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Is the given Graph a tree? Faster than below approach -

I was given a question during an interview and although my answer was accepted at the end they wanted a faster approach and I went blank..
Question :
Given an undirected graph, can you see if it's a tree? If so, return true and false otherwise.
A tree:
A - B
|
C - D
not a tree:
A
/ \
B - C
/
D
You'll be given two parameters: n for number of nodes, and a multidimensional array of edges like such: [[1, 2], [2, 3]], each pair representing the vertices connected by the edge.
Note:Expected space complexity : O(|V|)
The array edges can be empty
Here is My code: 105ms
def is_graph_tree(n, edges):
nodes = [None] * (n + 1)
for i in range(1, n+1):
nodes[i] = i
for i in range(len(edges)):
start_edge = edges[i][0]
dest_edge = edges[i][1]
if nodes[start_edge] != start_edge:
start_edge = nodes[start_edge]
if nodes[dest_edge] != dest_edge:
dest_edge = nodes[dest_edge]
if start_edge == dest_edge:
return False
nodes[start_edge] = dest_edge
return len(edges) <= n - 1

Here's one approach using a disjoint-set-union / union-find data structure:
def is_graph_tree(n, edges):
parent = list(range(n+1))
size = [1] * (n + 1)
for x, y in edges:
# find x (path splitting)
while parent[x] != x:
x, parent[x] = parent[x], parent[parent[x]]
# find y
while parent[y] != y:
y, parent[y] = parent[y], parent[parent[y]]
if x == y:
# Already connected
return False
# Union (by size)
if size[x] < size[y]:
x, y = y, x
parent[y] = x
size[x] += size[y]
return True
assert not is_graph_tree(4, [(1, 2), (2, 3), (3, 4), (4, 2)])
assert is_graph_tree(6, [(1, 2), (2, 3), (3, 4), (3, 5), (1, 6)])
The runtime is O(V + E*InverseAckermannFunction(V)), which better than O(V + E * log(log V)), so it's basically O(V + E).

Tim Roberts has posted a candidate solution, but this will work in the case of disconnected subtrees:
import queue
def is_graph_tree(n, edges):
# A tree with n nodes has n - 1 edges.
if len(edges) != n - 1:
return False
# Construct graph.
graph = [[] for _ in range(n)]
for first_vertex, second_vertex in edges:
graph[first_vertex].append(second_vertex)
graph[second_vertex].append(first_vertex)
# BFS to find edges that create cycles.
# The graph is undirected, so we can root the tree wherever we want.
visited = set()
q = queue.Queue()
q.put((0, None))
while not q.empty():
current_node, previous_node = q.get()
if current_node in visited:
return False
visited.add(current_node)
for neighbor in graph[current_node]:
if neighbor != previous_node:
q.put((neighbor, current_node))
# Only return true if the graph has only one connected component.
return len(visited) == n
This runs in O(n + len(edges)) time.

