Cache-Efficient Algorithms II

Transcription

1 6.172 Performace Egieerig of Software Systems LECTURE 9 Cache-Efficiet Algorithms II Charles E. Leiserso October 7, Charles E. Leiserso 1

2 Ideal-Cache Model Features Two-level hierarchy. Cache size of M bytes. Cache-lie legth of B bytes. Fully associative. Optimal, omisciet replacemet. P M/ B cache lies cache B memory Performace Measures work W (ordiary ruig time). cache misses Q Charles E. Leiserso 2

3 Mergig Two Sorted Arrays void Merge(double *C, double *A, double *B, it a, it b) { while (a>0 && b>0) { if (*A <= *B) { *C++ = *A++; a--; } else { *C++ = *B++; b--; } } while (a>0) { *C++ = *A++; a--; } while (b>0) { *C++ = *B++; b--; } } Charles E. Leiserso 3

4 Mergig Two Sorted Arrays void Merge(double *C, double *A, double *B, it a, it b) { while (a>0 && b>0) { if (*A <= *B) { *C++ = *A++; a--; } else { *C++ = *B++; b--; } } } while (a>0) { *C++ = *A++; a--; } while (b>0) { *C++ = *B++; b--; } Time to merge elemets = Θ() Charles E. Leiserso 4

5 Merge Sort void MergeSort(double *B, double *A, it ) { if (==1) { B[0] = A[0]; } else { double *C = malloc(*sizeof(double)); MergeSort(C, A, /2); MergeSort(C+/2, A+/2, -/2); Merge(B, C, C+/2, /2, -/2); } } Charles E. Leiserso 5

6 Merge Sort void MergeSort(double *B, double *A, it ) { if (==1) { B[0] = A[0]; } else { double *C = malloc(*sizeof(double)); MergeSort(C, A, /2); MergeSort(C+/2, A+/2, -/2); Merge(B, C, C+/2, /2, -/2); } } recursively sort recursively sort merge Charles E. Leiserso 6

7 Work of Merge Sort void MergeSort(double *B, double *A, it ) { if (==1) { B[0] = A[0]; } else { double *C = malloc(*sizeof(double)); MergeSort(C, A, /2); MergeSort(C+/2, A+/2, -/2); Merge(B, C, C+/2, /2, -/2); } } W () = Θ(1) if = 1, 2W (/2) + Θ() otherwise Charles E. Leiserso 7

8 Recursio tree Solve W() = 2W(/2) + Θ() Charles E. Leiserso 8

9 Recursio tree Solve W() = 2W(/2) + Θ(). W() 2010 Charles E. Leiserso 9

10 Recursio tree Solve W() = 2W(/2) + Θ(). W(/2) W(/2) 2010 Charles E. Leiserso 10

11 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 W(/4) W(/4) W(/4) W(/4) 2010 Charles E. Leiserso 11

12 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 /4 /4 /4 /4 Θ(1) 2010 Charles E. Leiserso 12

13 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 h = lg /4 /4 /4 /4 Θ(1) 2010 Charles E. Leiserso 13

14 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 h = lg /4 /4 /4 /4 Θ(1) 2010 Charles E. Leiserso 14

15 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 h = lg /4 /4 /4 /4 Θ(1) 2010 Charles E. Leiserso 15

16 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 h = lg /4 /4 /4 /4 Θ(1) 2010 Charles E. Leiserso 16

17 Recursio tree Solve W() = 2W(/2) + Θ(). /2 /2 h = lg /4 /4 /4 /4 Θ(1) #leaves = Θ() 2010 Charles E. Leiserso 17

18 Recursio tree Solve W() = 2W(/2) + Θ(). h = lg /2 /2 /4 /4 /4 /4 Θ(1) #leaves = Θ() W() Θ( lg ) 2010 Charles E. Leiserso 18

19 Now with Cachig Merge subroutie Merge sort Q() = Θ(/B). Θ(/B) if cm, cost c 1. Q () = 2Q (/2) + Θ(/B) otherwise Charles E. Leiserso 19

20 Recursio tree Q () = Θ(/B) if cm, cost c 1. 2Q (/2) + Θ(/B) otherwise. /B /B h=lg(/cm) /2B /2B /B /4B /4B /4B /4B /B Θ(M/B) #leaves = /cm Q() Θ(/B) Θ((/B) lg(/m)) 2010 Charles E. Leiserso 20

21 Bottom Lie for Merge Sort Q () = Θ(/B) if cm, cost c 1. 2Q (/2) + Θ(/B) otherwise. Θ( (/B) lg(/m) ) For M, we have lg(/m) lg, ad thus W()/Q() Θ(B). For M, we have lg(/m) Θ(1), ad thus W()/Q() Θ(B lg ) Charles E. Leiserso 21

22 Multiway Mergig Idea: Merge R < subarrays with a touramet. R Touramet takes Θ(R) work to produce the first output /R 7 7 lg R 2010 Charles E. Leiserso 22

