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 LIAO 1,2,3,4, Yicheng CHEN 1,2,3,4, Yu ZHANG 1,2,3,4,*, Hai JIN 1,2,3,4, Haikun LIU 1,2,3,4, Jin ZHAO 1,2,3,4
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  • ReceivedMay 16, 2018
  • AcceptedApr 18, 2019
  • PublishedAug 7, 2019


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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