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x0e0e?Rectangle 90 `V ( J Correctness and Completeness Cascading swaps? 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``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDAk1BCBom*[l ^zXf,L$uW}C<̛޷X 'qq nV۾ BD:&g ZލksIME2ᐣcK*CuvtbL'sʧ,J ~6V8^t_n˽ϳ]?|J= HCZA*\}K &%0?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK@B  s *1#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDAk1BCBom*[l ^zXf,L$uW}C<̛޷X 'qq nV۾ BD:&g ZލksIME2ᐣcK*CuvtbL'sʧ,J ~6V8^t_n˽ϳ]?|J= HCZA*\}K &%0?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK ___f3̙;/f9___PPT10i.E:+D=' = @B +"G$ N#F#.#(  8  F`I ?u`SMO&{ 4"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Xfdrs/shapexml.xmlUn7;,Gq! 6⤀j.kmnH,k^S=mZ$y|]َPA5rOj+ƇQ&0]Qe&__IBYG&k{?0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!/հ drs/shapexml.xmlUMo7 x-yU:\; B]w%䢇z=,("noa=7 <ݖZluXFPX8댥p0mPr|z'OF7bt.]#S)<-e))PQ1]el盅EFXR ͯ7w~۟/?ۙ)ܤk/pEIMyf)Cu=N$͏q$W/LF&`|(习GaȓbGvonj&jEJCJcvf0a$YY)`T/Uf|já9+~(!GWV袌W SB nlMcPJv|QBI >ʨƕR\Ys_`9X~ Nrwh.ZҭL执)+s\VeXW`VVOun+ߩjh`BKꡔ 僉eDܒ%yH"x(0LЗ4/ Y^9:Pp$J`v/i,jLf(i1*]^lU-Phvv7OZzy}kQsm'M'ÄxL%u7Q}1 ;h4 ΢^gzzφh+_wo;FmD3Xp1Pї(`cm}`;hP4PK!wdrs/downrev.xmlDAO@F&͘x1BTxC[֝ƃ'o78Y(At)h5Βˋ9fڝNEDXPABIeMy:7+=ܴ.Iƅ;z<F#EU- aAĸPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!/հ *drs/shapexml.xmlPK-!wdrs/downrev.xmlPK] `}K:2___PPT9  p* C    *   x`I 0e0e?Rectangle 2 `}  ( ^Experiments: Summary "&    zI 0e0e?Rectangle 15`  ( DThough Greedy[Halldrsson et al STOC94] also gives near-optimal results for most power law graphs, it cannot scale well on large graphs; and [Zeh02] s result is much worse Our greedy algorithm has high efficiency without losing much effectiveness One-K-Swap and Two-K-Swap improve independent set size to near-optimal, with limited memory cost and acceptable time cost Our algorithms outperform previous external algorithms, both in theory and in practice b "!ge"g4"   @`  ``l# ?Rectangle 4"@ T      ` ?Rectangle 5"@ T      R  ?Text Box 6"@` X( 2     `@h# ?Rectangle 9"@ T      b`i# ?Rectangle 10"@ T      bj# ?Rectangle 11"@ T      bn# ?Rectangle 13"@ T    ,r H:P_j^b  C $ ~T 16"f0  A$??$Full_Colour_ThumbPicture 5Full_Colour_ThumbKJH:P_jBIb*  A??ÞhyPicture 6https://www.cs.helsinki.fi/sites/default/files/pictures/hy.logodomain.com_.gif7H:P!H^b  ^A??Picture 8"@jPJ5 a ___f3̙;/f9___PPT10i.E:+D=' = @B +"` g_ G(  7  F, ?u`SMO&{ 4"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!kfdrs/shapexml.xmlUn7;,Wq! 6꤀b .kmnH,k^S=mR4y|]َPA5r懏j+ƇQ&0]Qe&_]IBYG&k{?0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!)|u  drs/shapexml.xmlUMo7 x Yևe!R4jgwFʒz*zr˹ Koot(ŇPr޼73|d]hR#yt Ĥ9^FYk()h*䓓_=.Gtݨrnd pL2c 𴴗*Ѓ@n (O*\˙e+9_ͬHIPPț>}/_E_>*zl֔]ljkZ?A&<\C-L$;qWߘ@CMl/d~bCa;ڋ;ؙˆgiި_I2:6Csk/!G^À++t^Drx5 3LC mMcPJK7|QLI >ʨƅR\YsU`Hwz=ӇEPT {w'U11V&tk$90E ~2ٶ +vlꩠ|;ukPW5 Lv~7ZKI`;]>XF-Yx "SI}IJu;yY_{+.AX)klӑ%MY ]_%t3&W&ċJ m1V}miٍ@k\ͯo3JvqNd/IPN|t2O[wzx|:IMI霸PKTYVE^y-IWsUw׎脙תoI$:|Ĩ%RY~‘qj'q_%_Ò9maof\iIh(k{~1\K{qKe -+C2HuKf'RPo$F⚁O43/=,=O}!8t]_;%}wO3< PK!R>5drs/downrev.xmlDN0DHH\P Rn+ PmIhl۴ь-VđoAH 6U^'s>QY#{X-//+{2>b|NM}.䧶guB)u'o^jjM|h秆Cf/cWYzsgwx}5<>5drs/downrev.xmlPK ] `}K:2___PPT9  p* C       xB 0e0e?Rectangle 2 `}  ( S Conclusions        xB 0e0e?Rectangle 40V ( @We develop three semi-external algorithms to find near-optimal independent set on massive graphs, all satisfying Low memory cost Low time and I/O complexity Near-optimal in theory and in practice Easy to implement We give non-trivial theoretical guarantees for our proposed algorithms, which proves to be near-optimal Experiments show that our algorithms have better performance and bounds than existing external algorithmsNr Ze Z Zr"e"" @`*p H:P_j^b C " ~T 9"f0  A$??