CSE 5311 Notes 12: Matrices

Transcription

1 CSE 5311 Noes 12: Marices Las updaed 11/12/15 9:12 AM) STRASSEN S MATRIX MULTIPLICATION Marix addiio: akes scalar addiios. Everyday arix uliply: p p Le = = p. akes p scalar uliplies ad -1)p scalar addiios o produce resul arix. Bes lower boud is Ω 2 ). For = 2: " A11 A12 " B 11 B12 " = C 11 C12 A 21 A 22 & B 21 B 22 & C 21 C 22 & is doe by everyday ehod usig: 8 scalar uliplies 4 scalar addiios bu Srasse s ehod CLRS, p.79) uses: 7 scalar uliplies 18 scalar addiios

2 2 Whe = 2 k : A ij ad B ij are 2 k 1 2 k 1 subarices o be uliplied recursively usig Srasse s ehod. Suppose = 4. Everyday ehod akes 4 3 = 64 scalar uliplies ad 16 3 = 48 scalar addiios. Srasse s ehod akes: 7 recursive 2 2 arix uliplies, each usig 7 scalar uliplies ad 18 scalar addiios arix addiios, each usig 4 scalar addiios. This gives 49 scalar uliplies ad 198 scalar addiios. Le Mk) = uber of scalar uliplies for 2 k 2 k = : ) =1 ) = 7 M 0 M 1 M k) = 7M k 1) = 7 k = 7 lg = lg Noe ha he cosa is 1. Le Pk) be he uber of addiios icludig subracios) doe for 2 k 2 k = : ) = 0 ) =18 P 0 P 1 ) =18 2 k 1 P k Le P ) 2 k & = P k P ) ) =18 2 & 2 + 7P k 1 ) Maser Mehod : ) =18 2k 2 & 2 ) ) ) = ) ) P 2 P 2 a = 7 b = 2 logb a = lg f ) = = O 2.81 ε & Case 1: P ) ) = Θ 2.81 & + 7P k 1) For he sae of research o fas arix uliplicaio: hp://dl.ac.org.ezproxy.ua.edu/ciaio.cf?doid=

3 KRONROD S ALGORITHM FOR BOOLEAN MATRIX MULTIPLICATION AKA Four Russias Mehod) 3 Boolea versio of arix uliplicaio subsiues for ad for + i uerical versio: for i=0;i<;i++) for j=0;j<;j++) c[i][j]=0; for k=0;k<;k++) if a[i][k] && b[k][j]) c[i][j]=1; break; ) c[ i] [ j] =1 k) a[ i] [ k] =1 b[ k] [ j] =1 Sice bis ca be packed io a bye, word, ec., a speed-up ay be obaied. Le = 8 ad suppose = 256 for uliplyig ogeher arices A ad B krorodchar5311.c): usiged char A[256][32],B[256][32],C[256][32]; where a upacked arix ca be packed wih: // Covers colu bi subscrip o subscrip for usiged char defie SUB2CHARsub) sub)/8) // Deeries bi posiio wihi usiged char for colu bi subscrip defie SUB2BITsub) sub)8) // Ges 0/1 value for a colu bi subscrip defie GETBITarr,row,col) arr[row][sub2charcol)]>>sub2bitcol))&1) // Ses bi o 1 defie SETBITarr,row,col) \ arr[row][sub2charcol)]=arr[row][sub2charcol)] 1<<SUB2BITcol))) // Clears bi o 0 defie CLEARBITarr,row,col) \ arr[row][sub2charcol)]=arr[row][sub2charcol)]&255-1<<sub2bitcol)))) for i=0; i<; i++) for j=0; j<; j++) if Aupacked[i][j]) SETBITA,i,j); else CLEARBITA,i,j); Row-orieed boolea arix uliplicaio is: 1. Fid posiios wih oes i row i of A i.e. colu idices). 2. Row i of C is he resul of or ig ogeher he rows of B for he posiios fro 1.

4 for i=0; i<; i++) for j=0; j</; j++) // Clear oupu row i C[i][j]=0; for j=0; j</; j++) for k=0; k<; k++) if A[i][j]>>k) & 1) // Exaie row i, bi *j+k for p=0; p</; p++) // Se correspodig bis C[i][p]=C[i][p] B[*j+k][p]; 4 Tie is Ο 3 & givig = 2,097,152 bye or s). Krorod s echique: Preprocessig + addressig o reduce cos of or ig ogeher rows B A 1. groups of cosecuive B rows group i is rows *i... *i+-1). 2. Each group is preprocessed by or ig ogeher all 2 possible cobiaios subse) of he rows usig 2 exra space for subse). Each of rows j 1 i subse are or ed wih row *i+j of B o obai rows 2 j... 2 j+1 1 i subse. 3. These resuls are he refereced by goig dow he correspodig colu of A usig he packed) bis as a subscrip o he exra able of or ed cobiaios.

5 Thus, uch of he effor for row or ig is rasfered o he preprocessig: 5 usiged char subse[256][32]; // Iiialize oupu arix for i=0; i<; i++) for j=0; j</; j++) C[i][j]=0; // Iiialize epy subse - sae for each group for i=0; i</; i++) subse[0][i]=0; // There are / groups of rows i B for i=0; i</; i++) ragesar=1; // Iclude each row o achieve cobiaios ORed ogeher for j=0; j<; j++) for k=ragesar; k<2*ragesar; k++) for q=0; q</; q++) // OR he subse wihou row j of his group wih row j // of his group. Firs row of group is B[*i]. subse[k][q]=subse[k-ragesar][q] B[*i+j][q]; ragesar*=2; // Updae resul rows based o his group of B rows for j=0; j<; j++) for q=0; q</; q++) C[j][q]=C[j][q] subse[a[j][i]][q]; Tie is Ο && = Ο & 2 givig 524,288 bye or s). If = lg, row-orieed gives Ο log 3 & ie while Krorod s ehod is Ο 3 & log 2. This echique will be applied i Noes 14 o loges coo subsequeces.