/* print the statistics about this floorplan.
* note that connects_file is NULL if wire
* information is already populated
*/
void print_flp_stats(flp_t *flp, RC_model_t *model,
char *l2_label, char *power_file,
char *connects_file)
{
double core, total, occupied; /* area */
double width, height, aspect, total_w, total_h;
double wire_metric;
double peak, avg; /* temperature */
double *power, *temp;
FILE *fp = NULL;
char str[STR_SIZE];
if (connects_file) {
fp = fopen(connects_file, "r");
if (!fp) {
sprintf(str, "error opening file %s\n", connects_file);
fatal(str);
}
flp_populate_connects(flp, fp);
}
power = hotspot_vector(model);
temp = hotspot_vector(model);
read_power(model, power, power_file);
core = get_core_area(flp, l2_label);
total = get_total_area(flp);
total_w = get_total_width(flp);
total_h = get_total_height(flp);
occupied = get_core_occupied_area(flp, l2_label);
width = get_core_width(flp, l2_label);
height = get_core_height(flp, l2_label);
aspect = (height > width) ? (height/width) : (width/height);
wire_metric = get_wire_metric(flp);
populate_R_model(model, flp);
steady_state_temp(model, power, temp);
peak = find_max_temp(model, temp);
avg = find_avg_temp(model, temp);
fprintf(stdout, "printing summary statistics about the floorplan\n");
fprintf(stdout, "total area:\t%g\n", total);
fprintf(stdout, "total width:\t%g\n", total_w);
fprintf(stdout, "total height:\t%g\n", total_h);
fprintf(stdout, "core area:\t%g\n", core);
fprintf(stdout, "occupied area:\t%g\n", occupied);
fprintf(stdout, "area utilization:\t%.3f\n", occupied / core * 100.0);
fprintf(stdout, "core width:\t%g\n", width);
fprintf(stdout, "core height:\t%g\n", height);
fprintf(stdout, "core aspect ratio:\t%.3f\n", aspect);
fprintf(stdout, "wire length metric:\t%.3f\n", wire_metric);
fprintf(stdout, "peak temperature:\t%.3f\n", peak);
fprintf(stdout, "avg temperature:\t%.3f\n", avg);
free_dvector(power);
free_dvector(temp);
if (fp)
fclose(fp);
}
/*
* this is the metric function used for the floorplanning.
* in order to enable a different metric, just change the
* return statement of this function to return an appropriate
* metric. The current metric used is a linear function of
* area (A), temperature (T) and wire length (W):
* lambdaA * A + lambdaT * T + lambdaW * W
* thermal model and power density are passed as parameters
* since temperature is used in the metric.
*/
double flp_evaluate_metric(flp_t *flp, RC_model_t *model, double *power,
double lambdaA, double lambdaT, double lambdaW)
{
double tmax, area, wire_length, width, height, aspect;
double *temp;
temp = hotspot_vector(model);
populate_R_model(model, flp);
steady_state_temp(model, power, temp);
tmax = find_max_temp(model, temp);
area = get_total_area(flp);
wire_length = get_wire_metric(flp);
width = get_total_width(flp);
height = get_total_height(flp);
if (width > height)
aspect = width / height;
else
aspect = height / width;
free_dvector(temp);
/* can return any arbitrary function of area, tmax and wire_length */
return (lambdaA * area + lambdaT * tmax + lambdaW * wire_length);
}
/* calculate avg sink temp for natural convection package model */
double calc_sink_temp_block(block_model_t *model, double *temp, thermal_config_t *config)
{
flp_t *flp = model->flp;
int i;
double sum = 0.0;
double width = get_total_width(flp);
double height = get_total_height(flp);
double spr_size = config->s_spreader*config->s_spreader;
double sink_size = config->s_sink*config->s_sink;
/* heatsink temperatures */
for (i=0; i < flp->n_units; i++)
if (temp[HSINK*flp->n_units+i] < 0)
fatal("negative temperature!\n");
else /* area-weighted average */
sum += temp[HSINK*flp->n_units+i]*(flp->units[i].width*flp->units[i].height);
for(i=SINK_C_W; i <= SINK_C_E; i++)
if (temp[i+NL*flp->n_units] < 0)
fatal("negative temperature!\n");
else
sum += temp[i+NL*flp->n_units]*0.25*(config->s_spreader+height)*(config->s_spreader-width);
for(i=SINK_C_N; i <= SINK_C_S; i++)
if (temp[i+NL*flp->n_units] < 0)
fatal("negative temperature!\n");
else
sum += temp[i+NL*flp->n_units]*0.25*(config->s_spreader+width)*(config->s_spreader-height);
for(i=SINK_W; i <= SINK_S; i++)
if (temp[i+NL*flp->n_units] < 0)
fatal("negative temperature!\n");
else
sum += temp[i+NL*flp->n_units]*0.25*(sink_size-spr_size);
return (sum / sink_size);
}
void create_RC_matrices(flp_t *flp, int omit_lateral)
{
int i, j, k = 0, n = flp->n_units;
int **border;
double **len, *gx, *gy, **g, *c_ver, **t, *gx_sp, *gy_sp;
double r_sp1, r_sp2, r_hs; /* lateral resistances to spreader and heatsink */
/* NOTE: *_mid - the vertical R/C from CENTER nodes of spreader