$Full_Colour_ThumbPicture 5Full_Colour_ThumbKJH:P_jBIb*  A??ÞhyPicture 6https://www.cs.helsinki.fi/sites/default/files/pictures/hy.logodomain.com_.gif7H:P!H^b   ^A??Picture 8"@jPJ5 a ___f3̙;/f9___PPT10i.E:+D=' = @B +"  r(  7  FB ?u`SMO&{ 4"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!kfdrs/shapexml.xmlUn7;,Wq! 6꤀b .kmnH,k^S=mR4y|]َPA5r懏j+ƇQ&0]Qe&_]IBYG&k{?0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!@ drs/shapexml.xmlUn7?,K+BAFi"=r-ɕ%=sO=豗o7:]YQEz(9oޛ>~)X+ =:Bajbr9y A˭rXF-Yx &3I}EJM;yYx+خA4IR4װݦ#K+&SJfLJk` 0c򇾻>ݧv7'^nI3|[I0"6^4.pɡr035drs/downrev.xmlDN0DHH\P Rn+ PmIhl۴ь-VđoAH 6U^'s>QY#{X-//+{2>b|NM}.䧶guB)u'o^jjM|h秆Cf/cWYzsgwx}5<>5drs/downrev.xmlPK ] `}K:2___PPT9  p* C       xC 0e0e?Rectangle 2 `}  ( KEND   @   x`C 0e0e?Rectangle 3 ` ( tThank you! Q & A    @`*p H:P_j^b C " ~T 9"f0  A$??$Full_Colour_ThumbPicture 5Full_Colour_ThumbKJH:P_jBIb*  A??ÞhyPicture 6https://www.cs.helsinki.fi/sites/default/files/pictures/hy.logodomain.com_.gif7H:P!H^b   ^A??Picture 8"@jPJ5 a ___f3̙;/f9___PPT10i.E:+D=' = @B +"  (  0   j@I 0e0e? h 1 `}  ( rReal-world dataset downloading      vI 0e0e?Q[`SMO&{ 2 ` ( Our preprocessed datasets are available at http://www.weizhewei.com/ Source datasets url: http://snap.stanford.edu/data/ca-AstroPh.html http://konect.uni-koblenz.de/networks/dblp-author http://konect.uni-koblenz.de/networks/youtube-links http://snap.stanford.edu/data/cit-Patents.html http://snap.stanford.edu/data/com-LiveJournal.html http://citeseerx.ist.psu.edu http://www.uniprot.org http://konect.uni-koblenz.de/networks/facebook-sg http://an.kaist.ac.kr/traces/WWW2010.html http://law.di.unimi.it/webdata/clueweb12/ FZ[""""P+   -  1  3  .  2      1  )  )   D(+DF([H(J(L( N("TP(VrR(tT(V(X( @`y  F`I ?u`SMO&{ 3"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!8[} drs/shapexml.xmlUn1#$)[D+ HH+hmxNjM‰+G3MS"v;c7yiL;_XLdfW f%hkW 30u"{ׯݭJ$jx5%Hܺ-Yrk Qi:N>q5r,ȉ"Kd_ \~۫7_^-i-y;'tUS?98h$>tV]u"%@$|h8'6#PsWn cy.([>&";>z"ݽ;lȢ3R0 1R }h\-];G1Rc41 ,ve"w4N4d071^8l7UF}G4NYR)!.bFpW5U 5\PѩLa-+CWj23pU@ulƺjh`B0zSJbs寉eDܒ%a Q03"x(0aE0g4TpRd:?t|2Ov.4x঑b79W4ӰUӖ ͓Q4c0Wjc,tndڿoZmZw2ĦR<0.4O^uMvEm@/bvZEi_ q"5nL^]{bV0u"\)MM(I1ՎoEqO\D/b (.9)F<ʾ FՍU̽IiݔkT/M5 XyNoIS=ĖƚiX",*"t7[&~RFBv/k_5JuBxl(~9;o+c*z}PK!/drs/downrev.xmlDMO0DHk C u+@ͅC $4^ۦG3z7_UG x60dKo >@IDz&gX../[57R ’:.Zʚwĩ0*mܵzewaƎk*30~y~l Wl~OAE-O_* F $d,A/~PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!8[} *drs/shapexml.xmlPK-!/drs/downrev.xmlPK`WF>___PPT9    2Towards Maximum Independent Sets on Massive Graphs*3 2C C 2       JI ?opGrS`SMO&{ 4"*$PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!> drs/shapexml.xmlU;oG?,5hӄOcSF깻=}\v(RATA ÝTt;"4˝ff<*6º舷8:1i/#1y)HEwxR\v"+Q钕PBhˌUqi/Nh)l_EJ@L!\V6&uP䥊JKwtQQ >ʰƕל]Ysߔ`gvvQOAQݝ.H*Z:[#ͩQ^ɾeH`ZVun+(ğ[h BKXJ"jIvFe0 BrH &KTd/k[\O:PH ¹m<}dQ$ +EH vマawwr*6[ Ӹ]〯?:Fj.)T&0} Fm$9!!)b>/v:]vm=hPK!B%Mdrs/downrev.xmlDN@E&䙸!2!!&n\P5yL{'.pysoYz݊;XlJS!>A@FQk # aZPlTJD!!tYێMi 1J*G׭O1񡦎j.ŏF|O0mv3ۛ~"p!/K~3|Fmv-FK_PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!> *drs/shapexml.xmlPK-!B%Mdrs/downrev.xmlPK] `}WF>___PPT9   *, C C      *p H:P_j^b C " ~T 9"f0  A$??$Full_Colour_ThumbPicture 5Full_Colour_ThumbKJH:P_jBIb*  A??ÞhyPicture 6https://www.cs.helsinki.fi/sites/default/files/pictures/hy.logodomain.com_.gif7H:P!H^b   ^A??Picture 8"@jPJ5 a ___f3̙;/f9___PPT10i.dc5+D=' = @B +"  o(     NI ?Rectangle 7_   |* "       r0e0e?Rectangle 2   (j   ~ I 0e0e?Rectangle 3 0  ( :1.Hello everyone, it s my pleasure to present our work. My name is Yu Liu and I m a PhD student from Renmin University of China. This is a joint work with Jiaheng, Hua, Xiaokui and Zhewei. It s about towards maximum independent sets on massive graphs. 