# how to reduce the time complexity?

I have a graph G=(V+E) and list of list Node where each sublist is subset of V. I want to find out nodes of each sublist that are not reachable from other nodes of that sublist and add edge between those nodes. Suppose,Node=[[1,2,3,4,5,7,8],[9,10],....] and in sublist=[1,2,3,4,5,7,8], {1,7,3,8} reachable from each other and {2,4,5} reachable from each other so an edge needs to be added between 1/7/3/8 and 2/4/5. I have used the following :

for each sublist in Node
SG=node_induced_subgraph(sublist) of G
C=connected_components(SG)
for each component c_i in C
add edge between c_i and c_{i+1}
end
end


complexity of node_induced_subgraph() is O(V+E) complexity of connected_components() is O(m+n) # m is no. of nodes and n is no. of edges in Sub-graph

So how to reduce the overall complexity?

• Can you edit your question to state the problem more clearly? I don't know what is meant by "list of list Node" or "inner list" or "nodes of inner list that are not reachable from other nodes" (which other nodes?). Where did you encounter this task? – D.W. Jul 7 at 16:55
• list of list is nested list like array of arrays @D.W. – rhl Jul 7 at 17:05
• That's not my confusion. I still remain confused after your edit. Several of my comments/questions remained unaddressed. I hope you'll make more substantial changes to explain the problem statement in more detail. – D.W. Jul 7 at 17:21
• In the pseudocode it looks like you want the nodes in the sublists to be connected passing only through nodes from the same sublist. Is that what the original problem wants to accomplish? The description could also be understood as making the nodes in each sublist reachable from each other, but on the given graph $G$. – plop Jul 7 at 18:40

I'm gonna address your question in two parts. First how to ensure reachability between nodes in one "sublist" with adding a minimal amount of edges (if you can add as many edges as you wanted you could go through the list and connect one element to its two neighboring elements - costing $$O(n)$$ edge addition operations, which is theoretically the fastest method...). If we want to add as few edges as possible we only want to connect the "connected components". I suppose your graph $$G$$ is stored as an adjacency list.

//reused arrays
visited := bool array of size |V| initialized to false
active  := bool array of size |V| initialized to false
//simple dfs
def dfs(node):
if visited[node] or not active[node]: return
visited[node] = true
for child in neighbors(node):
dfs(child)
//algo
def connect_node_list(list):
for node in list: active[node] = true
last := null
for node in list:
if not visited[node]:
dfs(node)
last = node
for node in list:
active[node]  = false
visited[node] = false


Now this algo has a runtime of $$O(n+|E|)$$ where $$n$$ is the length of list. Well to be honest it's the same as your algo I just want to highlight that the usage of bool arrays is very desirable in general; also I don't know how your "indcued subgraph" method works - but you should avoid building a new graph.

Now an idea that might be new: you can reduce the number of sublists by connecting those that have an overlapping element. (Of course this is only useful if they aren't necessarily distinct.) You can use a union-find-structure for that:

uf := union find structure with |V| elements
for list in sublists:
first = list[0]
for node in list:
uf.unify(first, node)
index := array of length |V|
counter := 0
for element in uf:
index[element]  = -1
if uf.parent(element) = element and uf.size(element) > 1:
index[element] = counter
counter++
new_sublists = list of <counter> many lists
for element in uf:
set = uf.find(element)
if index[set] != -1:

Now ´new_sublists´ only contains the necessary amount of sublists (also all sublists of size 1 are removed). The procedure takes $$O(max(|V|,N\cdot\alpha (|V|)))$$ where $$N=\sum_i \text{|sublists[i]|}$$. $$\alpha(|V|)$$ can basically be treated as a constant but you can read up on the union-find-structure article. If you are working with many overlapping node lists this algo should be executed first.