// ************************************************************************** // // // // eses eses // // eses eses // // eses eseses esesese eses Embedded Systems Group // // ese ese ese ese ese // // ese eseseses eseseses ese Department of Computer Science // // eses eses ese eses // // eses eseses eseseses eses University of Kaiserslautern // // eses eses // // // // ************************************************************************** // // The following module implements an algorithm to compute the transitive hull// // of a binary relation or a graph. It is based on an algorithm for boolean // // matrix multiplication that requires O(1) time using O(N^3) processors. // // This algorithm is used for iterative squaring of the given matrix so that // // in total O(log(N)) time with O(N^3) processors is required. // // Note that t is a memorized variable, so that it is sufficient to only set // // potentially new elements t[i][j] while those that are already true remain // // so automatically. Finally note that it is an CRCW algorithm since several // // writes to the same t[i][j] with the same value true are possible. // // ************************************************************************** // macro N = 9; module TransHull_OlogN([N][N]bool ?a, t,event !rdy) { // copy matrix a to t for(i=0..N-1) for(j=0..N-1) t[i][j] = a[i][j]; // iterative squaring for(h=0..log(N)-1) { // multiply matrix t with itself for(i=0..N-1) { for(j=0..N-1) for(k=0..N-1) if(t[i][k] and t[k][j]) next(t[i][j]) = true; } pause; } emit(rdy); } drivenby { // the assignments below construct the following graph: // // 3 // / \ // v v // 0 -> 1 <-> 2-> 4 <-> 5 <-> 6 // | | // v v // 7<| 8 // |_| // a[0][1] = true; a[1][2] = true; a[2][1] = true; a[2][4] = true; a[2][7] = true; a[3][2] = true; a[3][4] = true; a[4][5] = true; a[5][4] = true; a[5][6] = true; a[5][8] = true; a[6][5] = true; a[7][7] = true; await(rdy); for(i=0..N-1) for(j=0..N-1) assert((t[i][j] <-> (i==0 & (j==1 | j==2 | j==4 | j==5 | j==6 | j==7 | j==8) | i==1 & (j==1 | j==2 | j==4 | j==5 | j==6 | j==7 | j==8) | i==2 & (j==1 | j==2 | j==4 | j==5 | j==6 | j==7 | j==8) | i==3 & (j==1 | j==2 | j==4 | j==5 | j==6 | j==7 | j==8) | i==4 & ( j==4 | j==5 | j==6 | j==8) | i==5 & ( j==4 | j==5 | j==6 | j==8) | i==6 & ( j==4 | j==5 | j==6 | j==8) | i==7 & j==7) ) ); } drivenby t2 { // the assignments below construct the following graph that // is a worst case example // // 0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 // a[0][1] = true; a[1][2] = true; a[2][3] = true; a[3][4] = true; a[4][5] = true; a[5][6] = true; a[6][7] = true; a[7][8] = true; await(rdy); }