2.Here s the outline of my talk.      3380___PPT10.E ?7@(     p0e0e?{^opGrVP`SMO&{ 1   (    vI 0e0e?Yl`SMO&{ 2 0  ( b1.First I ll give the background of this talk, including the maximum independent set problem, previous internal and external memory algorithms computing independent set and computation models used in our paper; 2.Then 3 semi-external algorithms will be introduced, the greedy algorithm and two swap-based algorithms, followed by experimental evaluation, and finally our conclusion.~~ }     J`I ?opGrS`SMO&{ 3_   |* "     3380___PPT10.l0>  `(     p0e0e?{^opGrVP`SMO&{ 1   (    p@I 0e0e?Yl`SMO&{ 2"`ZPK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!0Y}tdrs/shapexml.xmlWn7;dNW-m"=]ڊKnI"r;P=sߦi_3]d@QÆA̐ WRr&'R(ټ0w|ss{!`r֨TnFv^%(l|RrB>[3[)g JȺ~W&@J%FPYϪ#*^O(T ^Jak?~?~ׯ?7(hf+ߘŤ[|5د>.=i-K |+M^ehPۢEQ_%?:@`ctu;):@'jpJ |yrz9=j;<[M`2ԚВJg L̬Ho7UHRpmgK`=%I*v^ɱ>Rk$Z|kB@nQF\/ ,2OAR3&6^2`_j ~mp0}GDõTEWPK!0drs/downrev.xmlDMO0D"qk* *9Ax٦I=8fFot/|g "3 [4߀FjL| 3BhC2)}ղ&i 1F*Gc^.d-5u&>4~XMZ,(Ӽ֟9=#\O O\+QBXIz~|Nv/FSPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!0Y}t*drs/shapexml.xmlPK-!0Idrs/downrev.xmlPKO 0  ( l1.The first question is, what is an independent set? Given graph G, it s a subset of the vertex set s.t. any 2 vertices in this set have NO edge between in G. A related notion is the maximal independent set shown in left figure, which means adding any more vertex into the set will make it not independent; the maximum independent set is the one with largest set size. 2.Punch line: These are the 2 types of independent set we care about.     J I ?opGrS`SMO&{ 3_   |* "     3380___PPT10.mf}a umU(     p0e0e?{^opGrVP`SMO&{ 1   (V   vI 0e0e?Yl`SMO&{ 2 0  ( 1.Computing maximum independent set has been proved NP-hard. Moreover, it s approximable hard, which means there s NO polynomial algorithm having constant approximation ratio. 2.But in practice, we DO care about a near-optimal independent set, for it s a interesting problem of graph theory, and many real-world applications favor a large independent set, such as labeling-based shortest path and distance query, and map labeling problem. 3.Let s first see if the existing algorithms work well on massive graphs.BLec Re      J+ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.нvi0 (     p0e0e?{^opGrVP`SMO&{ 1   (   v + 0e0e?Yl`SMO&{ 2 0  ( 1.There re a bunch of existing works on independent set, including exact algorithms, approximation algorithms such as the greedy algorithm, and various heuristic algorithms. They are all in-memory algorithms. 2.There re only few works on external memory, such as Zeh s work, it finds a maximal independent set but can be very worse than optimal.ZZ Z    J@+ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.ms b   (     p0e0e?{^opGrVP`SMO&{ 1   (   v + 0e0e?Yl`SMO&{ 2 0  ( .1.Here s a comparison of the state of art algorithms for independent set problem. The first algorithm has minimum time complexity to date finding the maximum independent set, and the second has best approximation ratio among polynomial algorithms. They are both impractical on large graphs. The in-memory greedy algorithm has acceptable CPU time complexity and approximation ratio, and perform well in practice. We ll see later why it cannot be extended to external setting. The last one is a external memory algorithm, but has NO acceptable theoretical gurantee. 