* and heatsink. *_per - the vertical R/C from PERIPHERAL (n,s,e,w) nodes
*/
double r_sp_per, r_hs_mid, r_hs_per, c_sp_per, c_hs_mid, c_hs_per;
double gn_sp=0, gs_sp=0, ge_sp=0, gw_sp=0;
double w_chip = get_total_width (flp); /* x-axis */
double l_chip = get_total_height (flp); /* y-axis */
border = imatrix(n, 4);
len = matrix(n, n); /* len[i][j] = length of shared edge bet. i & j */
gx = vector(n); /* lumped conductances in x direction */
gy = vector(n); /* lumped conductances in y direction */
gx_sp = vector(n); /* lateral conductances in the spreader layer */
gy_sp = vector(n);
g = matrix(NL*n+EXTRA, NL*n+EXTRA); /* g[i][j] = conductance bet. nodes i & j */
c_ver = vector(NL*n+EXTRA); /* vertical capacitance */
b = matrix(NL*n+EXTRA, NL*n+EXTRA); /* B, C, INVA and INVB are (NL*n+EXTRA)x(NL*n+EXTRA) matrices */
c = matrix(NL*n+EXTRA, NL*n+EXTRA);
inva = matrix(NL*n+EXTRA, NL*n+EXTRA);
invb = matrix(NL*n+EXTRA, NL*n+EXTRA);
t = matrix (NL*n+EXTRA, NL*n+EXTRA); /* copy of B */
/* compute the silicon fitting factor - see pg 10 of the UVA CS tech report - CS-TR-2003-08 */
factor_chip = C_FACTOR * ((SPEC_HEAT_INT / SPEC_HEAT_SI) * (w_chip + 0.88 * t_interface) \
* (l_chip + 0.88 * t_interface) * t_interface / ( w_chip * l_chip * t_chip) + 1);
/* fitting factor for interface - same rationale as above */
factor_int = C_FACTOR * ((SPEC_HEAT_CU / SPEC_HEAT_INT) * (w_chip + 0.88 * t_spreader) \
* (l_chip + 0.88 * t_spreader) * t_spreader / ( w_chip * l_chip * t_interface) + 1);
/*printf("fitting factors : %lf, %lf\n", factor_chip, factor_int); */
/* gx's and gy's of blocks */
for (i = 0; i < n; i++) {
/* at the silicon layer */
if (omit_lateral) {
gx[i] = gy[i] = 0;
}
else {
gx[i] = 1.0/getr(K_SI, flp->units[i].height, flp->units[i].width, l_chip, t_chip);
gy[i] = 1.0/getr(K_SI, flp->units[i].width, flp->units[i].height, w_chip, t_chip);
}
/* at the spreader layer */
gx_sp[i] = 1.0/getr(K_CU, flp->units[i].height, flp->units[i].width, l_chip, t_spreader);
gy_sp[i] = 1.0/getr(K_CU, flp->units[i].width, flp->units[i].height, w_chip, t_spreader);
}
/* shared lengths between blocks */
for (i = 0; i < n; i++)
for (j = i; j < n; j++)
len[i][j] = len[j][i] = get_shared_len(flp, i, j);
/* lateral R's of spreader and sink */
r_sp1 = getr(K_CU, (s_spreader+3*w_chip)/4.0, (s_spreader-w_chip)/4.0, w_chip, t_spreader);
r_sp2 = getr(K_CU, (3*s_spreader+w_chip)/4.0, (s_spreader-w_chip)/4.0, (s_spreader+3*w_chip)/4.0, t_spreader);
r_hs = getr(K_CU, (s_sink+3*s_spreader)/4.0, (s_sink-s_spreader)/4.0, s_spreader, t_sink);
/* vertical R's and C's of spreader and sink */
r_sp_per = RHO_CU * t_spreader * 4.0 / (s_spreader * s_spreader - w_chip*l_chip);
c_sp_per = factor_pack * SPEC_HEAT_CU * t_spreader * (s_spreader * s_spreader - w_chip*l_chip) / 4.0;
r_hs_mid = RHO_CU * t_sink / (s_spreader*s_spreader);
c_hs_mid = factor_pack * SPEC_HEAT_CU * t_sink * (s_spreader * s_spreader);
r_hs_per = RHO_CU * t_sink * 4.0 / (s_sink * s_sink - s_spreader*s_spreader);
c_hs_per = factor_pack * SPEC_HEAT_CU * t_sink * (s_sink * s_sink - s_spreader*s_spreader) / 4.0;
/* short the R's from block centers to a particular chip edge */
for (i = 0; i < n; i++) {
if (eq(flp->units[i].bottomy + flp->units[i].height, l_chip)) {
gn_sp += gy_sp[i];
border[i][2] = 1; /* block is on northern border */
}
if (eq(flp->units[i].bottomy, 0)) {
gs_sp += gy_sp[i];
border[i][3] = 1; /* block is on southern border */
}
if (eq(flp->units[i].leftx + flp->units[i].width, w_chip)) {
ge_sp += gx_sp[i];
border[i][1] = 1; /* block is on eastern border */
}
if (eq(flp->units[i].leftx, 0)) {
gw_sp += gx_sp[i];
border[i][0] = 1; /* block is on western border */
}
}
/* overall R and C between nodes */
for (i = 0; i < n; i++) {
double area = (flp->units[i].height * flp->units[i].width);
/*
* amongst functional units in the various layers
* resistances in the interface layer are assumed
* to be infinite
*/
for (j = 0; j < n; j++) {
double part = 0, part_sp = 0;
if (is_horiz_adj(flp, i, j)) {
part = gx[i] / flp->units[i].height;
part_sp = gx_sp[i] / flp->units[i].height;
}
else if (is_vert_adj(flp, i,j)) {
part = gy[i] / flp->units[i].width;
part_sp = gy_sp[i] / flp->units[i].width;
}
g[i][j] = part * len[i][j];
g[HSP*n+i][HSP*n+j] = part_sp * len[i][j];
}
/* vertical g's in the silicon layer */
g[i][IFACE*n+i]=g[IFACE*n+i][i]=2.0/(RHO_SI * t_chip / area);
/* vertical g's in the interface layer */
g[IFACE*n+i][HSP*n+i]=g[HSP*n+i][IFACE*n+i]=2.0/(RHO_INT * t_interface / area);
/* vertical g's in the spreader layer */
