SCIENTIA SINICA Informationis, Volume 49, Issue 8: 988-1004(2019) https://doi.org/10.1360/N112018-00125

An efficient incremental strongly connected components algorithm for evolving directed graphs

Xiaofei LIAO1,2,3,4, Yicheng CHEN1,2,3,4, Yu ZHANG1,2,3,4,*, Hai JIN1,2,3,4, Haikun LIU1,2,3,4, Jin ZHAO1,2,3,4
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  • ReceivedMay 16, 2018
  • AcceptedApr 18, 2019
  • PublishedAug 7, 2019


The strongly connected components (SCC) algorithm can contract a directed graph into a directed acyclic graph and is widely used in directed graph analysis applications, such as reachability queries. A variety of SCC algorithms for static directed graphs have been proposed but such algorithms require non-negligible runtime overheads to repeatedly perform computations on an entire graph in response to the frequent changes in the evolving directed graphs that are ubiquitous in the real world. In general, evolving directed graphs are often evolving with minor changes (less than 5%).It allows us to compute SCC in an evolving directed graph on the basis of incremental computations in order to reduce the response time. This paper proposes Inc-SCC, an efficient incremental SCC algorithm for evolving directed graphs, reducing the data access and computation overhead of the algorithm by eliminating unnecessary computations, and using the disjoint feature of SCC for parallel processing to improve the performance of the SCC algorithm. We propose a heuristic optimization method to further speed up the convergence of Inc-SCC. Experiments show that Inc-SCC can be used to enhance the timeliness for evolving directed graphs. When the number of the changed edges of the entire directed graph is 5%, the speedup of Inc-SCC over the existing algorithm is from 2.8 to 12 times. When the number of thechanged edges of an entire directed graph is 0.5%, the speedup of Inc-SCC over the existing algorithm is from 2.9 to 12 times.

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  • Figure 1

    The first type of deleted edges: no change in SCC structure

  • Figure 2

    The second type of deleted edges: recomputation in SCC with structure changes

  • Figure 3

    The second type of deleted edges: no recomputation in SCC with structure changes

  • Figure 4

    The CSR format of an evolving graph

  • Figure 5

    Synchronous iteration

  • Figure 6

    Lock-free solution for data conflicts

  • Figure 7

    Performance curves of Inc-SCC when the ratio of changed edges of the entire graph is different on four different datasets. (a) Email-EuAll; (b) Wiki-Talk; (c) Web-NotreDame; (d) Web-Stanford

  • Figure 8

    Performance curves of Inc-SCC when the ratio of nodes in the largest SCC of the entire graph is different

  • Figure 9

    Performance curves of Inc-SCC with graphs continuing changing on two different datasets. (a) Web-NotreDame; (b) Wiki-Talk

  • Figure 10

    The scalability of Inc-SCC on six different datasets. (a) Email-EuAll;(b) Soc-Epinions1; (c) Web-Berstan; (d) Web-NotreDame; (e) Web-Stanford; (f) Wiki-Talk


    Algorithm 1 Incremental SCC algorithm

    Require:$G_{t_{i}}$, SCCMAP($G_{t_{i-1}}$), increments, $G_{t_{i-1}}$; Output: SCCMAP($G_{t_{i}}$);

    SCCMAP($G_{t_{i}})\Leftarrow$ $\emptyset$;

    $G_{\rm~new}~\Leftarrow$ Preprocess(SCCMAP($G_{t_{i-1}}$), increments, $G_{t_{i-1}}$);

    SCCMAP($G_{t_{i}}$) $\Leftarrow~$ SCCMAP($G_{t_{i}}$) $\cup$ ($G_{\rm~new}$);

    SCCMAP($G_{t_{i}}$) $\Leftarrow~$ SCCMAP($G_{t_{i}}$) $\cup$ LocalColoring($G_{\rm~new}$);