2.Punch line: So previous algorithms either do not scale well, or have NO theoretical guarantee, they are all unacceptable for massive graphs. So, it s a really hard problem?       J@+ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.l; (     p0e0e?{^opGrVP`SMO&{ 1   (   v`. 0e0e?Yl`SMO&{ 2 0  ( v1& which means there exist some worst case graph instances. << ;     J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.@57 (     p0e0e?{^opGrVP`SMO&{ 1   (   v`. 0e0e?Yl`SMO&{ 2 0  ( r1.Since the randomness and some properties of the real-world graph prohibit the worst case instance in most cases.ss r     J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.Rg  0(     p0e0e?{^opGrVP`SMO&{ 1   (   v`. 0e0e?Yl`SMO&{ 2 0  ( .1.The number of vertices with degree k is in inverse proportion to k to the power of beta. If we plot the degree vs. number of vertices in log-log scale, it would be a straight line. 2.I ll briefly introduce the model we used.      J . ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2O P(     p0e0e?{^opGrVP`SMO&{ 1   (   v` . 0e0e?Yl`SMO&{ 2 0  ( 1.The model has 2 parameters, alpha and beta, where alpha controls the graph size and beta controls the skewness of degree distribution.       J . ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2F p(     p0e0e?{^opGrVP`SMO&{ 1   (   v` . 0e0e?Yl`SMO&{ 2 0  ( 1.To generate a graph by ACL model given alpha and beta, we first compute the number of vertices of degree i, for i from 1 to maximum degree.    J . ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2 rjR(     p0e0e?{^opGrVP`SMO&{ 1   (S   v I 0e0e?Yl`SMO&{ 2 0  ( 91.Then for a vertex of degree i, generate i copies of it.:: :   J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2 bZB(     p0e0e?{^opGrVP`SMO&{ 1   (C   v@. 0e0e?Yl`SMO&{ 2 0  ( )1.Then do random matching between copies.** *   JM ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2  j(     p0e0e?{^opGrVP`SMO&{ 1   (k   v. 0e0e?Yl`SMO&{ 2 0  ( Q1.At last, merge copies into vertices, 2 vertices have edge if their copies have.RR R   J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2 (     p0e0e?{^opGrVP`SMO&{ 1   (   v. 0e0e?Yl`SMO&{ 2 0  ( #1.The ACL model has many advantages comparing to other models. 2.Under power law random graph model, the independent set problem has better performance ratio comparing to arbitrary graph.      J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2 -% (     p0e0e?{^opGrVP`SMO&{ 1   (   v. 0e0e?Yl`SMO&{ 2 0  ( P1.For arbitrary graph, the best polynomial and trivial maximal independent set may have performance arbitrary worse than optimal. For power law graph, there still exists a significant gap between the trivial mis and optimal independent set, but narrower than arbitrary graph, and we can design efficient polynomial algorithms very close to optimal. 2.However, real-world graphs may go very large. Wwhen the graph is too large to fit into memory, we need external memory computation model.      J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.p 2 og0O(     p0e0e?{^opGrVP`SMO&{ 1   (P   v. 0e0e?Yl`SMO&{ 2 0  ( *1.In external computation model, data transfer between memory and disk is by block, and random access is I/O-expensive. So we use sequential scan and external memory sort instead. 2.For graph problems, the external memory model requires memory usage much smaller than graph size, whereas semi-external model permit memory consumption being a small constant times the size of vertex set. 3.And our goal is&       J. ?opGrS`SMO&{ 3_   |* "     3380___PPT10.  