Tarjan's algorithm to determine wheter a directed graph has a cycle












0












$begingroup$


I want to know if a directed graph has a cycle; something like



1->2->3->2 ...
1->2->3->4->3...
1->1->1->1...


So, I'm considering the Tarjan's strongly connected algorithm.



Do you think that with this algorithm I'll be able to know if some directed graph has a cycle?



Thanks!










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$endgroup$








  • 1




    $begingroup$
    If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
    $endgroup$
    – A.E
    Sep 2 '14 at 20:09










  • $begingroup$
    I want to stop my algorithm when I found the first cycle
    $endgroup$
    – Tomi
    Sep 2 '14 at 20:11










  • $begingroup$
    Tarzan's algorithm. :-)
    $endgroup$
    – Lucian
    Sep 2 '14 at 20:12
















0












$begingroup$


I want to know if a directed graph has a cycle; something like



1->2->3->2 ...
1->2->3->4->3...
1->1->1->1...


So, I'm considering the Tarjan's strongly connected algorithm.



Do you think that with this algorithm I'll be able to know if some directed graph has a cycle?



Thanks!










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
    $endgroup$
    – A.E
    Sep 2 '14 at 20:09










  • $begingroup$
    I want to stop my algorithm when I found the first cycle
    $endgroup$
    – Tomi
    Sep 2 '14 at 20:11










  • $begingroup$
    Tarzan's algorithm. :-)
    $endgroup$
    – Lucian
    Sep 2 '14 at 20:12














0












0








0





$begingroup$


I want to know if a directed graph has a cycle; something like



1->2->3->2 ...
1->2->3->4->3...
1->1->1->1...


So, I'm considering the Tarjan's strongly connected algorithm.



Do you think that with this algorithm I'll be able to know if some directed graph has a cycle?



Thanks!










share|cite|improve this question









$endgroup$




I want to know if a directed graph has a cycle; something like



1->2->3->2 ...
1->2->3->4->3...
1->1->1->1...


So, I'm considering the Tarjan's strongly connected algorithm.



Do you think that with this algorithm I'll be able to know if some directed graph has a cycle?



Thanks!







graph-theory algorithms computer-science






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 2 '14 at 20:06









TomiTomi

4151517




4151517








  • 1




    $begingroup$
    If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
    $endgroup$
    – A.E
    Sep 2 '14 at 20:09










  • $begingroup$
    I want to stop my algorithm when I found the first cycle
    $endgroup$
    – Tomi
    Sep 2 '14 at 20:11










  • $begingroup$
    Tarzan's algorithm. :-)
    $endgroup$
    – Lucian
    Sep 2 '14 at 20:12














  • 1




    $begingroup$
    If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
    $endgroup$
    – A.E
    Sep 2 '14 at 20:09










  • $begingroup$
    I want to stop my algorithm when I found the first cycle
    $endgroup$
    – Tomi
    Sep 2 '14 at 20:11










  • $begingroup$
    Tarzan's algorithm. :-)
    $endgroup$
    – Lucian
    Sep 2 '14 at 20:12








1




1




$begingroup$
If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
$endgroup$
– A.E
Sep 2 '14 at 20:09




$begingroup$
If you just want to detect whether or not it has any cycle, use DFS. If you want to find all cycles, then I think Tarjan's is a good choice. See stackoverflow.com/questions/546655/finding-all-cycles-in-graph
$endgroup$
– A.E
Sep 2 '14 at 20:09












$begingroup$
I want to stop my algorithm when I found the first cycle
$endgroup$
– Tomi
Sep 2 '14 at 20:11




$begingroup$
I want to stop my algorithm when I found the first cycle
$endgroup$
– Tomi
Sep 2 '14 at 20:11












$begingroup$
Tarzan's algorithm. :-)
$endgroup$
– Lucian
Sep 2 '14 at 20:12




$begingroup$
Tarzan's algorithm. :-)
$endgroup$
– Lucian
Sep 2 '14 at 20:12










2 Answers
2






active

oldest

votes


















0












$begingroup$

A slight modification to Tarjan's algorithm will serve you well. Run it as normal, but modify the DFS routine to return True if a node is visited more than once.



Equivalently, return True as soon as you find a strongly connected component with more than one cycle.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
    $endgroup$
    – Fabio Somenzi
    May 19 '17 at 4:30



















0












$begingroup$

Tarjan's algorithm may be used to detect cycles in a directed graph as follows: When the root node of a (maximal) strongly connected component (SCC) is detected, the nodes in the SCC are popped from the stack. At that point, if more than one node is popped, then it is known that the graph contains a cycle. If the SCC contains a single node, one has to check whether that node has a "self-loop" (which gives a cycle) or not (that is, the node makes up a trivial SCC).