g[HSP*n+i][NL*n+SP_B]=g[NL*n+SP_B][HSP*n+i]=2.0/(RHO_CU * t_spreader / area);
/* C's from functional units to ground */
c_ver[i] = factor_chip * SPEC_HEAT_SI * t_chip * area;
/* C's from interface portion of the functional units to ground */
c_ver[IFACE*n+i] = factor_int * SPEC_HEAT_INT * t_interface * area;
/* C's from spreader portion of the functional units to ground */
c_ver[HSP*n+i] = factor_pack * SPEC_HEAT_CU * t_spreader * area;
/* lateral g's from block center (spreader layer) to peripheral (n,s,e,w) spreader nodes */
g[HSP*n+i][NL*n+SP_N]=g[NL*n+SP_N][HSP*n+i]=2.0*border[i][2]/((1.0/gy_sp[i])+r_sp1*gn_sp/gy_sp[i]);
g[HSP*n+i][NL*n+SP_S]=g[NL*n+SP_S][HSP*n+i]=2.0*border[i][3]/((1.0/gy_sp[i])+r_sp1*gs_sp/gy_sp[i]);
g[HSP*n+i][NL*n+SP_E]=g[NL*n+SP_E][HSP*n+i]=2.0*border[i][1]/((1.0/gx_sp[i])+r_sp1*ge_sp/gx_sp[i]);
g[HSP*n+i][NL*n+SP_W]=g[NL*n+SP_W][HSP*n+i]=2.0*border[i][0]/((1.0/gx_sp[i])+r_sp1*gw_sp/gx_sp[i]);
}
/* max slope (max_power * max_vertical_R / vertical RC time constant) for silicon */
max_slope = MAX_PD / (factor_chip * t_chip * SPEC_HEAT_SI);
/* vertical g's and C's between central nodes */
/* between spreader bottom and sink bottom */
g[NL*n+SINK_B][NL*n+SP_B]=g[NL*n+SP_B][NL*n+SINK_B]=2.0/r_hs_mid;
/* from spreader bottom to ground */
c_ver[NL*n+SP_B]=c_hs_mid;
/* from sink bottom to ground */
c_ver[NL*n+SINK_B] = factor_pack * c_convec;
/* g's and C's from peripheral(n,s,e,w) nodes */
for (i = 1; i <= 4; i++) {
/* vertical g's between peripheral spreader nodes and spreader bottom */
g[NL*n+SP_B-i][NL*n+SP_B]=g[NL*n+SP_B][NL*n+SP_B-i]=2.0/r_sp_per;
/* lateral g's between peripheral spreader nodes and peripheral sink nodes */
g[NL*n+SP_B-i][NL*n+SINK_B-i]=g[NL*n+SINK_B-i][NL*n+SP_B-i]=2.0/(r_hs + r_sp2);
/* vertical g's between peripheral sink nodes and sink bottom */
g[NL*n+SINK_B-i][NL*n+SINK_B]=g[NL*n+SINK_B][NL*n+SINK_B-i]=2.0/r_hs_per;
/* from peripheral spreader nodes to ground */
c_ver[NL*n+SP_B-i]=c_sp_per;
/* from peripheral sink nodes to ground */
c_ver[NL*n+SINK_B-i]=c_hs_per;
}
/* calculate matrices A, B such that A(dT) + BT = POWER */
for (i = 0; i < NL*n+EXTRA; i++) {
for (j = 0; j < NL*n+EXTRA; j++) {
if (i==j) {
inva[i][j] = 1.0/c_ver[i];
if (i == NL*n+SINK_B) /* sink bottom */
b[i][j] += 1.0 / r_convec;
for (k = 0; k < NL*n+EXTRA; k++) {
if ((g[i][k]==0.0)||(g[k][i])==0.0)
continue;
else
/* here is why the 2.0 factor comes when calculating g[][] */
b[i][j] += 1.0/((1.0/g[i][k])+(1.0/g[k][i]));
}
} else {
inva[i][j]=0.0;
if ((g[i][j]==0.0)||(g[j][i])==0.0)
b[i][j]=0.0;
else
b[i][j]=-1.0/((1.0/g[i][j])+(1.0/g[j][i]));
}
}
}
/* we are always going to use the eqn dT + A^-1 * B T = A^-1 * POWER. so, store C = A^-1 * B */
matmult(c, inva, b, NL*n+EXTRA);
/* we will also be needing INVB so store it too */
copy_matrix(t, b, NL*n+EXTRA, NL*n+EXTRA);
matinv(invb, t, NL*n+EXTRA);
/* dump_vector(c_ver, NL*n+EXTRA); */
/* dump_matrix(g, NL*n+EXTRA, NL*n+EXTRA); */
/* dump_matrix(c, NL*n+EXTRA, NL*n+EXTRA); */
/* cleanup */
free_matrix(t, NL*n+EXTRA);
free_matrix(g, NL*n+EXTRA);
free_matrix(len, n);
free_imatrix(border, n);
free_vector(c_ver);
free_vector(gx);
free_vector(gy);
free_vector(gx_sp);
free_vector(gy_sp);
}
/*
* wrap the rim strips around. each edge has rim blocks
* equal to the number of blocks abutting that edge. at
* the four corners, the rim blocks are extended by the
* rim thickness in a clockwise fashion
*/
int flp_wrap_rim(flp_t *flp, double rim_thickness)
{
double width, height;
int i, j = 0, k, n = flp->n_units;
unit_t *unit;
width = get_total_width(flp) + 2 * rim_thickness;
height = get_total_height(flp) + 2 * rim_thickness;
flp_translate(flp, rim_thickness, rim_thickness);
for (i = 0; i < n; i++) {
/* shortcut */
unit = &flp->units[i];
/* block is on the western border */
if (eq(unit->leftx, rim_thickness)) {
sprintf(flp->units[n+j].name, "%s_%s",
RIM_LEFT_STR, unit->name);
flp->units[n+j].width = rim_thickness;
flp->units[n+j].height = unit->height;
flp->units[n+j].leftx = 0;
flp->units[n+j].bottomy = unit->bottomy;
/* northwest corner */
if (eq(unit->bottomy + unit->height, height-rim_thickness))
flp->units[n+j].height += rim_thickness;
j++;
}
/* block is on the eastern border */
if (eq(unit->leftx + unit->width, width-rim_thickness)) {
sprintf(flp->units[n+j].name, "%s_%s",
RIM_RIGHT_STR, unit->name);
flp->units[n+j].width = rim_thickness;
flp->units[n+j].height = unit->height;
flp->units[n+j].leftx = unit->leftx + unit->width;
flp->units[n+j].bottomy = unit->bottomy;
/* southeast corner */
if (eq(unit->bottomy, rim_thickness)) {
flp->units[n+j].height += rim_thickness;