wP_(     p0e0e?{^opGrVP`SMO&{ 1   (`   v / 0e0e?Yl`SMO&{ 2 0  ( :1.We compute independent set in semi-external settings, with constant being sufficiently small, i.e., 2 or 3. 2.The algorithm should have low CPU time and I/O complexity, find near-optimal independent set and have non-trivial theoretical bounds. 3.So I ll begin talking our algorithms.      J/ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.jk ps(     p0e0e?{^opGrVP`SMO&{ 1   (t   v / 0e0e?Yl`SMO&{ 2 0  ( N1.First we look at the in-memory greedy algorithm because it s a very efficient and effective algorithm in practical, we ll see why it cannot be extended to external memory. 2.The reason is, when a vertex is selected, we remove it and its neighbors from the graph, and update degree of neighbors of these vertices. So next time the vertex of smallest degree can be anywhere in the file no matter how the file is sorted. So random accesses are inevitable. In worst case, there can be O(|V|) random accesses, that s impractical for external algorithms.(( '     J@/ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.q@*/ 5-(     p0e0e?{^opGrVP`SMO&{ 1   (   v . 0e0e?Yl`SMO&{ 2 0  ( X1.Then I ll give our semi-external greedy algorithm. The difference is we do not update degree of any vertex, but only sort adjacency lists of vertices by degree in advance. 2.Clearly we can see all vertices with State equal to IS form an independent set.     J@/ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.q@*/  p(       p0e0e?{^opGrVP`SMO&{ 1   (q    v/ 0e0e?Yl`SMO&{ 2 0  ( K1.The I/O complexity of greedy algorithm is dominated by the sorting. And since we only sort |V| adjacency lists of different length, the complexity can be improved to the following. 2.Since each vertex only uses 1 bit to distinguish IS or NIS, the algorithm use |V| bits, or |V|/L memory units given each memory unit has length L.LL K      J/ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.q@*/ q 3 +  (     p0e0e?{^opGrVP`SMO&{ 1   (   v/ 0e0e?Yl`SMO&{ 2 0  ( V1.We managed to compute an expected independent set size by semi-external greedy algorithm, see this formula. 2.Basically, the idea is we compute the probability of a vertex with degree I links all its edges to higher degree vertices. So this vertex is in independent set by greedy. Then we get the expected independent set size. In most cases it s a lower bound. 3.This table shows the expected performance ratio of greedy algorithm. It s computed by the lower bound over an upper bound of maximum independent set, using a modified version of the algorithm in this. Later we ll give the experimental evaluation. 4.However, real-world graphs may not perfectly described by ACL model, so we design 2 swap-based algorithms to improve the performance of greedy.      J/ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.siv QI1(     p0e0e?{^opGrVP`SMO&{ 1   (2   v AB 0e0e?Yl`SMO&{ 2 0  ( t 1.The first algorithm is called One-K-Swap, and it s inspired by this work. Intuitively, if & 2.We call v the IS neighbor of a and b.       JAB ?opGrS`SMO&{ 3_   |* "     3380___PPT10.`uo (     p0e0e?{^opGrVP`SMO&{ 1   (   v DB 0e0e?Yl`SMO&{ 2 0  ( 41.The algorithm needs rounds of iteration and in each round we need 3 sequential scans. & 2.Actually 1st scan and 3rd scan can be merged. This is because the maximum degree of an IS vertex should be much smaller than the block size, so its adjacency list can be put into main memory and easily read again. 