Graph theory problem that requires transitive reduction

Revision en11, by Lance_HAOH, 2017-12-16 05:46:23

Hi. I am trying to solve this problem.

For convenience, I have summarized the problem statement below (based on my understanding):

Given a directed graph with N vertices and E edges (with cycles and not necessarily connected), find the minimum number of edges that we need to retain such that connectivity between vertices is retained as given in the original graph.

Input size: 1 ≤ N, E ≤ 2e5
Time limit: 1s

For example, for the following graph:

graph

We should retain the edges:

0 -> 1
1 -> 2
1 -> 3

So we must use a minimum of 3 edges.

Note that:
0 -> 2 is redundant as we can use the path 0 -> 1 -> 2 to get from 0 to 2.
0 -> 3 is redundant as we can use the path 0 -> 1 -> 3 to get from 0 to 3. (Thanks filippos for catching this mistake!)

This problem seems to be asking for the transitive reduction of a graph which can only be found in at least O(n2) based on what I found on Google and this clearly wouldn't pass the time limit. Furthermore, the list of AC solutions suggests that there is a linear time solution to this problem.

Could someone please advise me on how to solve this problem?

UPD Managed to solve this problem (thank you please_delete_account for your incredible patience and explanation!).

I indeed had a wrong understanding of this problem and overly constrained it.

For readers who are looking for the solution to this problem, I have posted my solution below:

My approach
My accepted code
Tags #graph, #algorithms, #connectivity, #dunjudgeme, #teleportation

History

 
 
 
 
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  Rev. Lang. By When Δ Comment
en11 English Lance_HAOH 2017-12-16 05:46:23 2672 Tiny change: 'ans);\n}\n</spoile' -> 'ans);\n}\n\n</spoile'
en10 English Lance_HAOH 2017-12-14 01:10:01 6 Tiny change: '\n0 -> 1\n0 -> 3\n1 -> 2\n~~~~~\n\' -> '\n0 -> 1\n1 -> 2\n1 -> 3\n~~~~~\n\'
en9 English Lance_HAOH 2017-12-14 01:08:24 16 Tiny change: ' (Thanks [filippos] ' -> ' (Thanks [user:filippos] '
en8 English Lance_HAOH 2017-12-14 01:07:37 160 Tiny change: '\n1 -> 2\n1 -> 3\n~~~~~\n\' -> '\n1 -> 2\n<strike>1 -> 3</strike>\n~~~~~\n\'
en7 English Lance_HAOH 2017-12-13 16:47:46 14 Tiny change: 'me limit. However, the list' -> 'me limit. Furthermore, the list'
en6 English Lance_HAOH 2017-12-13 16:45:44 0 Tiny change: 'olutions suggests that ther' -> 'olutions seems to suggest that ther' (published)
en5 English Lance_HAOH 2017-12-13 16:41:42 289 Tiny change: 'olutions seems to suggest that ther' -> 'olutions suggests that ther'
en4 English Lance_HAOH 2017-12-13 16:34:58 186
en3 English Lance_HAOH 2017-12-13 16:32:15 313
en2 English Lance_HAOH 2017-12-13 16:29:13 137
en1 English Lance_HAOH 2017-12-13 16:27:44 311 Initial revision (saved to drafts)