You could approach this from the perspective of tree leaves. Every leaf node in a tree will have exactly one edge connected to it. So, if you count the number of edges for each nodes, you can get the list of leaves (i.e. the ones with only one edge).
Then, take the linked node from these leaves and reduce their edge count by one (as if you were removing all the leaves from the tree. That will give you a new set of leaves corresponding to the parents of the original leaves. Repeat the process until you have no more leaves.
[EDIT] checking that the number of edges is N-1 eliminiates the need to do the multi-root check because there will be another discrepancy (e.g. double link, missing node) in the graph if there are multiple 'roots' or a disconnected subtree
If the graph is a tree, this process should eliminate all nodes from the node counts (i.e. they will all be flagged as leaves at some point).
Using the Counter class (from collections) will make this relatively easy to implement:
from collections import Counter
def isTree(N,E):
if N==1 and not E: return True # root only is a tree
if len(E) != N-1: return False # a tree has N-1 edges
counts = Counter(n for ab in E for n in ab) # edge counts per node
if len(counts) != N : return False # unlinked nodes
while True:
leaves = {n for n,c in counts.items() if c==1} # new leaves
if not leaves:break
for a,b in E: # subtract leaf counts
if counts[a]>1 and b in leaves: counts[a] -= 1
if counts[b]>1 and a in leaves: counts[b] -= 1
for n in leaves: counts[n] = -1 # flag leaves in counts
return all(c==-1 for c in counts.values()) # all must become leaves
output:
G = [[1,2],[1,3],[4,5],[4,6]]
print(isTree(6,G)) # False (disconnected sub-tree)
G = [[1,2],[1,3],[1,4],[2,3],[5,6]]
print(isTree(6,G)) # False (doubly linked node 3)
G = [[1,2],[2,6],[3,4],[5,1],[2,3]]
print(isTree(6,G)) # True
G = [[1,2],[2,3]]
print(isTree(3,G)) # True
G = [[1,2],[2,3],[3,4]]
print(isTree(4,G)) # True
G = [[1,2],[1,3],[2,5],[2,4]]
print(isTree(6,G)) # False (missing node)
Space complexity is O(N) because the counts dictionary has one entry per node(vertex) with an integer as value. Time complexity will be O(ExL) where E is the number of edges and L is the number of levels in the tree. The worts case time is O(E^2) for a tree where all parents have only one child node. However, since the initial condition is for E to be less than V, the worst case will actually be O(V^2)
Note that this algorithm makes no assumption on edge order or numerical relationships between node numbers. The root (last node to be made a leaf) found by this algorithm is not necessarily the only possible root given that, unless the nodes have an implicit cardinality relationship (or edges have an order), there could be ambiguous scenarios:
[1,2],[2,3],[2,4] could be:
1 2 3
|_2 OR |_1 OR |_2
|_3 |_3 |_1
|_4 |_4 |_4
If a cardinality relationship between node numbers or an order of edges can be relied upon, the algorithm could potentially be made more time efficient (because we could easily determine which node is the root and start from there).
[EDIT2] Alternative method using groups.
When the number of edges is N-1, if the graph is a tree, all nodes should be reachable from any other node. This means that, if we form groups of reachable nodes for each node and merge them together based on the edges, we should end up with a single group after going through all the edges.
Here is the modified function based on that approach:
def isTree(N,E):
if N==1 and not E: return True # root only is a tree
if len(E) != N-1: return False # a tree has N-1 edges
groups = {n:[n] for ab in E for n in ab} # each node in its own group
if len(groups) != N : return False # unlinked nodes
for a,b in E:
groups[a].extend(groups[b]) # merge groups
for n in groups[b]: groups[n] = groups[a] # update nodes' groups
return len(set(map(id,groups.values()))) == 1 # only one group when done
Given that we start out with fewer edges than nodes and that group merging will consume at most 2x a group size (so also < N), the space complexity will remain O(V). The time complexity will also be O(V^2) at for the worts case scenarios

You don't even need to know how many edges there are:
def is_graph_tree(n, edges):
seen = set()
for a,b in edges:
b = max(a,b)
if b in seen:
return False
seen.add(b)
return True
a = [[1,2],[2,3],[3,4]]
print(is_graph_tree(0,a))
b = [[1,2],[1,3],[2,3],[2,4]]
print(is_graph_tree(0,b))
Now, this WON'T catch the case of disconnected subtrees, but that wasn't in the problem description...

Representing a tree as glued half-edges

I have a tree, given e.g. as a networkx object. In order to inpput it into a black-box algorithm I was given, I need to save it in the following strange format:
Traverse the tree in a clockwise order. As I pass through one side of an edge, I label it incrementally. Then I want to save for each edge the labels of its two sides.
For example, a star will become a list [(0,1),(2,3),(4,5),...] and a path with 3 vertices will be [(0,3),(1,2)].
I am stumped with implementing this. How can this be done? I can use any library.

I'll answer this without reference to any library.
You would need to perform a depth-first traversal, and log the (global) incremental number before you visit a subtree, and also after you visited it. Those two numbers make up the tuple that you have to prepend to the result you get from the subtree traversal.
Here is an implementation that needs the graph to be represented as an adjacency list. The main function needs to get the root node and the adjacency list
def iter_naturals(): # helper function to produce sequential numbers
n = 0
while True:
yield n
n += 1
def half_edges(root, adj):
visited = set()
sequence = iter_naturals()
def dfs(node):
result = []
visited.add(node)
for child in adj[node]:
if child not in visited:
forward = next(sequence)
path = dfs(child)
backward = next(sequence)
result.extend([(forward, backward)] + path)
return result
return dfs(root)
Here is how you can run it for the two examples you mentioned. I have just implemented those graphs as adjacency lists, where nodes are identified by their index in that list:
Example 1: a "star":
The root is the parent of all other nodes
adj = [
[1,2,3], # 1,2,3 are children of 0
[],
[],
[]
]
print(half_edges(0, adj)) # [(0, 1), (2, 3), (4, 5)]
Example 2: a single path with 3 nodes
adj = [
[1], # 1 is a child of 0
[2], # 2 is a child of 1
[]
]
print(half_edges(0, adj)) # [(0, 3), (1, 2)]

