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˜uCUDA¿1Æ·»{ p∗ O¯˘E˘§E˘ ` 'J4«CUDA¢y¿1Æ·»{" §'O˜uDijkstra! Bellman-Ford! ∆- Stepping!Sparse Matrix-Vector Bellman-Ford"'˜kØu†;DijkstraBellman-Ford'O?1 ¿1zU?§Øc5UZ∆-Stepping¿1{¢yCUDA†U?§J ˜uSparse Matrix-VectorBellman-Ford§'O¢yCSRELLCUDA{" 3¢'§ CUDA{c5U‘Boost¥¥«Æ·»{?1’§y†'{3m mkr¿§3Œ5U‘‡†w" ’c Æ·»§Dijkstra§Bellman-Ford, ∆-Stepping§Sparse Matrix-Vector, CSR Bellman-Ford, ELL Bellman-Ford, CUDA§GPU§¿1?§ 1 {0 Cc5§¿1O•u—"NVIDIA!Microsoft!Intelœi'O gC¿1O†" 32011c§A⁄kO¯))l! )P! “ ¯!O¯^ı?n"CUˆ¯⁄ı?n#¿ |"~X§NVIDIA32010cØ£˜†Tegra 2V?n" D¥?nŒ?n6Y’§ª/?n£GPU⁄¿1O3C Ac⁄#Vg" @ˇGPU?§4¡§§SU–«n/ “\Œ§,§S\S“kX" GPUˆ{?n2:Œ " N`4¡" §SI˘SOpenGL‰DirectX" ˇd§@ˇGPUO¿ 61" 32006c§NVIDIAœ1–˜uCUDAe£Compute Uniﬁed Device Archi- tecture⁄GPU))GeForce 8800 GTX" CUDAeGPUOO«# ‹§ƒ U 3 ˇ ^ O ¥ B" T e ƒGPUU ) ß E , O ﬂ K§§ „CUDA-8e£ISA⁄–9GPUS¿1O" mu

g2007c–5§–CUDA C˜:A^§S§…O4J," ~ X˘ª! O6N˜˘! ‚˘+–ˇLGPU¿1O5 UJ,"'ØuÆ·»ﬂK§J˜uCUDA†¿1{§ØuG1 {§3?J," '{e'|Xe" 12o(L’uÆ·»§’ «{‘" 130˜uDijkstra{CUDA" 140Bellman- FordCUDA¿1{" 315¥§£ª∆-Stepping{CUDAU?" 16J ˜uSparse Matrix-VectorBellman-Ford¿1{" 3¢'§’ ª4«¿1{Boost¥¥G1¿1Æ·»{5U" 2 ’ kªG = (V, E)§¥|V | = n§|E| = m" 4s ∈ V L!:" z^ >e ∈ EDK›§^›…ŒL«c : E → R"3øp§•‰´^· »›·»¥⁄k>›" Øu:Æ·»ﬂK·Ol:s8I !:v›^·»" ıŒÆ·»{ocsÆ·»Œ|d§d(v)Llsv›§ ¿{zg?1t£relax⁄#d" {m'§d(s) ← 0, d(v) ← ∞" zg t^>e(v, w) ∈ E§rd(w)d(w)d(v) + c(v, w) ¥§vk!:I #§{(" XJsv§K{(d(v) = ∞" ˜ud#“§ı ŒÆ·»{'«label-settinglabel-correcting"Label-setting{£~X Dijkstra⁄zgØÆ·»ﬁ†(‰!:v’>?1t§⁄–Label-settingI tm^>"Label-correcting{£~XBellman-Ford⁄zgØu(‰Æ·» !:v‹?1t§⁄–L§UtLm^>" Cherkassky [32]o(c«Æ·»{" ¥§†;: Æ·»{)Dijkstra [1, 14, 19]! Bellman-Ford [5, 14, 23]! –9˜uØu“…Œ A [6, 15, 7]" ˜uø†;{U?„ı" ~XØuDijkstraU? )˜uMin-Max Heap [18]! Pairing Heap [20]! Fibonacci Heap [21]{" 5 ‘ø{J‘kŁ£Q⁄¥s·»Æ!:vmE,O(logn)§u ·{NmE,O(nlogn + m)" ,§ [24] [25]'OØBellman-FordJ U ? {" d uA { ·Dijkstra z § h(x) = 0(Ø u “ … Œ h(x) ≤ d(x, y) + h(y))§AzDijkstra{" AA^Ø2§~XiZ ¥ˇ·{§IS˘" A{’3uØu“…Œh(x)O" 3ﬂK¥§!:m›§vkı&E§Øu“…ŒJO§⁄ –A••ﬂK" Cc§Æ·»¿1{§~XCrauser [30]!Eager [31]!∆-Stepping [17] J" ¥§Crauser5U’›‰§Io3‘kŁ§Ømƒ§