23 Multiway Mergig Idea: Merge R < subarrays with a touramet. R Touramet takes Θ(R) work to produce the first output /R 7 7 lg R 2010 Charles E. Leiserso 23

24 Multiway Mergig R Idea: Merge R < subarrays with a touramet. Touramet takes Θ(R) work to produce the first output Subsequet outputs cost 6 Θ(lg R) per elemet /R 7 lg R 2010 Charles E. Leiserso 24

25 Multiway Mergig R Idea: Merge R < subarrays with a touramet. Touramet takes Θ(R) work to produce the first output Subsequet outputs cost 6 Θ(lg R) per elemet /R 7 lg R 2010 Charles E. Leiserso 25

26 Multiway Mergig R Idea: Merge R < subarrays with a touramet. Touramet takes Θ(R) work to produce the first output. 28 Subsequet outputs cost 14 Θ(lg R) per elemet /R 7 lg R 7 Total work mergig = Θ(R+ lg R) = Θ( lg R) Charles E. Leiserso 26

27 Multiway Merge Sort W () = Θ(1) if = 1, R W (/R) + Θ( lg R) otherwise. log R recursio tree lg R R (/R) lg R (/R) lg R (/R) lg R R lg R lg R (/R 2 ) lg R (/R 2 ) lg R (/R 2 ) lg R lg R #leaves = Θ(1) Θ() W()= Θ(( lg R) log R +) Same as biary merge sort. = Θ(( lg R)(lg )/lg R+) = Θ( lg ) 2010 Charles E. Leiserso 27

28 Cache Recurrece Assume that R < cm/b for suff. small cost c 1. Cosider the R-way mergig of cotiguous arrays of total size. If R < cm/b, the etire touramet plus 1 block from each array ca fit i cache. Q () Θ(/B). R-way merge sort Θ(/B) if < cm, Q() R Q(/R) + Θ(/B) otherwise Charles E. Leiserso 28

29 Cache Aalysis Q() Θ(/B) if < cm, R Q(/R) + Θ(/B) otherwise. log R (/cm) /BR 2 Θ(M/B) recursio tree /B /BR R /BR /BR 2 /BR 2 R /B /BR /B /B #leaves = /cm Θ(/B) Q() = Θ((/B) log R (/M)) 2010 Charles E. Leiserso 29

30 Tue We have Q () = Θ((/B) log R (/M)), which decreases as R icreases. Choose R as big as possible R = Θ(M/B). By the tall-cache assumptio ad the fact that log (/ M) = Θ((lg )/lg M), we have M Q () = Θ((/B) log M/ B(/ M)) = Θ((/B) log M (/ M)) = Θ(( lg )/B lg M). Hece, we have W()/Q() Θ(B lg M) Charles E. Leiserso 30

31 Multiway versus Biary Merge Sort We have versus Q multiway () = Θ(( lg )/B lg M) Q biary () = Θ((/B) lg (/M)). If M, the lg (/M) lg, ad thus multiway merge sort saves a factor of Θ(lgM) i cache misses. Eample L1-cache: M = 2 15, B = savigs. L2-cache: M = 2 18, B = savigs. L3-cache: M = 2 23, B = savigs Charles E. Leiserso 31

32 Optimal Cache-Oblivious Sortig Fuelsort [FLPR99] 1. Recursively sort 1/3 groups of 2/3 items. 2. Merge the sorted groups with a 1/3 -fuel. A k-fuel merges k 3 items i k sorted lists, icurrig at most cache misses. Thus, fuelsort icurs cache misses, which is asymptotically optimal [AV88] Charles E. Leiserso 32

33 Costructio of a k-fuel Subfuels i cotiguous storage. Buffers i cotiguous storage. Refill buffers o demad. Space = O(k2). buffers k k k3/2 k k k3 items Cache misses = O(k + (k3/b)(1+logmk)) Tall-cache assumptio: M = Ω(B2) Charles E. Leiserso 33

34 Heat Diffusio 2D heat equatio Let u(t,,y) = temperature at time t at poit (,y). u t = α 2 u y u 2 α is the thermal diffusivity. Ackowledgmet These stecil slides were heavily ispired by origials due to Matteo Frigo Charles E. Leiserso 34

35 2D Heat-Diffusio Simulatio Before After 2010 Charles E. Leiserso 35

36 1D Heat Equatio u t 2 α u Charles E. Leiserso 36

37 Fiite-Differece Approimatio 2010 Charles E. Leiserso α u t u, Δ /2) Δ, ( /2) Δ, ( ), ( u t u t t u. ) (Δ ) Δ, ( ), ( 2 ) Δ, ( Δ /2) Δ, )( / ( /2) Δ, )( / ( ), ( 2 2 u t u t u t t u t u t u 2, Δ ), ( ), Δ ( ), ( t u t t u t t t u. ) (Δ ) Δ, ( ), ( 2 ) Δ, ( α Δ ), ( ), Δ ( 2 u t u t u t t u t t u t The 1D heat equatio thus reduces to