I found this great built-in function dfs_labeled_edges in networkx. From there it is a breeze.
def get_new_encoding(G):
dfs = [(v[0],v[1]) for v in nx.dfs_labeled_edges(G, source=1) if v[0]!=v[1] and v[2]!="nontree"]
dfs_ind = sorted(range(len(dfs)), key=lambda k: dfs[k])
new_tree_encoding = [(dfs_ind[i],dfs_ind[i+1]) for i in range(0,len(dfs_ind),2)]
return new_tree_encoding

Find connected branches from list of line segments

Problem
I have a list of line segments:
exampleLineSegments = [(1,2),(2,3),(3,4),(4,5),(5,6),(4,7),(8,7)]
These segments include the indices of the corresponding point in a separate array.
From this sublist, one can see that there is a branching point (4). So three different branches are emerging from this branching point.
(In other, more specific problems, there might be / are multiple branching points for n branches.)
Target
My target is to get a dictionary including information about the existing branches, so e.g.:
result = { branch_1: [1,2,3,4],
branch_2: [4,5,6],
branch_3: [4,7,8]}
Current state of work/problems
Currently, I am identifying the branch points first by setting up a dictionary for each point and checking for each entry if there are more than 2 neighbor points found. This means that there is a branching point.
Afterwards I am crawling through all points emerging from these branch points, checking for successors etc.
In these functions, there are a some for loops and generally an intensive "crawling". This is not the cleanest solution and if the number of points increasing, the performance is not so good either.
Question
What is the best / fastest / most performant way to achieve the target in this case?

I think you can achieve it by following steps:
use a neighbors dict to store the graph
find all branch points, which neighbors count > 2
start from every branch point, and use dfs to find all the paths
from collections import defaultdict
def find_branch_paths(exampleLineSegments):
# use dict to store the graph
neighbors = defaultdict(list)
for p1, p2 in exampleLineSegments:
neighbors[p1].append(p2)
neighbors[p2].append(p1)
# find all branch points
branch_points = [k for k, v in neighbors.items() if len(v) > 2]
res = []
def dfs(cur, prev, path):
# reach the leaf
if len(neighbors[cur]) == 1:
res.append(path)
return
for neighbor in neighbors[cur]:
if neighbor != prev:
dfs(neighbor, cur, path + [neighbor])
# start from all the branch points
for branch_point in branch_points:
dfs(branch_point, None, [branch_point])
return res
update an iteration version, for big data, which may cause a recursion depth problem:
def find_branch_paths(exampleLineSegments):
# use dict to store the graph
neighbors = defaultdict(list)
for p1, p2 in exampleLineSegments:
neighbors[p1].append(p2)
neighbors[p2].append(p1)
# find all branch points
branch_points = [k for k, v in neighbors.items() if len(v) > 2]
res = []
# iteration way to dfs
stack = [(bp, None, [bp]) for bp in branch_points]
while stack:
cur, prev, path = stack.pop()
if len(neighbors[cur]) == 1 or (prev and cur in branch_points):
res.append(path)
continue
for neighbor in neighbors[cur]:
if neighbor != prev:
stack.append((neighbor, cur, path + [neighbor]))
return res
test and output:
print(find_branch_paths([(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (4, 7), (8, 7)]))
# output:
# [[4, 3, 2, 1], [4, 5, 6], [4, 7, 8]]
Hope that helps you, and comment if you have further questions. : )
UPDATE: if there are many branch points, the path will grow exponentially. So if you only want distinct segments, you can end the path when encounter another branch point.
change this line
if len(neighbors[cur]) == 1:
to
if len(neighbors[cur]) == 1 or (prev and cur in branch_points):

Calculating the number of graphs created and the number of vertices in each graph from a list of edges

