X-Git-Url: http://info.iut-bm.univ-fcomte.fr/pub/gitweb/simgrid.git/blobdiff_plain/5707630f4aaebdc50607bd6896677388eb373499..493fc5668a6e4819bd820f86093ca3a7f0909d5a:/src/surf/lagrange.c

diff --git a/src/surf/lagrange.c b/src/surf/lagrange.c
index e656c701ed..008c2108c5 100644
--- a/src/surf/lagrange.c
+++ b/src/surf/lagrange.c
@@ -17,61 +17,115 @@
 #include <math.h>
 #endif
 
+#define VEGAS_SCALING 1000.0
 
-XBT_LOG_NEW_DEFAULT_SUBCATEGORY(surf_lagrange, surf, "Logging specific to SURF (lagrange)");
+
+XBT_LOG_NEW_DEFAULT_SUBCATEGORY(surf_lagrange, surf,
+				"Logging specific to SURF (lagrange)");
+XBT_LOG_NEW_SUBCATEGORY(surf_lagrange_dichotomy, surf,
+			"Logging specific to SURF (lagrange dichotomy)");
 
 /*
- * Local prototypes to implement the lagrangian optimization with optimal step, also called dicotomi.
+ * Local prototypes to implement the lagrangian optimization with optimal step, also called dichotomy.
  */
-//solves the proportional fairness using a lagrange optimizition with dicotomi step
-void   lagrange_solve       (lmm_system_t sys);
-//computes the value of the dicotomi using a initial values, init, with a specific variable or constraint
-double dicotomi(double init, double diff(double, void*), void *var_cnst, double min_error);
+//solves the proportional fairness using a lagrange optimizition with dichotomy step
+void lagrange_solve(lmm_system_t sys);
+//computes the value of the dichotomy using a initial values, init, with a specific variable or constraint
+double dichotomy(double init, double diff(double, void *), void *var_cnst,
+		 double min_error);
 //computes the value of the differential of variable param_var applied to mu  
-double partial_diff_mu      (double mu, void * param_var);
+double partial_diff_mu(double mu, void *param_var);
 //computes the value of the differential of constraint param_cnst applied to lambda  
-double partial_diff_lambda  (double lambda, void * param_cnst);
+double partial_diff_lambda(double lambda, void *param_cnst);
 //auxiliar function to compute the partial_diff
 double diff_aux(lmm_variable_t var, double x);
 
 
+static int __check_kkt(xbt_swag_t cnst_list, xbt_swag_t var_list, int warn)
+{
+  xbt_swag_t elem_list = NULL;
+  lmm_element_t elem = NULL;
+  lmm_constraint_t cnst = NULL;
+  lmm_variable_t var = NULL;
+
+  double tmp;
+
+  //verify the KKT property for each link
+  xbt_swag_foreach(cnst, cnst_list) {
+    tmp = 0;
+    elem_list = &(cnst->element_set);
+    xbt_swag_foreach(elem, elem_list) {
+      var = elem->variable;
+      if (var->weight <= 0)
+	continue;
+      tmp += var->value;
+    }
+
+    if (double_positive(tmp - cnst->bound)) {
+      if (warn)
+	WARN3
+	    ("The link (%p) is over-used. Expected less than %f and got %f",
+	     cnst, cnst->bound, tmp);
+      return 0;
+    }
+    DEBUG3("Checking KKT for constraint (%p): sat = %f, lambda = %f ",
+	   cnst, tmp - cnst->bound, cnst->lambda);
+  }
+
+  //verify the KKT property of each flow
+  xbt_swag_foreach(var, var_list) {
+    if (var->bound < 0 || var->weight <= 0)
+      continue;
+    DEBUG3("Checking KKT for variable (%p): sat = %f mu = %f", var,
+	   var->value - var->bound, var->mu);
+
+    if (double_positive(var->value - var->bound)) {
+      if (warn)
+	WARN3
+	    ("The variable (%p) is too large. Expected less than %f and got %f",
+	     var, var->bound, var->value);
+      return 0;
+    }
+  }
+  return 1;
+}
+
 void lagrange_solve(lmm_system_t sys)
 {
   /*
    * Lagrange Variables.
    */
-  int max_iterations= 10000;
-  double epsilon_min_error  = 1e-6;
-  double dicotomi_min_error = 1e-6;
+  int max_iterations = 100;
+  double epsilon_min_error = MAXMIN_PRECISION;
+  double dichotomy_min_error = 1e-20;
   double overall_error = 1;
 