38 3-Poit Stecil t Update rule A stecil computatio updates each poit i a array by a fied patter, called a stecil. boudary αδt u[ t + 1][ ] = u[ t][ ] + ( u[ t][ + 1] 2u [ t][ ] + u[ t][ 1]), 2 (Δ) 2010 Charles E. Leiserso 38

39 Cache Behavior of Loopig double u[2][n]; // eve-odd trick static ilie double kerel(double * u) { retur u[0] + ALPHA * (u[-1] - 2*u[0] + u[1]); } for (it t = 1; t < T-1; ++t) { // time loop for(it = 1; < N-1; ++) // space loop u[(t+1)%2][] = kerel( &u[t%2][] ); u Assumig LRU, if N > M, the Q = Θ(NT/B) Charles E. Leiserso 39

40 Cache-Oblivious 3-Poit Stecil t1 t height t0 Recursively traverse trapezoidal regios of space-time poits (t,) such that t0 t < t1 0+d0(t-t0) < 1+d1(t-t0) d0, d1 { 1, 0, 1} width Charles E. Leiserso 40 1

41 Base Case If height = 1, compute all space-time poits i the trapezoid. Ay order of computatio is valid, sice o poit depeds o aother. t 2010 Charles E. Leiserso 41

42 Space Cut If width 2 height, cut the trapezoid with a lie of slope -1 through the ceter. Traverse the trapezoid o the left first, ad the the oe o the right. t 2010 Charles E. Leiserso 42

43 Time Cut If width < 2 height, cut the trapezoid with a horizotal lie through the ceter. Traverse the bottom trapezoid first, ad the the top oe. t 2010 Charles E. Leiserso 43

44 C Implemetatio void trapezoid(it t0, it t1, it 0, it d0, it 1, it d1) { it lt = t1 - t0; if (lt == 1) { for (it = 0; < 1; ++) kerel(t, ); } else if (lt > 1) { if (2 * (1-0) + (d1 - d0) * lt >= 4 * lt) { it m = (2 * (0 + 1) + (2 + d0 + d1) * lt) / 4; trapezoid(t0, t1, 0, d0, m, -1); trapezoid(t0, t1, m, -1, 1, d1); } else { it halflt = lt / 2; trapezoid(t0, t0 + halflt, 0, d0, 1, d1); trapezoid(t0 + halflt, t1, 0 + d0 * halflt, d0, 1 + d1 * halflt, d1); } } } 2010 Charles E. Leiserso 44

45 Cache Aalysis recursio tree Θ(1) W(/2) Θ(1) W(/2) Θ(1) Θ(1) Θ(1) Θ(1) Each leaf represets Θ(hw) poits where h = Θ(w). Each leaf icurs Θ(w/B) misses where w = Θ(M). Θ(NT/hw) leaves. #iteral odes = #leaves - 1 do ot cotribute substatially to Q. Q = Θ(NT/hw) Θ(w/B) = Θ(NT/M 2 ) Θ(M/B) = Θ(NT/MB) Charles E. Leiserso 45

46 Simulatio: 3-Poit Stecil Rectagular regio N = 95 T = 87 Loopig Fully associative LRU cache B = 4 poits M = 32 Cache-hit latecy = 1 cycle Cache-miss latecy = 10 cycles Trapezoid 2010 Charles E. Leiserso 46

47 Loopig v. Trapezoid for Real 2010 Charles E. Leiserso 47

48 Other C-O Algorithms Matri Traspositio/Additio Straightforward recursive algorithm. Θ(1+m/B) Strasse s Algorithm Θ( + 2 /B + lg 7 /BM (lg 7)/2-1 ) Straightforward recursive algorithm. Fast Fourier Trasform Θ(1 + (/B)(1 + log )) M Variat of Cooley-Tukey [CT65] usig cache-oblivious matri traspose. LUP-Decompositio Θ(1 + 2 /B + 3 /BM 1/2 ) Recursive algorithm due to Siva Toledo [T97] Charles E. Leiserso 48

49 C-O Data Structures Ordered-File Maiteace O(1 + (lg 2 )/B) INSERT/DELETE or delete aywhere i file while maitaiig O(1)-sized gaps. Amortized boud [BDFC00], later improved i [BCDFC02]. B-Trees INSERT/DELETE: O(1+logB+1+(lg 2 )/ B) SEARCH: O(1+log B+1 ) TRAVERSE: O(1+k/B) Solutio [BDFC00] with later simplificatios [BDIW02], [BFJ02]. Priority Queues O(1+(1/B)log / (/ )) M B B Fuel-based solutio [BF02]. Geeral scheme based o buffer trees [ABDHMM02] supports INSERT/DELETE Charles E. Leiserso 49

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