Given a list of edges such as, edges = [[1,2],[2,3],[3,1],[4,5]]
I need to find how many graphs are created, by this I mean how many groups of components are created by these edges. Then get the number of vertices in the group of components.
However, I am required to be able to handle 10^5 edges, and i am currently having trouble completing the task for large number of edges.
My algorithm is currently getting the list of edges= [[1,2],[2,3],[3,1],[4,5]] and merging each list as set if they have a intersection, this will output a new list that now contains group components such as , graphs = [[1,2,3],[4,5]]
There are two connected components : [1,2,3] are connected and [4,5] are connected as well.
I would like to know if there is a much better way of doing this task.
def mergeList(edges):
sets = [set(x) for x in edges if x]
m = 1
while m:
m = 0
res = []
while sets:
common, r = sets[0], sets[1:]
sets = []
for x in r:
if x.isdisjoint(common):
sets.append(x)
else:
m = 1
common |= x
res.append(common)
sets = res
return sets
I would like to try doing this in a dictionary or something efficient, because this is toooo slow.

A basic iterative graph traversal in Python isn't too bad.
import collections
def connected_components(edges):
# build the graph
neighbors = collections.defaultdict(set)
for u, v in edges:
neighbors[u].add(v)
neighbors[v].add(u)
# traverse the graph
sizes = []
visited = set()
for u in neighbors.keys():
if u in visited:
continue
# visit the component that includes u
size = 0
agenda = {u}
while agenda:
v = agenda.pop()
visited.add(v)
size += 1
agenda.update(neighbors[v] - visited)
sizes.append(size)
return sizes

Do you need to write your own algorithm? networkx already has algorithms for this.
To get the length of each component try
import networkx as nx
G = nx.Graph()
G.add_edges_from([[1,2],[2,3],[3,1],[4,5]])
components = []
for graph in nx.connected_components(G):
components.append([graph, len(graph)])
components
# [[set([1, 2, 3]), 3], [set([4, 5]), 2]]

You could use Disjoint-set data structure:
edges = [[1,2],[2,3],[3,1],[4,5]]
parents = {}
size = {}
def get_ancestor(parents, item):
# Returns ancestor for a given item and compresses path
# Recursion would be easier but might blow stack
stack = []
while True:
parent = parents.setdefault(item, item)
if parent == item:
break
stack.append(item)
item = parent
for item in stack:
parents[item] = parent
return parent
for x, y in edges:
x = get_ancestor(parents, x)
y = get_ancestor(parents, y)
size_x = size.setdefault(x, 1)
size_y = size.setdefault(y, 1)
if size_x < size_y:
parents[x] = y
size[y] += size_x
else:
parents[y] = x
size[x] += size_y
print(sum(1 for k, v in parents.items() if k == v)) # 2
In above parents is a dict where vertices are keys and ancestors are values. If given vertex doesn't have a parent then the value is the vertex itself. For every edge in the list the ancestor of both vertices is set the same. Note that when current ancestor is queried the path is compressed so following queries can be done in O(1) time. This allows the whole algorithm to have O(n) time complexity.
Update
In case components are required instead of just number of them the resulting dict can be iterated to produce it:
from collections import defaultdict
components = defaultdict(list)
for k, v in parents.items():
components[v].append(k)
print(components)
Output:
defaultdict(<type 'list'>, {3: [1, 2, 3], 5: [4, 5]})

BFS in the nodes of a graph

Graph
I am trying to perform BFS on this graph starting from node 16. But my code is giving erroneous output. Can you please help me out. Thanks.
visited_nodes = set()
queue = [16]
pardaught = dict()
exclu = list()
path = set()
for node in queue:
path.add(node)
neighbors = G.neighbors(node)
visited_nodes.add(node)
queue.remove(node)
queue.extend([n for n in neighbors if n not in visited_nodes])
newG = G.subgraph(path)
nx.draw(newG, with_labels=True)
My output is:
Output

The cause of your problem is that you are removing things from (the start of) queue while looping through it. As it loops it steps ahead, but because the element is removed from the start, the list "steps" one in the opposite direction. The net result is that it appears to jump 2 at a time. Here's an example:
integer_list = [1,2,3]
next_int = 4
for integer in integer_list:
print integer
integer_list.remove(integer)
integer_list.append(next_int)
next_int += 1
Produces output
1
3
5

path should be a list, not set since set has no order.
That should work:
visited_nodes = set()
path = []
queue = [16]
while queue:
node = queue.pop(0)
visited_nodes.add(node)
path.append(node)
for neighbor in G.neighbors(node):
if neighbor in visited_nodes:
continue
queue.append(neighbor)