flp->units[n+j].bottomy = 0;
}
j++;
}
/* block is on the northern border */
if (eq(unit->bottomy + unit->height, height-rim_thickness)) {
sprintf(flp->units[n+j].name, "%s_%s",
RIM_TOP_STR, unit->name);
flp->units[n+j].width = unit->width;
flp->units[n+j].height = rim_thickness;
flp->units[n+j].leftx = unit->leftx;
flp->units[n+j].bottomy = unit->bottomy + unit->height;
/* northeast corner */
if (eq(unit->leftx + unit->width, width-rim_thickness))
flp->units[n+j].width += rim_thickness;
j++;
}
/* block is on the southern border */
if (eq(unit->bottomy, rim_thickness)) {
sprintf(flp->units[n+j].name, "%s_%s",
RIM_BOTTOM_STR, unit->name);
flp->units[n+j].width = unit->width;
flp->units[n+j].height = rim_thickness;
flp->units[n+j].leftx = unit->leftx;
flp->units[n+j].bottomy = 0;
/* southwest corner */
if (eq(unit->leftx, rim_thickness)) {
flp->units[n+j].width += rim_thickness;
flp->units[n+j].leftx = 0;
}
j++;
}
}
flp->n_units += j;
/* update all the rim wire densities */
for(i=n; i < n+j; i++)
for(k=0; k <= i; k++)
flp->wire_density[i][k] = flp->wire_density[k][i] = 0;
return j;
}
/*
* wrap the L2 around this floorplan. L2's area information
* is obtained from flp_desc. memory for L2 and its arms has
* already been allocated in the flp. note that flp & flp_desc
* have L2 hidden beyond the boundary at this point
*/
void flp_wrap_l2(flp_t *flp, flp_desc_t *flp_desc)
{
/*
* x is the width of the L2 arms
* y is the height of the bottom portion
*/
double x, y, core_width, core_height, total_side, core_area, l2_area;
unit_t *l2, *l2_left, *l2_right;
/* find L2 dimensions so that the total chip becomes a square */
core_area = get_total_area(flp);
core_width = get_total_width(flp);
core_height = get_total_height(flp);
/* flp_desc has L2 hidden beyond the boundary */
l2_area = flp_desc->units[flp_desc->n_units].area;
total_side = sqrt(core_area + l2_area);
/*
* width of the total chip after L2 wrapping is equal to
* the width of the core plus the width of the two arms
*/
x = (total_side - core_width) / 2.0;
y = total_side - core_height;
/*
* we are trying to solve the equation
* (2*x+core_width) * (y+core_height)
* = l2_area + core_area
* for x and y. it is possible that the values
* turnout to be negative if we restrict the
* total chip to be a square. in that case,
* theoretically, any value of x in the range
* (0, l2_area/(2*core_height)) and the
* corresponding value of y or any value of y
* in the range (0, l2_area/core_width) and the
* corresponding value of x would be a solution
* we look for a solution with a reasonable
* aspect ratio. i.e., we constrain kx = y (or
* ky = x depending on the aspect ratio of the
* core) where k = WRAP_L2_RATIO. solving the equation
* with this constraint, we get the following
*/
if ( x <= 0 || y <= 0.0) {
double sum;
if (core_width >= core_height) {
sum = WRAP_L2_RATIO * core_width + 2 * core_height;
x = (sqrt(sum*sum + 8*WRAP_L2_RATIO*l2_area) - sum) / (4*WRAP_L2_RATIO);
y = WRAP_L2_RATIO * x;
} else {
sum = core_width + 2 * WRAP_L2_RATIO * core_height;
y = (sqrt(sum*sum + 8*WRAP_L2_RATIO*l2_area) - sum) / (4*WRAP_L2_RATIO);
x = WRAP_L2_RATIO * y;
}
total_side = 2 * x + core_width;
}
/* fix the positions of core blocks */
flp_translate(flp, x, y);
/* restore the L2 blocks */
flp->n_units += (L2_ARMS+1);
/* copy L2 info again from flp_desc but from beyond the boundary */
copy_l2_info(flp, flp->n_units-L2_ARMS-1, flp_desc,
flp_desc->n_units, flp_desc->n_units);
/* fix the positions of the L2 blocks. connectivity
* information has already been fixed (in flp_placeholder).
* bottom L2 block - (leftx, bottomy) is already (0,0)
*/
l2 = &flp->units[flp->n_units-1-L2_ARMS];
l2->width = total_side;
l2->height = y;
l2->leftx = l2->bottomy = 0;
/* left L2 arm */
l2_left = &flp->units[flp->n_units-L2_ARMS+L2_LEFT];
l2_left->width = x;
l2_left->height = core_height;
l2_left->leftx = 0;
l2_left->bottomy = y;
/* right L2 arm */
l2_right = &flp->units[flp->n_units-L2_ARMS+L2_RIGHT];
l2_right->width = x;
l2_right->height = core_height;
l2_right->leftx = x + core_width;
l2_right->bottomy = y;
}
/* creates matrices B and invB: BT = Power in the steady state.
* NOTE: EXTRA nodes: 4 heat spreader peripheral nodes, 4 heat
* sink inner peripheral nodes, 4 heat sink outer peripheral
* nodes(north, south, east and west) and 1 ambient node.
*/
void populate_R_model_block(block_model_t *model, flp_t *flp)
{
/* shortcuts */
double **b = model->b;
double *gx = model->gx, *gy = model->gy;
double *gx_int = model->gx_int, *gy_int = model->gy_int;