Øuz!:o›" EagerKI\lookaheadºŒ§^5zg ˇØl3lookaheadS!:" lookahead‹{¿15§KH{g Œ‹\¿Nı" ØulookaheadJK·ˇª" ⁄–Eager {˙6ulookahead J§¿›‰" ∆-Stepping·¿1Æ·»{¥˙ p" §ºŒ∆§3{1igt§ˇØd›3i × ∆(i + 1) × ∆ m!:t" [29] [27, 8]gØ∆-Stepping{?1¢y" c·˜u9th DIMACSCray MTA-2.¯AO‘zŁ§ˆ{3˚ˇ?1t" —'d(s) ← 0§d(v) ← ∞§¥v = s" { ı1n − 1t–Ø⁄k!:vsÆ·»§mE,O(n2)" Algorithm 1 CUDA Dijkstra (a) Initial: foreach v ∈ V do d[v] ← ∞ end d[s] ← 0 Q ← V while Q = φ do (b) ExtractMin: u ← {v : v ∈ Q ∧ ∀w ∈ Q, d(v) ≤ d(w)} if d[u] = ∞ then break remove u from Q (c) Relax: foreach (u, v) ∈ E do if d[u] + c(u, v) < d[v] then d[v] ← d[u] + c(u, v) end end Algorithm1CUDA Dijkstra{e"¥§Initial!ExtractMin!Relax

U¿11" InitialmE,O(n/t)§ExtractMinO(n2/t)§RelaxO(m/t)§ ¥tL«…Œ¥§(thread)oŒ"⁄–{mE,O(n/t+n2/t+m/t) = O(n2/t)" 3.2 Reduction 3¿1?§¥§Reduction [13, 10]·Nı¿1˜" ØuExtractMin§• –^Reduction5\" Reduction˜g·Œ–'£{? n§zg~5§˛(^{(" n§t ≥ n§ ExtractMin mE,O(logn)§duŒmˇ&ﬂK§¢S$ œ" 3 G 1 Dijkstra { ¥ExtractMinI H { Œ | g§ m E , O(n)" Ø uDijkstra O(logn) U ? { [18, 20, 21] · ˜ u ‘ k Ł ¢ yExtractMin § ‘ k Ł Œ ( ¿ • ¿ 1 {§ ⁄ – • ^ Reduction { ¢ yExtractMin" ‘:3u?§N·§ØuDijkstra'{Œ(U?= "o⁄$§Im?1‘kŁ;"JmE,U 3O(logn)" 3 [10]¥07«Reduction{§ØReduction?/?1«‘z" •J—«§¿ØThreadsPerBlock(TPB)ºŒ?15U’" ^Thrust¥ [11]¥Reduction! CPULoopHeap?1’" lª1–w§ 3Œ5107§256! 512! 1024TPB’Loop(-O2‘z) ﬂ§ﬂ512TPB\ ’1.65" Thrust\’·6.07" †‘zLoop$k56790nsı" ReductionØuHeap·œ§ø·du§mˇ&§ŒD·ﬂKE ⁄" –*§Œ5§Loop‘‡§Œ5CReduction‘ ‡§⁄–¿1?§•Œ" 1 10 102 128(TPB) 40.82 35.92 39.63 256(TPB) 48.56 57.37 43.96 512(TPB) 42.64 47.32 41.2 1024(TPB) 43.81 43.76 45.52 103 43.12 43.05 41.55 49.77 104 59.4 50.49 45.97 46.74 105 106 107 145.4 1033.26 8033.73 112.43 556.43 4839.28 92.04 470.487 4075.55 108.75 734.59 6475.09 Thrust 324.8 65.97 73.04 513.12 453.91 450.91 Loop Loop(-O2) Heap 0 0 0 0 0 0 0 0 0 0 0 0 150 630 0 0 0 0 498.5 5620 630 1105.1 56790 6710 0.124 0.1311 L 1: ExtractMin Time(ns)"Œ5l1107§?11000gExtractMin†"

