How to find connected components?

Question

I'm writing a function get_connected_components for a class Graph:

def get_connected_components(self):
    path=[]
    for i in self.graph.keys():
        q=self.graph[i]
        while q:
            print(q)
            v=q.pop(0)
            if not v in path:
                path=path+[v]
    return path

My graph is:

{0: [(0, 1), (0, 2), (0, 3)], 1: [], 2: [(2, 1)], 3: [(3, 4), (3, 5)], \
4: [(4, 3), (4, 5)], 5: [(5, 3), (5, 4), (5, 7)], 6: [(6, 8)], 7: [], \
8: [(8, 9)], 9: []}

where the keys are the nodes and the values are the edge. My function gives me this connected component:

[(0, 1), (0, 2), (0, 3), (2, 1), (3, 4), (3, 5), (4, 3), (4, 5), (5, 3), \
(5, 4), (5, 7), (6, 8), (8, 9)]

But I would have two different connected components, like:

[[(0, 1), (0, 2), (0, 3), (2, 1), (3, 4), (3, 5), (4, 3), (4, 5), \
(5, 3), (5, 4), (5, 7)],[(6, 8), (8, 9)]]

I don't understand where I made the mistake. Can anyone help me?

Spelter

Note that your representation include redundant information, eg. in 3: [(3, 4), (3, 5)]. We already know that the edge is starting from 3! – Conoscenti

Do you suggest me to change the values in the dict and put only the node connected and no the edges? – Spelter

BTW instead of for i in self.graph.keys(): q=self.graph[i] you can for (i, q) in self.graph.iteritems() – Packsaddle

How can you expect to get a result like you want? The only way that you ever modify path is with the statement path = path + [v], which adds an edge to the list. If you want to create a list of lists of edges, then you need to have code that can make more than one list of edges, and add them to the list of list of edges... – Lissa

Is there a reason you're creating your own graph? The awesome networkx library has a connected components algorithm built-in. – Irrepealable

Yes I want create my own graph to improve my skills in python programming – Spelter

@Irrepealable As a reader of this post and many others, I would always welcome new implementations even naive rather than referring to an existing package. – Sweptwing

@Sweptwing I always ask because a lot of times it's homework... – Irrepealable

Answer 1

Let's simplify the graph representation:

myGraph = {0: [1,2,3], 1: [], 2: [1], 3: [4,5],4: [3,5], 5: [3,4,7], 6: [8], 7: [],8: [9], 9: []}

Here we have the function returning a dictionary whose keys are the roots and whose values are the connected components:

def getRoots(aNeigh):
    def findRoot(aNode,aRoot):
        while aNode != aRoot[aNode][0]:
            aNode = aRoot[aNode][0]
        return (aNode,aRoot[aNode][1])
    myRoot = {} 
    for myNode in aNeigh.keys():
        myRoot[myNode] = (myNode,0)  
    for myI in aNeigh: 
        for myJ in aNeigh[myI]: 
            (myRoot_myI,myDepthMyI) = findRoot(myI,myRoot) 
            (myRoot_myJ,myDepthMyJ) = findRoot(myJ,myRoot) 
            if myRoot_myI != myRoot_myJ: 
                myMin = myRoot_myI
                myMax = myRoot_myJ 
                if  myDepthMyI > myDepthMyJ: 
                    myMin = myRoot_myJ
                    myMax = myRoot_myI
                myRoot[myMax] = (myMax,max(myRoot[myMin][1]+1,myRoot[myMax][1]))
                myRoot[myMin] = (myRoot[myMax][0],-1) 
    myToRet = {}
    for myI in aNeigh: 
        if myRoot[myI][0] == myI:
            myToRet[myI] = []
    for myI in aNeigh: 
        myToRet[findRoot(myI,myRoot)[0]].append(myI) 
    return myToRet  

Let's try it:

print getRoots(myGraph)

{8: [6, 8, 9], 1: [0, 1, 2, 3, 4, 5, 7]}

Raymonderaymonds

Answer 2

I like this algorithm:

def connected_components(neighbors):
    seen = set()
    def component(node):
        nodes = set([node])
        while nodes:
            node = nodes.pop()
            seen.add(node)
            nodes |= neighbors[node] - seen
            yield node
    for node in neighbors:
        if node not in seen:
            yield component(node)