double *gx_sp = model->gx_sp, *gy_sp = model->gy_sp;
double *gx_hs = model->gx_hs, *gy_hs = model->gy_hs;
double *g_amb = model->g_amb;
double **len = model->len, **g = model->g, **lu = model->lu;
int **border = model->border;
int *p = model->p;
double t_chip = model->config.t_chip;
double r_convec = model->config.r_convec;
double s_sink = model->config.s_sink;
double t_sink = model->config.t_sink;
double s_spreader = model->config.s_spreader;
double t_spreader = model->config.t_spreader;
double t_interface = model->config.t_interface;
double k_chip = model->config.k_chip;
double k_sink = model->config.k_sink;
double k_spreader = model->config.k_spreader;
double k_interface = model->config.k_interface;
int i, j, n = flp->n_units;
double gn_sp=0, gs_sp=0, ge_sp=0, gw_sp=0;
double gn_hs=0, gs_hs=0, ge_hs=0, gw_hs=0;
double r_amb;
double w_chip = get_total_width (flp); /* x-axis */
double l_chip = get_total_height (flp); /* y-axis */
/* sanity check on floorplan sizes */
if (w_chip > s_sink || l_chip > s_sink ||
w_chip > s_spreader || l_chip > s_spreader) {
//print_flp(flp);
//print_flp_fig(flp);
//fatal("inordinate floorplan size!\n");
}
if(model->flp != flp || model->n_units != flp->n_units ||
model->n_nodes != NL * flp->n_units + EXTRA)
fatal("mismatch between the floorplan and the thermal model\n");
/* gx's and gy's of blocks */
for (i = 0; i < n; i++) {
/* at the silicon layer */
if (model->config.block_omit_lateral) {
gx[i] = gy[i] = 0;
}
else {
gx[i] = 1.0/getr(k_chip, flp->units[i].width / 2.0, flp->units[i].height * t_chip);
gy[i] = 1.0/getr(k_chip, flp->units[i].height / 2.0, flp->units[i].width * t_chip);
}
/* at the interface layer */
gx_int[i] = 1.0/getr(k_interface, flp->units[i].width / 2.0, flp->units[i].height * t_interface);
gy_int[i] = 1.0/getr(k_interface, flp->units[i].height / 2.0, flp->units[i].width * t_interface);
/* at the spreader layer */
gx_sp[i] = 1.0/getr(k_spreader, flp->units[i].width / 2.0, flp->units[i].height * t_spreader);
gy_sp[i] = 1.0/getr(k_spreader, flp->units[i].height / 2.0, flp->units[i].width * t_spreader);
/* at the heatsink layer */
gx_hs[i] = 1.0/getr(k_sink, flp->units[i].width / 2.0, flp->units[i].height * t_sink);
gy_hs[i] = 1.0/getr(k_sink, flp->units[i].height / 2.0, flp->units[i].width * t_sink);
}
/* shared lengths between blocks */
for (i = 0; i < n; i++)
for (j = i; j < n; j++)
len[i][j] = len[j][i] = get_shared_len(flp, i, j);
/* package R's */
populate_package_R(&model->pack, &model->config, w_chip, l_chip);
/* short the R's from block centers to a particular chip edge */
for (i = 0; i < n; i++) {
if (eq(flp->units[i].bottomy + flp->units[i].height, l_chip)) {
gn_sp += gy_sp[i];
gn_hs += gy_hs[i];
border[i][2] = 1; /* block is on northern border */
} else
border[i][2] = 0;
if (eq(flp->units[i].bottomy, 0)) {
gs_sp += gy_sp[i];
gs_hs += gy_hs[i];
border[i][3] = 1; /* block is on southern border */
} else
border[i][3] = 0;
if (eq(flp->units[i].leftx + flp->units[i].width, w_chip)) {
ge_sp += gx_sp[i];
ge_hs += gx_hs[i];
border[i][1] = 1; /* block is on eastern border */
} else
border[i][1] = 0;
if (eq(flp->units[i].leftx, 0)) {
gw_sp += gx_sp[i];
gw_hs += gx_hs[i];
border[i][0] = 1; /* block is on western border */
} else
border[i][0] = 0;
}
/* initialize g */
zero_dmatrix(g, NL*n+EXTRA, NL*n+EXTRA);
zero_dvector(g_amb, n+EXTRA);
/* overall Rs between nodes */
for (i = 0; i < n; i++) {
double area = (flp->units[i].height * flp->units[i].width);
/* amongst functional units in the various layers */
for (j = 0; j < n; j++) {
double part = 0, part_int = 0, part_sp = 0, part_hs = 0;
if (is_horiz_adj(flp, i, j)) {
part = gx[i] / flp->units[i].height;
part_int = gx_int[i] / flp->units[i].height;
part_sp = gx_sp[i] / flp->units[i].height;
part_hs = gx_hs[i] / flp->units[i].height;
}
else if (is_vert_adj(flp, i,j)) {
part = gy[i] / flp->units[i].width;
part_int = gy_int[i] / flp->units[i].width;
part_sp = gy_sp[i] / flp->units[i].width;
part_hs = gy_hs[i] / flp->units[i].width;
}
g[i][j] = part * len[i][j];
g[IFACE*n+i][IFACE*n+j] = part_int * len[i][j];
g[HSP*n+i][HSP*n+j] = part_sp * len[i][j];
g[HSINK*n+i][HSINK*n+j] = part_hs * len[i][j];
}
/* the 2.0 factor in the following equations is
* explained during the calculation of the B matrix
*/
/* vertical g's in the silicon layer */
g[i][IFACE*n+i]=g[IFACE*n+i][i]=2.0/getr(k_chip, t_chip, area);
/* vertical g's in the interface layer */
g[IFACE*n+i][HSP*n+i]=g[HSP*n+i][IFACE*n+i]=2.0/getr(k_interface, t_interface, area);
/* vertical g's in the spreader layer */
g[HSP*n+i][HSINK*n+i]=g[HSINK*n+i][HSP*n+i]=2.0/getr(k_spreader, t_spreader, area);