4 CUDA Bellman-Ford 4.1 Bellman-Ford{ Bellman-Ford{uDijkstra§§%g·Øª¥z^>ØA!:?1 t§vk!:I#§Æulabel-correcting{"–y†ıtn − 1 {(§mE,O(nm)" Algorithm 2 CUDA Bellman-Ford (a) Initial: foreach v ∈ V do d[v] ← ∞ end d[s] ← 0 for i ← 1 to |V | − 1 do (b) Relax: foreach (u, v) ∈ E do if d[u] + c(u, v) < d[v] then d[v] ← d[u] + c(u, v) end end Algorithm2{e§InitialRelax'Dijkstra" § •3{¥^«ØBellman-Ford~^U?^Łzg#!:"3e gRelaxL§¥JŁ˜!:§¿Ø’>?1t§Ł" 5 ‘un − 1{U(§⁄–{mE,uO(nm)" {Øu>t E,O(m/t)§⁄–¿1Bellman-Ford{E,O(n ∗ (m/t)) = O(nm/t)" 4.2 Ł Bellman-Ford†;¢y·|^ŁIt!:"ø«Ł¢y ‹’{ﬂ§t!:Œun§ªtﬁ†("GPU¥ ŁIv–eA:1⁄ﬂ\!J"mE,3O(1)"2⁄GPUŁ ¥UﬂE£CPU¥?1" ˜u–:§•ƒ^Ł§¿dŁ\"S¥" øQy GPU3O(1)m\§CPUU3O(1)mJ" ŁŒ(· nŒ|"¥kheadtail" tail⁄·aqW! :§Œ§⁄–Ł¥ın − 1Œ"head = tail§Ł"duŒ |n§⁄–zg#pp ← (p + 1) mod n"ŁUØ—/|^– c!:¢m#!:ƒ^"øŁØumƒ§Ł?§E

,§Ø\!U3O(1)⁄§Ø•GPU¥Ł¢y"3'¥§ ØGPUŁ??1U?" du¿1{§ﬂı§ØuŁ‹)N ıˇ&!ﬂK§‡7$˙" ⁄–§{Ł¢yŒ|§zBlockØ AŁ§?J,{˙" 4.3 Large Label LastSmall Label First‘z [26]JØuBellman-FordLarge Label Last(LLL)Small Label First(SLF)‘ z" LLLŁ˜1§Łn§davg = 1 v=1 d(v)" edist(i) > davgKi\ n Ł§Øe§Ø,iƒd(i) ≤ x§Ki Ł?1t" SLF\\!:·j§Ł˜i§ed(j) < d(i)§Kj\Ł˜§˜K\Ł "SLF ƒJp15%20%§SLF + LLL Jp50%" n 5 CUDA ∆-Stepping 5.1 ∆-Stepping{ ∆-Stepping [17]˜uDijkstra?1¿1U?"G1Dijkstra·§31i {ˇØd(v)3i × ∆(i + 1) × ∆mv?1t"∆-Stepping^Bucket[i]51i {It!:§Bucket = φ{(" dutgŒ(‰5§⁄–∆- SteppingÆuLabel-correcting{" ∆ß‰zgBucket8§⁄–∆J⁄ {¿1zgt˙Kˇf§3¡•‹?Ø∆?1?" Algorithm3CUDA ∆-Stepping{e§{¥Initial!Add To Request! Relax–¿11" 5.2 Initial E¥z^>·Æ§⁄–Øu>—'z–¿11§heavy(v)light(v) —'z3O(m/t)mS⁄" 3'{¥§duS! w–9?§B |5ˇ§Initial¢y·3Relax¥¿¢y§3…Œ¥=c(v, w) > ∆‰c(v, w) ≤ ∆5(‰ØA>a." ª~§–Øulightheavy⁄^ Sm§duheavylight3—'z;/)§mˇ&! ﬂK–¿/" 3relax¥z§Øc(v, w)⁄O\mØ§⁄ –{N‹$" 5.3 Add To Request {§Req(v, x):Ø§¥vLI#!:§x·#l" duReq·(‰§Æ0§m" XJ3§S—'‹2 × m £vx⁄SReq§uª§mL" ,§Req5( ØuO··¶§IosizeC5Req§mk

