Algoritmi Avanzati a.a.2014/2015 Prof.ssa Rossella Petreschi Lezione n°2 Algoritmi Avanzati a.a.2014/2015 Prof.ssa Rossella Petreschi 1
Somme prefisse Sequenziale PrefixSum(A, n) begin for i = 1 to n-1 do A[ i ] = A[ i ] + A[ i-1 ] end Tempo O(n) 2 5 1 3 6 7 8 11 13 19 20
Somme prefisse su P-RAM P-RAM EREW con n processori PrefixSum(A, n) begin for i = 0 to log n -1 do for j = 0 to n-1 -2i pardo Pj: A[ j+2i ] = A[ j ] + A[ j+2i ] end Tempo Parallelo O(log n) 2 5 1 3 6 7 4 8 11 12 9 13 19 20
Somme prefisse su mesh Ogni processore ha il valore da sommare in xi,j begin for i = 0 to R-1 pardo for j = 1 to C-1 do Pi,j: xi,j = xi,j + xi,j-1 P0,C-1: su0,C-1 = 0 for i = 1 to R-1 do Pi,C-1: sui,C-1 = xi-1,C-1 + sui-1,C-1 for i = 1 to R-1 pardo for j = C-2 downto 0 do Pi,j: sui,j = sui,j+1 for j = 0 to C-1 pardo Pi,j: xi,j = xi,j + sui,j end Tempo parallelo R+C 4
Somme prefisse su albero binario Le n foglie hanno già il valore da sommare nella variabile xi begin for i = log n -1 downto 0 do for j = 2i to 2i+1-1 pardo Pj: xj = x2j + x2j+1 P1: sp1 = 0 for i = 0 to log n -1 do Pj: sp2j = spj sp2j+1 = spj + x2j for i = n to 2n-1 pardo Pi: xi = xi + spi end Tempo parallelo logaritmico X1+…+x4 X1+x2 X3+x4 X5+…+x8 X5+x6 X7+x8 X1+…+x8 x1 x2 x3 x5 x6 x7 x8 P2 P1 P4 P8 P5 P6 P7 P3 P15 sp = 0 X1+…+x8 sp = 0 sp = X1+…+x4 X1+…+x4 X5+…+x8 sp = 0 X1+x2 sp = X1+…+x6 X1+x2 X3+x4 X5+x6 X7+x8 sp = 0 x1 sp = X1+…+x7 x1 x2 x3 x4 x5 x6 x7 x8 5
Concorrenza in lettura e scrittura PRAM EREW: Exclusive Read, Exclusive Write; CREW: Concurrent Read, Exclusive Write; ERCW: Exclusive Read, Concurrent Write; CRCW: Concurrent Read, Concurrent Write; Memoria P0 P1 P2 Pn-1
Broadcast su P-RAM EREW Si adoperano n processori Broadcast(x) begin P0: A[ 0 ] = x for i = 0 to log n -1 do for j = 2i to 2i+1-1 pardo Pj: A[ j ] = A[ j-2i ] end Tempo parallelo logaritmico x