/* vertical g's in the heatsink core layer */
/* vertical R to ambient: divide r_convec proportional to area */
r_amb = r_convec * (s_sink * s_sink) / area;
g_amb[i] = 1.0 / (getr(k_sink, t_sink, area) + r_amb);
/* lateral g's from block center (spreader layer) to peripheral (n,s,e,w) spreader nodes */
g[HSP*n+i][NL*n+SP_N]=g[NL*n+SP_N][HSP*n+i]=2.0*border[i][2] /
((1.0/gy_sp[i])+model->pack.r_sp1_y*gn_sp/gy_sp[i]);
g[HSP*n+i][NL*n+SP_S]=g[NL*n+SP_S][HSP*n+i]=2.0*border[i][3] /
((1.0/gy_sp[i])+model->pack.r_sp1_y*gs_sp/gy_sp[i]);
g[HSP*n+i][NL*n+SP_E]=g[NL*n+SP_E][HSP*n+i]=2.0*border[i][1] /
((1.0/gx_sp[i])+model->pack.r_sp1_x*ge_sp/gx_sp[i]);
g[HSP*n+i][NL*n+SP_W]=g[NL*n+SP_W][HSP*n+i]=2.0*border[i][0] /
((1.0/gx_sp[i])+model->pack.r_sp1_x*gw_sp/gx_sp[i]);
/* lateral g's from block center (heatsink layer) to peripheral (n,s,e,w) heatsink nodes */
g[HSINK*n+i][NL*n+SINK_C_N]=g[NL*n+SINK_C_N][HSINK*n+i]=2.0*border[i][2] /
((1.0/gy_hs[i])+model->pack.r_hs1_y*gn_hs/gy_hs[i]);
g[HSINK*n+i][NL*n+SINK_C_S]=g[NL*n+SINK_C_S][HSINK*n+i]=2.0*border[i][3] /
((1.0/gy_hs[i])+model->pack.r_hs1_y*gs_hs/gy_hs[i]);
g[HSINK*n+i][NL*n+SINK_C_E]=g[NL*n+SINK_C_E][HSINK*n+i]=2.0*border[i][1] /
((1.0/gx_hs[i])+model->pack.r_hs1_x*ge_hs/gx_hs[i]);
g[HSINK*n+i][NL*n+SINK_C_W]=g[NL*n+SINK_C_W][HSINK*n+i]=2.0*border[i][0] /
((1.0/gx_hs[i])+model->pack.r_hs1_x*gw_hs/gx_hs[i]);
}
/* g's from peripheral(n,s,e,w) nodes */
/* vertical g's between peripheral spreader nodes and center peripheral heatsink nodes */
g[NL*n+SP_N][NL*n+SINK_C_N]=g[NL*n+SINK_C_N][NL*n+SP_N]=2.0/model->pack.r_sp_per_y;
g[NL*n+SP_S][NL*n+SINK_C_S]=g[NL*n+SINK_C_S][NL*n+SP_S]=2.0/model->pack.r_sp_per_y;
g[NL*n+SP_E][NL*n+SINK_C_E]=g[NL*n+SINK_C_E][NL*n+SP_E]=2.0/model->pack.r_sp_per_x;
g[NL*n+SP_W][NL*n+SINK_C_W]=g[NL*n+SINK_C_W][NL*n+SP_W]=2.0/model->pack.r_sp_per_x;
/* lateral g's between peripheral outer sink nodes and center peripheral sink nodes */
g[NL*n+SINK_C_N][NL*n+SINK_N]=g[NL*n+SINK_N][NL*n+SINK_C_N]=2.0/(model->pack.r_hs + model->pack.r_hs2_y);
g[NL*n+SINK_C_S][NL*n+SINK_S]=g[NL*n+SINK_S][NL*n+SINK_C_S]=2.0/(model->pack.r_hs + model->pack.r_hs2_y);
g[NL*n+SINK_C_E][NL*n+SINK_E]=g[NL*n+SINK_E][NL*n+SINK_C_E]=2.0/(model->pack.r_hs + model->pack.r_hs2_x);
g[NL*n+SINK_C_W][NL*n+SINK_W]=g[NL*n+SINK_W][NL*n+SINK_C_W]=2.0/(model->pack.r_hs + model->pack.r_hs2_x);
/* vertical g's between inner peripheral sink nodes and ambient */
g_amb[n+SINK_C_N] = g_amb[n+SINK_C_S] = 1.0 / (model->pack.r_hs_c_per_y+model->pack.r_amb_c_per_y);
g_amb[n+SINK_C_E] = g_amb[n+SINK_C_W] = 1.0 / (model->pack.r_hs_c_per_x+model->pack.r_amb_c_per_x);
/* vertical g's between outer peripheral sink nodes and ambient */
g_amb[n+SINK_N] = g_amb[n+SINK_S] = g_amb[n+SINK_E] =
g_amb[n+SINK_W] = 1.0 / (model->pack.r_hs_per+model->pack.r_amb_per);
/* calculate matrix B such that BT = POWER in steady state */
/* non-diagonal elements */
for (i = 0; i < NL*n+EXTRA; i++)
for (j = 0; j < i; j++)
if ((g[i][j] == 0.0) || (g[j][i] == 0.0))
b[i][j] = b[j][i] = 0.0;
else
/* here is why the 2.0 factor comes when calculating g[][] */
b[i][j] = b[j][i] = -1.0/((1.0/g[i][j])+(1.0/g[j][i]));
/* diagonal elements */
for (i = 0; i < NL*n+EXTRA; i++) {
/* functional blocks in the heat sink layer */
if (i >= HSINK*n && i < NL*n)
b[i][i] = g_amb[i%n];
/* heat sink peripheral nodes */
else if (i >= NL*n+SINK_C_W)
b[i][i] = g_amb[n+i-NL*n];
/* all other nodes that are not connected to the ambient */
else
b[i][i] = 0.0;
/* sum up the conductances */
for(j=0; j < NL*n+EXTRA; j++)
if (i != j)
b[i][i] -= b[i][j];
}
/* compute the LUP decomposition of B and store it too */
copy_dmatrix(lu, b, NL*n+EXTRA, NL*n+EXTRA);
/*
* B is a symmetric positive definite matrix. It is
* symmetric because if a node A is connected to B,
* then B is also connected to A with the same R value.
* It is positive definite because of the following
* informal argument from Professor Lieven Vandenberghe's
* lecture slides for the spring 2004-2005 EE 103 class
* at UCLA: http://www.ee.ucla.edu/~vandenbe/103/chol.pdf
* x^T*B*x = voltage^T * (B*x) = voltage^T * current
* = total power dissipated in the resistors > 0
* for x != 0.
*/
lupdcmp(lu, NL*n+EXTRA, p, 1);
/* done */
model->flp = flp;
model->r_ready = TRUE;
}
/* creates 2 matrices: invA, C: dT + A^-1*BT = A^-1*Power,
* C = A^-1 * B. note that A is a diagonal matrix (no lateral
* capacitances. all capacitances are to ground). also note that
* it is stored as a 1-d vector. so, for computing the inverse,
* inva[i] = 1/a[i] is just enough.