  • Table 1   Graph datasets
    Dataset Nodes Edges SCCs Nodes in the largest SCC
    Soc-epinions1 75888 508837 42185 32223 (42.5%)
    Email-euall 265214 420025 231000 34203 (12.9%)
    Web-stanford 281903 2312497 29919 150527 (53.4%)
    Web-notreDame 325729 1497134 203609 53968 (16.6%)
    Web-Berstan 685230 7600595 109407 334856 (48.9%)
    Wiki-Talk 2394385 5021410 2281879 111881 (4%)

    Algorithm 2 Preprocess

    Require:SCCMAP($G_{t_{i-1}}$); increments; $G_{t_{i-1}}$;


    DAG $\Leftarrow$ Contract(SCCMAP($G_{t_{i-1}}$), $G_{t_{i-1}}$);

    for each edge $e~\in$ increments which is added in parallel

    if ${\rm~SCCMAP}_{e.{\rm~src}}~\neq~{\rm~SCCMAP}_{e.{\rm~dst}}$ then

    DAG $\Leftarrow$ DAG $\cup~e$;

    end if

    end for

    SCCMAP($G_{t_{i-1}}$) $\Leftarrow$ LocalColoring(DAG);


    for each edge $e~\in$ increments which is deleted in parallel

    if ${\rm~SCCMAP}_{e.{\rm~src}}={\rm~SCCMAP}_{e.{\rm~dst}}$ then

    OldSCC $\Leftarrow~{\rm~SCCMAP}_{e.{\rm~src}}$;


    end if

    end for

  • Table 2   Time cost of each part in Inc-SCC when the ratio of changed edges of the entire graph snapshot is 0.5%
    Dataset Preprocess (ms) Process (ms) LocalFBS (ms) LocalColoring (ms)
    Soc-epinions1 1.135 5.708 5.448 0.260
    Email-euall 2.628 4.502 4.261 0.241
    Web-stanford 2.628 4.502 4.261 0.241
    Web-notredame 3.117 11.438 10.380 1.058
    Web-Berstan 6.946 55.542 48.880 6.662
    Wiki-Talk 19.199 68.166 64.032 3.134
  • Table 3   Speedup of Inc-SCC over baseline
    Dataset SCO (ms) Inc-SCC (ms) Speedup
    Soc-epinions1 46.746 5.708 8.190
    Email-euall 28.100 4.502 6.242
    Web-stanford 120.951 35.729 3.385
    Web-notredame 51.147 11.438 4.472
    Web-Berstan 160.622 55.542 2.897
    Wiki-Talk 825.124 68.166 12.104

    Algorithm 3 LocalFBS


    Output:SCCMAP($G_{t_{i}}$); $G_{\rm~new}$;

    SCCMAP($G_{t_{i}}$) $\Leftarrow$ SCCMAP($G_{t_{i-1}}$)$\setminus~G_{\rm~new}$;

    for all OldSCC in $G_{\rm~new}$ in parallel

    Select root $\in$ OldSCC;



    NewSCC(root) $\Leftarrow$ $D~\cap~P$;

    SCCMAP($G_{t_{i}}$) $\Leftarrow~$ SCCMAP($G_{t_{i}}$) $\cup$ NewSCC(root);


    end for


    Algorithm 4 LocalColoring



    while $G_{\rm~new}~\neq~\emptyset$ do

    for all OldSCC $\in~G_{\rm~new}$ in parallel

    Init colors;

    while colors have changed do

    for edge e $\in$ OldSCC in parallel

    if colorse.dst $>$ colorse.src then

    colorse.dst $\Leftarrow$ colorse.src;

    end if

    end for

    end while

    for all vertex root = colorsroot in parallel

    NewSCC(root)$\Leftarrow$ BS(OldSCC, root);

    SCCMAP($G_{t_{i}}$) $\Leftarrow~$ SCCMAP($G_{t_{i}}$) $\cup$ NewSCC(root);


    end for

    end for

    end while

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