3.However, to make the idea of One-K-Swap work in external setting, we design data structure like this: Z@Z   "* "*$""     JDB ?opGrS`SMO&{ 3_   |* "     3380___PPT10.gk<w 0(     p0e0e?{^opGrVP`SMO&{ 1   (   vFB 0e0e?Yl`SMO&{ 2 0  ( %1.We store an ISN array in memory of length |V|. For 1-k-swap candidate, it stores the IS neighbor; for IS vertex it stores number of adjacency candidates. So the memory cost is |V| units.       JGB ?opGrS`SMO&{ 3_   |* "     3380___PPT10. {sP [(       p0e0e?{^opGrVP`SMO&{ 1   (\    v`IB 0e0e?Yl`SMO&{ 2 0  ( 61.We also have to solve some questions, like Q1, Q2 and Q3. 2.I ll take Question 1 as example. Once reading a 1-k-swap candidate, we can find how many candidates having same IS neighbors in its adjacency list. If this value is smaller than the number stored by its IS neighbor, it means that at least 1 other candidate is independent to this vertex. So if the vertex has no adjacent IS vertex except its ISN, it can do 1-k-swap; or else we should update its IS neigbor s info(ISN decrease by 1) 3.Then we see the performance of One-K-Swap.       JJB ?opGrS`SMO&{ 3_   |* "     3380___PPT10.@ay p$(  $ $  p0e0e?{^opGrVP`SMO&{ 1   ( $  v`LB 0e0e?Yl`SMO&{ 2 0  ( 1.We may care about how many rounds it will take since this affects the complexity of the algorithm. Theoretically, the can be cascading swaps, means& 2.But in practice, as we discussed before, power law and other real-world graph properties almost prevent this from happening. Usually 2 rounds of iteration leave the un-done 1-k-swaps less than 2-3%, so the algorithm can stop to get a trade-off between efficiency and effectiveness. 3.So the I/O complexity is&      $ JMB ?opGrS`SMO&{ 3_   |* "     3380___PPT10.;  y(a(  ( (  p0e0e?{^opGrVP`SMO&{ 1   (b (  v/ 0e0e?Yl`SMO&{ 2 0  ( 1.We only give some intuition on how large the independent set can be increased by One-K-Swap. This formula shows the expected increments of 1st round after greedy algorithm. We analyze the degree relationship between 1-k-swap candidates and there is neighbors, and prove they are both very small w.h.p. 2.For the One-K-Swap based on result of random algorithm, it can have significant amount of swaps. The intuition is that, if unfortunately we include some vertex of large degree into independent set, then it s very possible it has 2 independent 1-k-swap candidates, so a swap is performed. The final result is similar to a greedy algorithm s. @ ( "     ( J@J ?opGrS`SMO&{ 3_   |* "     3380___PPT10.;z %,(  , ,  p0e0e?{^opGrVP`SMO&{ 1   ( ,  vBJ 0e0e?Yl`SMO&{ 2 0  ( H1.Due to the time limit, I ll just give some high-level ideas of Two-K-Swap and don t go into details. 2.Firstly as in One-K-Swap, we find all swap skeletons, or patterns of minimal subgraph can do 2-k-swap. They have the following 3 types.     , JCJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.vAO$| 0(  0 0  p0e0e?{^opGrVP`SMO&{ 1   ( 0  vEJ 0e0e?Yl`SMO&{ 2 0  ( _2.Then we need to solve some difficulties in designing a semi-external swap algorithm, such as:`` _    0 JFJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.`roS *"4 (  4 4  p0e0e?{^opGrVP`SMO&{ 1   (  4  vHJ 0e0e?Yl`SMO&{ 2 0  ( M3.And the data structure is a little more complex than One-K-Swap, and needs more spaces. 4.We store 1-k-swap and 2-k-swap candidates in ISN, labels and conflict graph as hash map with limited size, and conflict graph is elastic.     4 JIJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.w{ OG8/(  8 8  p0e0e?