*/
void populate_C_model_block(block_model_t *model, flp_t *flp)
{
/* shortcuts */
double *inva = model->inva, **c = model->c;
double **b = model->b;
double *a = model->a;
double t_chip = model->config.t_chip;
double c_convec = model->config.c_convec;
double s_sink = model->config.s_sink;
double t_sink = model->config.t_sink;
double t_spreader = model->config.t_spreader;
double t_interface = model->config.t_interface;
double p_chip = model->config.p_chip;
double p_sink = model->config.p_sink;
double p_spreader = model->config.p_spreader;
double p_interface = model->config.p_interface;
double c_amb;
double w_chip, l_chip;
int i, n = flp->n_units;
if (!model->r_ready)
fatal("R model not ready\n");
if (model->flp != flp || model->n_units != flp->n_units ||
model->n_nodes != NL * flp->n_units + EXTRA)
fatal("different floorplans for R and C models!\n");
w_chip = get_total_width (flp); /* x-axis */
l_chip = get_total_height (flp); /* y-axis */
/* package C's */
populate_package_C(&model->pack, &model->config, w_chip, l_chip);
/* functional block C's */
for (i = 0; i < n; i++) {
double area = (flp->units[i].height * flp->units[i].width);
/* C's from functional units to ground */
a[i] = getcap(p_chip, t_chip, area);
/* C's from interface portion of the functional units to ground */
a[IFACE*n+i] = getcap(p_interface, t_interface, area);
/* C's from spreader portion of the functional units to ground */
a[HSP*n+i] = getcap(p_spreader, t_spreader, area);
/* C's from heatsink portion of the functional units to ground */
/* vertical C to ambient: divide c_convec proportional to area */
c_amb = C_FACTOR * c_convec / (s_sink * s_sink) * area;
a[HSINK*n+i] = getcap(p_sink, t_sink, area) + c_amb;
}
/* C's from peripheral(n,s,e,w) nodes */
/* from peripheral spreader nodes to ground */
a[NL*n+SP_N] = a[NL*n+SP_S] = model->pack.c_sp_per_y;
a[NL*n+SP_E] = a[NL*n+SP_W] = model->pack.c_sp_per_x;
/* from center peripheral sink nodes to ground
* NOTE: this treatment of capacitances (and
* the corresponding treatment of resistances
* in populate_R_model) as parallel (series)
* is only approximate and is done in order
* to avoid creating an extra layer of nodes
*/
a[NL*n+SINK_C_N] = a[NL*n+SINK_C_S] = model->pack.c_hs_c_per_y +
model->pack.c_amb_c_per_y;
a[NL*n+SINK_C_E] = a[NL*n+SINK_C_W] = model->pack.c_hs_c_per_x +
model->pack.c_amb_c_per_x;
/* from outer peripheral sink nodes to ground */
a[NL*n+SINK_N] = a[NL*n+SINK_S] = a[NL*n+SINK_E] = a[NL*n+SINK_W] =
model->pack.c_hs_per + model->pack.c_amb_per;
/* calculate A^-1 (for diagonal matrix A) such that A(dT) + BT = POWER */
for (i = 0; i < NL*n+EXTRA; i++)
inva[i] = 1.0/a[i];
/* we are always going to use the eqn dT + A^-1 * B T = A^-1 * POWER. so, store C = A^-1 * B */
diagmatmult(c, inva, b, NL*n+EXTRA);
/* done */
model->c_ready = TRUE;
}
/* creates 3 matrices: invA, B, C: dT + A^-1*BT = A^-1*Power,
* C = A^-1 * B. note that A is a diagonal matrix (no lateral
* capacitances. all capacitances are to ground). so, inva[i][i]
* (= 1/a[i][i]) is just enough.
*
* NOTE: EXTRA nodes: 1 chip bottom, 5 spreader and 5 heat sink nodes
* (north, south, east, west and bottom).
*/
void create_RC_matrices(flp_t *flp, int omit_lateral)
{
int i, j, k = 0, n = flp->n_units;
int **border;
double **len, *gx, *gy, **g, *c_ver, **t;
double r_sp1, r_sp2, r_hs; /* lateral resistances to spreader and heatsink */
/* NOTE: *_mid - the vertical R/C from center nodes of spreader
* and heatsink. *_ver - the vertical R/C from peripheral (n,s,e,w) nodes
*/
double r_sp_mid, r_sp_ver, r_hs_mid, r_hs_ver, c_sp_mid, c_sp_ver, c_hs_mid, c_hs_ver;
double gn=0, gs=0, ge=0, gw=0;
double w_chip = get_total_width (flp); /* x-axis */
double l_chip = get_total_height (flp); /* y-axis */
FILE *fp_b,*fp_c,*fp_inva,*fp_invb;
fp_b=fopen("B","w");
fp_c=fopen("C","w");
fp_invb=fopen("invB","w");
fp_inva=fopen("invA","w");
border = imatrix(n, 4);
len = matrix(n, n); /* len[i][j] = length of shared edge bet. i & j */
gx = vector(n); /* lumped conductances in x direction */
gy = vector(n); /* lumped conductances in y direction */
g = matrix(n+EXTRA, n+EXTRA); /* g[i][j] = conductance bet. nodes i & j */
c_ver = vector(n+EXTRA); /* vertical capacitance */
b = matrix(n+EXTRA, n+EXTRA); /* B, C, INVA and INVB are (n+EXTRA)x(n+EXTRA) matrices */
c = matrix(n+EXTRA, n+EXTRA);
inva = matrix(n+EXTRA, n+EXTRA);
invb = matrix(n+EXTRA, n+EXTRA);
t = matrix (n+EXTRA, n+EXTRA); /* copy of B */
/* compute the silicon fitting factor - see pg 10 of the UVA CS tech report - CS-TR-2003-08 */
factor_chip = C_FACTOR * ((SPEC_HEAT_CU / SPEC_HEAT_SI) * (w_chip + 0.88 * t_spreader) \
* (l_chip + 0.88 * t_spreader) * t_spreader / ( w_chip * l_chip * t_chip) + 1);
/* gx's and gy's of blocks */
for (i = 0; i < n; i++) {
gx[i] = 1.0/getr(K_SI, flp->units[i].height, flp->units[i].width, l_chip);
gy[i] = 1.0/getr(K_SI, flp->units[i].width, flp->units[i].height, w_chip);
}
/* shared lengths between blocks */
for (i = 0; i < n; i++)
for (j = i; j < n; j++)
len[i][j] = len[j][i] = get_shared_len(flp, i, j);
/* lateral R's of spreader and sink */
