successor = stack.pop() follow[successor] = follow[node] connected_component.append(successor) if successor == node: break component = tuple(connected_component) # storing the result result.append(component) for v in initFollow: follow[v] = initFollow[v] #(*init!*) for node in self.v.keys(): if node not in lowlinks: computeFollow(node) return follow

#### Diff to Previous Revision

--- revision 1 2015-12-22 10:46:28 +++ revision 2 2015-12-22 10:49:59 @@ -1,4 +1,4 @@ -# Graph Algorithms using basic pythos Constructs. +# Graph Algorithms using basic python Constructs. # Narayana Chikkam, Dec, 22, 2015. from collections import defaultdict