Algorithm 3 CUDA ∆-Stepping (a) Initial: foreach v ∈ V do heavy(v) ← {(v, w) ∈ E : c(v, w) > ∆} light(v) ← {(v, w) ∈ E : c(v, w) ≤ ∆} d(v) ← ∞ end relax(s, 0) i ← 0 while B = φ do S ← φ while B[i] = φ do (b1) Add To Request: Req ← {(w, d(v) + c(v, w)) : v ∈ B[i] ∧ (v, w) ∈ light(v)} S ← S ∪ B[i] B[i] ← φ (c1) Relax: foreach (v, x) ∈ Req do relax(v, w) end end (b2) Add To Request: Req ← {(w, d(v) + c(v, w)) : v ∈ S ∧ (v, w) ∈ heavy(v)} (c2) Relax: foreach (v, x) ∈ Req do relax(v, x) end i ← i + 1 end Procedure relax(v, x) if x < d(v) then B[d(v)/∆] ← B[d(v)/∆] v B[x/∆] ← B[x/∆] ∪ v d(v) ← x

§U?1#§¿1Ø$" §§mˇ&! ﬂK·m·¶ " u·§•^ReqI!:{§=#!:wc!:v" 3 Relax¥?1A?U=" ø—?kn1§!m" ’c k2 × mmƒ§{ıInm£=!:Œ8⁄" 1§${E ,"ØuReqO\Iv·˜ÆuB[i]\\§{¢ydRelax⁄"§ ~›§•^˜uReductionpreﬁx sum [12]{OReqIndex§Ø A!:3Req¥§ØReq#–¿11§dd)ßˇ&!ﬂK§ Jp¿1˙" 5.4 Relax ˜k§Øuz§tidreqSizeØ3Req¥ØA!:ØAI#! :Œ"reqSizeaquJCompressed Sparse Row¥Cp¢Œ|"reqSize[i]P „ci − 1!:Lo" 3#?1c(v, w)5(‰heavy‰light>§–?1 Aheavy‰lightª#" lightªmE,·O(l/t)§heavyO(h/t)§ ¥lh'O·lightheavy#>Œ§t§oŒ" 5.5 BucketŒ( 3{¥§,’:·B[i]Œ(" {¥§Bucket8a." ø (Ø•^uC++ STL¥seta§ØuCUDA?§K¿•" •3relaxF" ØBucket?1O(1)m\! " duCUDAJłseta.§⁄–Oø Œ(·~]§S! m! ˇ&ﬂKU⁄·¶" ˜ uøﬂK§•Ø{?1U?–•^uCUDA?§" c[*B[i]–uy§ z!:v3mUÆuBucket§3B[i]B[j]„v„§ ¥i = j" ,3¥ØuB = φB[i] = φ·OBucketI˜ﬂ K" 3•{¥§OBIndex(v)Œ^5v⁄ÆBucket¢" ø —?k1§!S" BIndexInmŒ|=§Øu {B[i]8a.⁄^m£max(p)/∆" ¥§p ∈ P §P L⁄kU ·»⁄!Nı" 1§B" Øu8B[i]«O·~E, §Uˆ{y3O(1)mS⁄"ØuBIndex(v)Ø!\!–9 S¿UyO(1)" 5.6 ∆J ∆–kn«J1)∆ = M EDIAN (c(e))§2)∆ = 1 n e∈E c(e)§3)∆ = max{c(e) : e ∈ E}/max{d : d ∈ Degree}" §Meyer [17]y†re-insertionoŒ3|P∆|§re- relaxationoŒ3|P2∆|" ¥§|P∆|LL·»∆8" Øu? ∆§phases Œ‰3dc/∆lmax§¥dc = max{d(v) : d(v) < ∞}" ⁄–§

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