   /*
    * Variables to manipulate the data structure proposed to model the maxmin
    * fairness. See docummentation for more details.
    */
-  xbt_swag_t elem_list  = NULL;
-  lmm_element_t elem    = NULL;
-
-  xbt_swag_t cnst_list  = NULL;
+  xbt_swag_t cnst_list = NULL;
   lmm_constraint_t cnst = NULL;
-  
-  xbt_swag_t var_list   = NULL;
-  lmm_variable_t var    = NULL;
+
+  xbt_swag_t var_list = NULL;
+  lmm_variable_t var = NULL;
 
   /*
    * Auxiliar variables.
    */
-  int iteration=0;
-  double tmp=0;
+  int iteration = 0;
+  double tmp = 0;
   int i;
-   
+
 
   DEBUG0("Iterative method configuration snapshot =====>");
   DEBUG1("#### Maximum number of iterations       : %d", max_iterations);
-  DEBUG1("#### Minimum error tolerated            : %e", epsilon_min_error);  
-  DEBUG1("#### Minimum error tolerated (dicotomi) : %e", dicotomi_min_error);
+  DEBUG1("#### Minimum error tolerated            : %e",
+	 epsilon_min_error);
+  DEBUG1("#### Minimum error tolerated (dichotomy) : %e",
+	 dichotomy_min_error);
 
-  if ( !(sys->modified))
+  if (!(sys->modified))
     return;
 
   /* 
@@ -79,46 +133,52 @@ void lagrange_solve(lmm_system_t sys)
    * Associate an index in the swag variables. Initialize mu.
    */
   var_list = &(sys->variable_set);
-  i=0;
-  xbt_swag_foreach(var, var_list) {    
-    if((var->bound > 0.0) || (var->weight <= 0.0)){
-      DEBUG1("#### NOTE var(%d) is a boundless variable", i);
+  i = 0;
+  xbt_swag_foreach(var, var_list) {
+    if ((var->bound < 0.0) || (var->weight <= 0.0)) {
+      DEBUG1("#### NOTE var(%d) is a boundless (or inactive) variable", i);
       var->mu = -1.0;
-    } else{ 
-      var->mu =   1.0;
+    } else {
+      var->mu = 1.0;
       var->new_mu = 2.0;
     }
-    DEBUG2("#### var(%d)->mu :  %e", i, var->mu);
-    DEBUG2("#### var(%d)->weight: %e", i, var->weight);
+    DEBUG3("#### var(%d) %p ->mu :  %e", i, var, var->mu);
+    DEBUG3("#### var(%d) %p ->weight: %e", i, var, var->weight);
+    DEBUG3("#### var(%d) %p ->bound: %e", i, var, var->bound);
     i++;
   }
 
   /* 
    * Initialize lambda.
    */
-  cnst_list=&(sys->active_constraint_set); 
-  xbt_swag_foreach(cnst, cnst_list){
+  cnst_list = &(sys->active_constraint_set);
+  xbt_swag_foreach(cnst, cnst_list) {
     cnst->lambda = 1.0;
     cnst->new_lambda = 2.0;
     DEBUG2("#### cnst(%p)->lambda :  %e", cnst, cnst->lambda);
   }
-  
+
   /*
    * While doesn't reach a minimun error or a number maximum of iterations.
    */
-  while(overall_error > epsilon_min_error && iteration < max_iterations){
-   
-    iteration++;
-    DEBUG1("************** ITERATION %d **************", iteration);    
+  while (overall_error > epsilon_min_error && iteration < max_iterations) {
+    int dual_updated=0;
 