Said otherwise, if the directed graph has no cycles, Tarjan's algorithm will only find SCCs consisting of individual nodes without self-loops.





If the cycle needs to be identified and the SCC found has more than one node, one can perform a depth-first search from the SCC root, limited to the nodes that are in that SCC and until the root of the SCC is re-entered.



In fact, in practice, cycle detection in directed graphs is performed by two nested depth-first searches, without concern for the enumeration of SCCs, especially when cycles have to go through some designated nodes.



When the "outer" DFS is about to retreat from a (designated) node $v$, a second DFS is run from $v$ to see if some node $u$ currently on the stack of the outer DFS may be reached from $v$.



If that is the case, we have a cycle reachable from the node from which the first DFS started. Such a cycle goes through both $v$, the node from which the outer DFS is about to retreat, and $u$, the node on the stack of the outer DFS found during the inner DFS. We know we can reach $v$ from $u$ because $u$ is on the stack of the outer DFS. We know we can reach $u$ from $v$ because the inner DFS says so.





The nested DFS algorithm is not asymptotically faster than Tarjan's algorithm, but often performs better.






share|cite|improve this answer









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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    A slight modification to Tarjan's algorithm will serve you well. Run it as normal, but modify the DFS routine to return True if a node is visited more than once.



    Equivalently, return True as soon as you find a strongly connected component with more than one cycle.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
      $endgroup$
      – Fabio Somenzi
      May 19 '17 at 4:30
















    0












    $begingroup$

    A slight modification to Tarjan's algorithm will serve you well. Run it as normal, but modify the DFS routine to return True if a node is visited more than once.



    Equivalently, return True as soon as you find a strongly connected component with more than one cycle.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
      $endgroup$
      – Fabio Somenzi
      May 19 '17 at 4:30














    0












    0








    0





    $begingroup$

    A slight modification to Tarjan's algorithm will serve you well. Run it as normal, but modify the DFS routine to return True if a node is visited more than once.



    Equivalently, return True as soon as you find a strongly connected component with more than one cycle.






    share|cite|improve this answer









    $endgroup$



    A slight modification to Tarjan's algorithm will serve you well. Run it as normal, but modify the DFS routine to return True if a node is visited more than once.



    Equivalently, return True as soon as you find a strongly connected component with more than one cycle.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Sep 2 '14 at 20:24









    A.EA.E

    1,6901722




    1,6901722












    • $begingroup$
      Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
      $endgroup$
      – Fabio Somenzi
      May 19 '17 at 4:30


















    • $begingroup$
      Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
      $endgroup$
      – Fabio Somenzi
      May 19 '17 at 4:30
















    $begingroup$
    Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
    $endgroup$
    – Fabio Somenzi
    May 19 '17 at 4:30




    $begingroup$
    Consider the directed graph with four vertices $a,b,c,d$ and four edges $(a,b), (a,c), (b,d), (c,d)$. The graph has no cycles, but DFS from $a$ visits $d$ twice.
    $endgroup$
    – Fabio Somenzi
    May 19 '17 at 4:30











    0












    $begingroup$

    Tarjan's algorithm may be used to detect cycles in a directed graph as follows: When the root node of a (maximal) strongly connected component (SCC) is detected, the nodes in the SCC are popped from the stack. At that point, if more than one node is popped, then it is known that the graph contains a cycle. If the SCC contains a single node, one has to check whether that node has a "self-loop" (which gives a cycle) or not (that is, the node makes up a trivial SCC).



    Said otherwise, if the directed graph has no cycles, Tarjan's algorithm will only find SCCs consisting of individual nodes without self-loops.





    If the cycle needs to be identified and the SCC found has more than one node, one can perform a depth-first search from the SCC root, limited to the nodes that are in that SCC and until the root of the SCC is re-entered.



    In fact, in practice, cycle detection in directed graphs is performed by two nested depth-first searches, without concern for the enumeration of SCCs, especially when cycles have to go through some designated nodes.



    When the "outer" DFS is about to retreat from a (designated) node $v$, a second DFS is run from $v$ to see if some node $u$ currently on the stack of the outer DFS may be reached from $v$.



    If that is the case, we have a cycle reachable from the node from which the first DFS started. Such a cycle goes through both $v$, the node from which the outer DFS is about to retreat, and $u$, the node on the stack of the outer DFS found during the inner DFS. We know we can reach $v$ from $u$ because $u$ is on the stack of the outer DFS. We know we can reach $u$ from $v$ because the inner DFS says so.