r_sp1 = getr(K_CU, (s_spreader+3*w_chip)/4.0, (s_spreader-w_chip)/4.0, w_chip);
r_sp2 = getr(K_CU, (3*s_spreader+w_chip)/4.0, (s_spreader-w_chip)/4.0, (s_spreader+3*w_chip)/4.0);
r_hs = getr(K_CU, (s_sink+3*s_spreader)/4.0, (s_sink-s_spreader)/4.0, s_spreader);
/* vertical R's and C's of spreader and sink */
r_sp_mid = RHO_CU * t_spreader / (w_chip * l_chip);
c_sp_mid = factor_pack * SPEC_HEAT_CU * t_spreader * (w_chip * l_chip);
r_sp_ver = RHO_CU * t_spreader * 4.0 / (s_spreader * s_spreader - w_chip*l_chip);
c_sp_ver = factor_pack * SPEC_HEAT_CU * t_spreader * (s_spreader * s_spreader - w_chip*l_chip) / 4.0;
r_hs_mid = RHO_CU * t_sink / (s_spreader*s_spreader);
c_hs_mid = factor_pack * SPEC_HEAT_CU * t_sink * (s_spreader * s_spreader);
r_hs_ver = RHO_CU * t_sink * 4.0 / (s_sink * s_sink - s_spreader*s_spreader);
c_hs_ver = factor_pack * SPEC_HEAT_CU * t_sink * (s_sink * s_sink - s_spreader*s_spreader) / 4.0;
/* short the R's from block centers to a particular chip edge */
for (i = 0; i < n; i++) {
if (eq(flp->units[i].bottomy + flp->units[i].height, l_chip)) {
gn += gy[i];
border[i][2] = 1; /* block is on northern border */
}
if (eq(flp->units[i].bottomy, 0)) {
gs += gy[i];
border[i][3] = 1; /* block is on southern border */
}
if (eq(flp->units[i].leftx + flp->units[i].width, w_chip)) {
ge += gx[i];
border[i][1] = 1; /* block is on eastern border */
}
if (eq(flp->units[i].leftx, 0)) {
gw += gx[i];
border[i][0] = 1; /* block is on western border */
}
}
/* overall R and C between nodes */
for (i = 0; i < n; i++) {
/* amongst functional units */
for (j = 0; j < n; j++) {
double part = 0;
if (!omit_lateral) {
if (is_horiz_adj(flp, i, j)){
part = gx[i] / flp->units[i].height;
printf("%d %d horiz adj\n",i,j);
}
else if (is_vert_adj(flp, i,j)) {
part = gy[i] / flp->units[i].width;
printf("%d %d vert adj\n",i,j);
}
}
g[i][j] = part * len[i][j];
}
/* C's from functional units to ground */
c_ver[i] = factor_chip * SPEC_HEAT_SI * t_chip * flp->units[i].height * flp->units[i].width;
/* lateral g's from block center to peripheral (n,s,e,w) spreader nodes */
g[i][n+SP_N]=g[n+SP_N][i]=2.0*border[i][2]/((1.0/gy[i])+r_sp1*gn/gy[i]);
g[i][n+SP_S]=g[n+SP_S][i]=2.0*border[i][3]/((1.0/gy[i])+r_sp1*gs/gy[i]);
g[i][n+SP_E]=g[n+SP_E][i]=2.0*border[i][1]/((1.0/gx[i])+r_sp1*ge/gx[i]);
g[i][n+SP_W]=g[n+SP_W][i]=2.0*border[i][0]/((1.0/gx[i])+r_sp1*gw/gx[i]);
/* vertical g's from block center to chip bottom */
g[i][n+CHIP_B]=g[n+CHIP_B][i]=2.0/(RHO_SI * t_chip / (flp->units[i].height * flp->units[i].width));
}
/* max slope (1/vertical RC time constant) for silicon */
max_slope = 1.0 / (factor_chip * t_chip * t_chip * RHO_SI * SPEC_HEAT_SI);
/* vertical g's and C's between central nodes */
/* between chip bottom and spreader bottom */
g[n+CHIP_B][n+SP_B]=g[n+SP_B][n+CHIP_B]=2.0/r_sp_mid;
/* from chip bottom to ground */
c_ver[n+CHIP_B]=c_sp_mid;
/* between spreader bottom and sink bottom */
g[n+SINK_B][n+SP_B]=g[n+SP_B][n+SINK_B]=2.0/r_hs_mid;
/* from spreader bottom to ground */
c_ver[n+SP_B]=c_hs_mid;
/* from sink bottom to ground */
c_ver[n+SINK_B]=c_convec;
/* g's and C's from peripheral(n,s,e,w) nodes */
for (i = 1; i <= 4; i++) {
/* vertical g's between peripheral spreader nodes and spreader bottom */
g[n+SP_B-i][n+SP_B]=g[n+SP_B][n+SP_B-i]=2.0/r_sp_ver;
/* lateral g's between peripheral spreader nodes and peripheral sink nodes */
g[n+SP_B-i][n+SINK_B-i]=g[n+SINK_B-i][n+SP_B-i]=2.0/(r_hs + r_sp2);
/* vertical g's between peripheral sink nodes and sink bottom */
g[n+SINK_B-i][n+SINK_B]=g[n+SINK_B][n+SINK_B-i]=2.0/r_hs_ver;
/* from peripheral spreader nodes to ground */
c_ver[n+SP_B-i]=c_sp_ver;
/* from peripheral sink nodes to ground */
c_ver[n+SINK_B-i]=c_hs_ver;
}
/* calculate matrices A, B such that A(dT) + BT = POWER */
for (i = 0; i < n+EXTRA; i++) {
for (j = 0; j < n+EXTRA; j++) {
if (i==j) {
inva[i][j] = 1.0/c_ver[i];
if (i == n+SINK_B) /* sink bottom */
b[i][j] += 1.0 / r_convec;
for (k = 0; k < n+EXTRA; k++) {
if ((g[i][k]==0.0)||(g[k][i])==0.0)
continue;
else
/* here is why the 2.0 factor comes when calculating g[][] */
b[i][j] += 1.0/((1.0/g[i][k])+(1.0/g[k][i]));
}
} else {
inva[i][j]=0.0;
if ((g[i][j]==0.0)||(g[j][i])==0.0)
b[i][j]=0.0;
else
b[i][j]=-1.0/((1.0/g[i][j])+(1.0/g[j][i]));
}
}
}
/* we are always going to use the eqn dT + A^-1 * B T = A^-1 * POWER. so, store C = A^-1 * B */
matmult(c, inva, b, n+EXTRA);
/* we will also be needing INVB so store it too */
copy_matrix(t, b, n+EXTRA, n+EXTRA);
matinv(invb, t, n+EXTRA);
for (i = 0; i < n+EXTRA; i++) {
for (j = 0; j < n+EXTRA; j++) {
fprintf(fp_inva,"%f ",inva[i][j]);
fprintf(fp_invb,"%f ",invb[i][j]);
fprintf(fp_c,"%f ",c[i][j]);
fprintf(fp_b,"%f ",b[i][j]);
}
fprintf(fp_inva, "\n");
fprintf(fp_invb, "\n");
fprintf(fp_c , "\n");
fprintf(fp_b, "\n");
}
fclose(fp_inva);
fclose(fp_b);
fclose(fp_c);
fclose(fp_invb);
/* dump_vector(c_ver, n+EXTRA); */
/* dump_matrix(invb, n+EXTRA, n+EXTRA); */
/* dump_matrix(c, n+EXTRA, n+EXTRA); */
/* cleanup */
free_matrix(t, n+EXTRA);
free_matrix(g, n+EXTRA);
free_matrix(len, n);
free_imatrix(border, n);
free_vector(c_ver);
free_vector(gx);
free_vector(gy);
}