{^opGrVP`SMO&{ 1   (0 8  vKJ 0e0e?Yl`SMO&{ 2 0  ( r 1.Finally I ll show the performance analysis. Like One-K-Swap, in practice, 2 rounds of iteration leave the un-done 1-k-swaps and 2-k-swaps less than 2-3%. So the I/O complexity can be seen as& 2.Since Two-K-Swap detect all possible 1-k-swaps, in expectation,&      8 JLJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.w@,-    0< (  < <  p0e0e?{^opGrVP`SMO&{ 1   ( <  vNJ 0e0e?Yl`SMO&{ 2 0  ( h1.Experiments are conducted to evaluate our algorithms. First we generate synthetic power law graphs by ACL model, and vary beta, to see how our 3 algorithms perform. 2.This figure shows they are all near-perfect, and the approximation ratio is computed as dividing the independent set size of various algorithms by the upper bound we mentioned before. 3.Since Greedy is already very good, swap algorithms have only a little improvements. 4.The table shows the expected performance ratio(we showed it before), we can see when conducting sufficient many tests it s a lower bound, and the bound is quite tight. 5.Actually the independent set computed by various algorithms on power law graphs are very concentrated at some expected value, for this experiment we never see a time the result falls under the estimated value.55 4    < JPJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.zKv# 2*P@(  @ @  p0e0e?{^opGrVP`SMO&{ 1   ( @  vQJ 0e0e?Yl`SMO&{ 2 0  ( U2.Further it needs to test our algorithms on real-world datasets. These are the datasets we used, varied in size and average degree. 3.For example, Youtube is the social network of Youtube users and their connections; Patent is maintained by the National Bureau of Economic Research. This citation graph includes all citations made by patents granted between 1975 and 1999. Twitter10 contains a social network crawled from Twitter.com in July 2009. ClueWeb12 is the web graph crawled in 2012.     @ JSJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.z< pDx(  D D  p0e0e?{^opGrVP`SMO&{ 1   (y D  vTJ 0e0e?Yl`SMO&{ 2 0  ( S1.First we must prove our algorithms are practical, i.e., have acceptable time and memory cost. 2.For time cost, we can see the semi-greedy is fast, and 2 swap algorithms also have acceptable time cost. For large graphs, in fact this time is determined by how fast the disk speed is. 3.As a comparison, we can see the in-memory greedy algorithm is fast only when graphs can be fitted into memory, if not it can not finish in tens of hours due to large amount of random accesses. The existing external memory algorithm is also fast on all datasets, in compromise with the size of independent set.TT S    D JVJ ?opGrS`SMO&{ 3_   |* "     3380___PPT10.{`\  H(  H H  p0e0e?{^opGrVP`SMO&{ 1   ( H  vMB 0e0e?Yl`SMO&{ 2 0  ( 61.The memory cost has similar conclusions. See our greedy only use very limited memory, and memory cost of 2 swap-based algorithms are acceptable considering the whole graph size, 2.The in-memory greedy cost a lot of memory because it needs to read the whole graph, and implementation of data structures like Fibonacci heap has extra cost. 3.The previous external algorithm only use very small memory and it can be set. We set it in proportion to the graph size.     H J % ?opGrS`SMO&{ 3_   |* "     3380___PPT10.{`\2 xL`(  L L  p0e0e?{^opGrVP`SMO&{ 1   (a L  v% 0e0e?Yl`SMO&{ 2 0  ( I1.Most importantly, we care about how large a independent set these algorithms can get. Numbers in bold are the best, others shows their first significant bits different from the best. 2.In-memory greedy is always good on small graphs; but cannot work on large graphs; 3.Two-K-Swap after greedy is also nearly the best on all datasets. All One-K-Swap and Two-K-Swap algorithms have results close to the best in comparison. 