    The nested DFS algorithm is not asymptotically faster than Tarjan's algorithm, but often performs better.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Tarjan's algorithm may be used to detect cycles in a directed graph as follows: When the root node of a (maximal) strongly connected component (SCC) is detected, the nodes in the SCC are popped from the stack. At that point, if more than one node is popped, then it is known that the graph contains a cycle. If the SCC contains a single node, one has to check whether that node has a "self-loop" (which gives a cycle) or not (that is, the node makes up a trivial SCC).



      Said otherwise, if the directed graph has no cycles, Tarjan's algorithm will only find SCCs consisting of individual nodes without self-loops.





      If the cycle needs to be identified and the SCC found has more than one node, one can perform a depth-first search from the SCC root, limited to the nodes that are in that SCC and until the root of the SCC is re-entered.



      In fact, in practice, cycle detection in directed graphs is performed by two nested depth-first searches, without concern for the enumeration of SCCs, especially when cycles have to go through some designated nodes.



      When the "outer" DFS is about to retreat from a (designated) node $v$, a second DFS is run from $v$ to see if some node $u$ currently on the stack of the outer DFS may be reached from $v$.



      If that is the case, we have a cycle reachable from the node from which the first DFS started. Such a cycle goes through both $v$, the node from which the outer DFS is about to retreat, and $u$, the node on the stack of the outer DFS found during the inner DFS. We know we can reach $v$ from $u$ because $u$ is on the stack of the outer DFS. We know we can reach $u$ from $v$ because the inner DFS says so.





      The nested DFS algorithm is not asymptotically faster than Tarjan's algorithm, but often performs better.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Tarjan's algorithm may be used to detect cycles in a directed graph as follows: When the root node of a (maximal) strongly connected component (SCC) is detected, the nodes in the SCC are popped from the stack. At that point, if more than one node is popped, then it is known that the graph contains a cycle. If the SCC contains a single node, one has to check whether that node has a "self-loop" (which gives a cycle) or not (that is, the node makes up a trivial SCC).



        Said otherwise, if the directed graph has no cycles, Tarjan's algorithm will only find SCCs consisting of individual nodes without self-loops.





        If the cycle needs to be identified and the SCC found has more than one node, one can perform a depth-first search from the SCC root, limited to the nodes that are in that SCC and until the root of the SCC is re-entered.



        In fact, in practice, cycle detection in directed graphs is performed by two nested depth-first searches, without concern for the enumeration of SCCs, especially when cycles have to go through some designated nodes.



        When the "outer" DFS is about to retreat from a (designated) node $v$, a second DFS is run from $v$ to see if some node $u$ currently on the stack of the outer DFS may be reached from $v$.



        If that is the case, we have a cycle reachable from the node from which the first DFS started. Such a cycle goes through both $v$, the node from which the outer DFS is about to retreat, and $u$, the node on the stack of the outer DFS found during the inner DFS. We know we can reach $v$ from $u$ because $u$ is on the stack of the outer DFS. We know we can reach $u$ from $v$ because the inner DFS says so.





        The nested DFS algorithm is not asymptotically faster than Tarjan's algorithm, but often performs better.






        share|cite|improve this answer









        $endgroup$



        Tarjan's algorithm may be used to detect cycles in a directed graph as follows: When the root node of a (maximal) strongly connected component (SCC) is detected, the nodes in the SCC are popped from the stack. At that point, if more than one node is popped, then it is known that the graph contains a cycle. If the SCC contains a single node, one has to check whether that node has a "self-loop" (which gives a cycle) or not (that is, the node makes up a trivial SCC).



        Said otherwise, if the directed graph has no cycles, Tarjan's algorithm will only find SCCs consisting of individual nodes without self-loops.





        If the cycle needs to be identified and the SCC found has more than one node, one can perform a depth-first search from the SCC root, limited to the nodes that are in that SCC and until the root of the SCC is re-entered.



        In fact, in practice, cycle detection in directed graphs is performed by two nested depth-first searches, without concern for the enumeration of SCCs, especially when cycles have to go through some designated nodes.



        When the "outer" DFS is about to retreat from a (designated) node $v$, a second DFS is run from $v$ to see if some node $u$ currently on the stack of the outer DFS may be reached from $v$.



        If that is the case, we have a cycle reachable from the node from which the first DFS started. Such a cycle goes through both $v$, the node from which the outer DFS is about to retreat, and $u$, the node on the stack of the outer DFS found during the inner DFS. We know we can reach $v$ from $u$ because $u$ is on the stack of the outer DFS. We know we can reach $u$ from $v$ because the inner DFS says so.





        The nested DFS algorithm is not asymptotically faster than Tarjan's algorithm, but often performs better.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Oct 15 '18 at 17:55









        Fabio SomenziFabio Somenzi

        6,44321321




        6,44321321






























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