Not only is it short and elegant, but also fast. Use it like so (Python 2.7):

old_graph = {
    0: [(0, 1), (0, 2), (0, 3)],
    1: [],
    2: [(2, 1)],
    3: [(3, 4), (3, 5)],
    4: [(4, 3), (4, 5)],
    5: [(5, 3), (5, 4), (5, 7)],
    6: [(6, 8)],
    7: [],
    8: [(8, 9)],
    9: []}

edges = {v for k, vs in old_graph.items() for v in vs}
graph = defaultdict(set)

for v1, v2 in edges:
    graph[v1].add(v2)
    graph[v2].add(v1)

components = []
for component in connected_components(graph):
    c = set(component)
    components.append([edge for edges in old_graph.values()
                            for edge in edges
                            if c.intersection(edge)])

print(components)

The result is:

[[(0, 1), (0, 2), (0, 3), (2, 1), (3, 4), (3, 5), (4, 3), (4, 5), (5, 3), (5, 4), (5, 7)],
 [(6, 8), (8, 9)]]

Thanks, aparpara for spotting the bug.

Allodial

Strictly speaking, it is incorrect. E.g. it gives 2 components for {1: {2}, 2 : set(), 3: {2}}, while the accepted answer correctly gives 1. But it can be easily fixed by making the graph bi-directional before applying the algorithm. – Fakieh

@aparpara: fixed it. – Allodial

Answer 3

The previous answer is great. Anyway, it took to me a bit to understand what was going on. So, I refactored the code in this way that is easier to read for me. I leave here the code in case someone founds it easier too (it runs in python 3.6)

def get_all_connected_groups(graph):
    already_seen = set()
    result = []
    for node in graph:
        if node not in already_seen:
            connected_group, already_seen = get_connected_group(node, already_seen)
            result.append(connected_group)
    return result


def get_connected_group(node, already_seen):
        result = []
        nodes = set([node])
        while nodes:
            node = nodes.pop()
            already_seen.add(node)
            nodes = nodes or graph[node] - already_seen
            result.append(node)
        return result, already_seen


graph = {
     0: {0, 1, 2, 3},
     1: set(),
     2: {1, 2},
     3: {3, 4, 5},
     4: {3, 4, 5},
     5: {3, 4, 5, 7},
     6: {6, 8},
     7: set(),
     8: {8, 9},
     9: set()}

components = get_all_connected_groups(graph)
print(components)

Result:

Out[0]: [[0, 1, 2, 3, 4, 5, 7], [6, 8, 9]] 

Also, I simplified the input and output. I think it's a bit more clear to print all the nodes that are in a group

Mulford

"nodes = nodes or graph[node] - already_seen" should be "nodes.update(graph[node] - already_seen)" – Historiated

@Historiated if graph[5] = {3, 4, 5, 8}, shouldn't we get get one connected component – Buskined

I corrected the error in the line "nodes = nodes or graph[node] - already_seen" to 'nodes.update(n for n in graph[node] if n not in already_seen)' – Hiltan

Answer 4

If you represent the graph using an adjacency list, you can use this generator function (implementing BFS) to get all connected components:

from collections import deque

def connected_components(graph):
    seen = set()

    for root in range(len(graph)):
        if root not in seen:
            seen.add(root)
            component = []
            queue = deque([root])

            while queue:
                node = queue.popleft()
                component.append(node)
                for neighbor in graph[node]:
                    if neighbor not in seen:
                        seen.add(neighbor)
                        queue.append(neighbor)
            yield component

Demo:

graph = [
    [1, 2, 3],  # neighbors of node "0"
    [0, 2],     # neighbors of node "1"
    [0, 1],     # ...
    [0, 4, 5],
    [3, 5],
    [3, 4, 7],
    [8],
    [5],
    [9, 6],
    [8]
]

print(list(connected_components(graph)))  # [[0, 1, 2, 3, 4, 5, 7], [6, 8, 9]]

Dey