4.Our greedy is only a little worse than in-memory greedy 4.In almost all datasets, random algorithm has a gap with other algorithms, sometimes can be very worse.JJ J   L J% ?opGrS`SMO&{ 3_   |* "     3380___PPT10.zg  P(  P P  p0e0e?{^opGrVP`SMO&{ 1   ( P  v% 0e0e?Yl`SMO&{ 2 0  ( .1.We also compute the exact maximum independent set or its upperbound.(red-independence number; green-upperbound) 2.Many datasets, our algorithms have results very close to optimal or upper bound. 3.For some large graphs cannot get the exact optimal value, and with large average degree, the upper bound may not be tight.(Concluded from smaller graphs) e.g., Astroph 24,707 DBLP 317,629 -> 262,087 Youtube 889,375 8u  4    P J% ?opGrS`SMO&{ 3_   |* "     3380___PPT10.z{r\`*0isv8@djn%ryAx&3/U~Qr _ D  L Jێ•T"*6MH*I PVmfjntn~ڐ(" [RYt1ITJS, "(_.g4c9V(Oh+'0 px   D P \ ht|XOn Boosting Holism in XML Twig Pattern Matching Using 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,   $ , ȫʾ(4:3)$National Univeristy of Singapore+p#+& 4Arial Garamond WingdingsCalibriComic Sans MSTimes New RomanEdge1_Edge3Towards Maximum Independent Sets on Massive GraphsOutline0Background: The Maximum Independent Set Problem0Background: The Maximum Independent Set Problem>Background: Previous Internal and External Memory Algorithms>Background: Previous Internal and External Memory Algorithms)Background: Problem on Real-world Graphs)Background: Problem on Real-world Graphs)Background: Problem on Real-world Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs&Background: Models for Massive Graphs'Background: External Computation Model*Background: Our Goals(Problem Definition)$Our Semi-external Algorithm: Greedy$Our Semi-external Algorithm: Greedy$Our Semi-external Algorithm: Greedy$Our Semi-external Algorithm: Greedy(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: One-K-Swap(Our Semi-external Algorithm: Two-K-Swap(Our Semi-external Algorithm: Two-K-Swap(Our Semi-external Algorithm: Two-K-Swap(Our Semi-external Algorithm: Two-K-Swap!Experiments: On Synthetic GraphsExperiments: Datasets5Experiments: Various algorithms on Real-world graphs5Experiments: Various algorithms on Real-world graphs5Experiments: Various algorithms on Real-world graphs5Experiments: Various algorithms on Real-world graphsExperiments: Summary ConclusionsENDReal-world dataset downloading  õ õƬ+ (PX  _PID_HLINKSVersionA http://www.weizhewei.com/.http://snap.stanford.edu/data/ca-AstroPh.html2http://konect.uni-koblenz.de/networks/dblp-author4http://konect.uni-koblenz.de/networks/youtube-links/http://snap.stanford.edu/data/cit-Patents.html3http://snap.stanford.edu/data/com-LiveJournal.htmlhttp://citeseerx.ist.psu.edu/http://www.uniprot.org/2http://konect.uni-koblenz.de/networks/facebook-sg*http://an.kaist.ac.kr/traces/WWW2010.html*http://law.di.unimi.it/webdata/clueweb12/http://www.weizhewei.com/.http://snap.stanford.edu/data/ca-AstroPh.html2http://konect.uni-koblenz.de/networks/dblp-author4http://konect.uni-koblenz.de/networks/youtube-links/http://snap.stanford.edu/data/cit-Patents.html3http://snap.stanford.edu/data/com-LiveJournal.htmlhttp://citeseerx.ist.psu.edu/http://www.uniprot.org/2http://konect.uni-koblenz.de/networks/facebook-sg*http://an.kaist.ac.kr/traces/WWW2010.html*http://law.di.unimi.it/webdata/clueweb12/&_^;(dokirabbitholedokirabbithole  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