+    iteration++;
+    DEBUG1("************** ITERATION %d **************", iteration);
+    DEBUG0("-------------- Gradient Descent ----------");
     /*                       
      * Compute the value of mu_i
      */
     //forall mu_i in mu_1, mu_2, ..., mu_n
     xbt_swag_foreach(var, var_list) {
-      if((var->bound >= 0) && (var->weight > 0) ){
-	var->new_mu = dicotomi(var->mu, partial_diff_mu, var, dicotomi_min_error);
-	if(var->new_mu < 0) var->new_mu = 0;
+      if ((var->bound >= 0) && (var->weight > 0)) {
+	DEBUG1("Working on var (%p)", var);
+	var->new_mu =
+	    dichotomy(var->mu, partial_diff_mu, var, dichotomy_min_error);
+	dual_updated += (fabs(var->new_mu-var->mu)>dichotomy_min_error);
+	DEBUG2("dual_updated (%d) : %1.20f",dual_updated,fabs(var->new_mu-var->mu));
+	DEBUG3("Updating mu : var->mu (%p) : %1.20f -> %1.20f", var, var->mu, var->new_mu);
 	var->mu = var->new_mu;
       }
     }
@@ -128,8 +188,13 @@ void lagrange_solve(lmm_system_t sys)
      */
     //forall lambda_i in lambda_1, lambda_2, ..., lambda_n
     xbt_swag_foreach(cnst, cnst_list) {
-      cnst->new_lambda = dicotomi(cnst->lambda, partial_diff_lambda, cnst, dicotomi_min_error);
-      DEBUG2("====> cnst->lambda (%p) = %e", cnst, cnst->new_lambda);      
+      DEBUG1("Working on cnst (%p)", cnst);
+      cnst->new_lambda =
+	  dichotomy(cnst->lambda, partial_diff_lambda, cnst,
+		    dichotomy_min_error);
+      dual_updated += (fabs(cnst->new_lambda-cnst->lambda)>dichotomy_min_error);
+      DEBUG2("dual_updated (%d) : %1.20f",dual_updated,fabs(cnst->new_lambda-cnst->lambda));
+      DEBUG3("Updating lambda : cnst->lambda (%p) : %1.20f -> %1.20f", cnst, cnst->lambda, cnst->new_lambda);
       cnst->lambda = cnst->new_lambda;
     }
 
@@ -137,74 +202,69 @@ void lagrange_solve(lmm_system_t sys)
      * Now computes the values of each variable (\rho) based on
      * the values of \lambda and \mu.
      */
-    overall_error=0;   
+    DEBUG0("-------------- Check convergence ----------");
+    overall_error = 0;
     xbt_swag_foreach(var, var_list) {
-      if(var->weight <=0) 
+      if (var->weight <= 0)
 	var->value = 0.0;
       else {
 	//compute sigma_i + mu_i
 	tmp = 0;
-	for(i=0; i<var->cnsts_number; i++){
+	for (i = 0; i < var->cnsts_number; i++) {
 	  tmp += (var->cnsts[i].constraint)->lambda;
-	  if(var->bound > 0) 
-	    tmp+=var->mu;
 	}
+	if (var->bound > 0)
+	  tmp += var->mu;
+	DEBUG3("\t Working on var (%p). cost = %e; Df = %e", var, tmp,
+	       var->df);
 
 	//uses the partial differential inverse function
 	tmp = var->func_fpi(var, tmp);
 
 	//computes de overall_error using normalized value
-	if(overall_error <  (fabs(var->value - tmp)/tmp) ){
+	if (overall_error < (fabs(var->value - tmp)/tmp)) {
 	  overall_error = (fabs(var->value - tmp)/tmp);
 	}
-	
-	var->value = tmp;
-      }
-      DEBUG3("======> value of var (%p)  = %e, overall_error = %e", var, var->value, overall_error);       
-    }
-  }
 
+	if (overall_error < (fabs(var->value - tmp))) {
+	  overall_error = (fabs(var->value - tmp));
+	}
 
-  //verify the KKT property for each link
-  xbt_swag_foreach(cnst, cnst_list){
-    tmp = 0;
-    elem_list = &(cnst->element_set);
-    xbt_swag_foreach(elem, elem_list) {
-      var = elem->variable;
-      if(var->weight<=0) continue;
-      tmp += var->value;
+	var->value = tmp;
+	DEBUG3("New value of var (%p)  = %e, overall_error = %e", var,
+	       var->value, overall_error);
+      }
     }
-  
-    tmp = tmp - cnst->bound;
 
-    if(tmp > epsilon_min_error){
-      WARN3("The link (%p) doesn't match the KKT property, expected less than %e and got %e", cnst, epsilon_min_error, tmp);
+    if (!__check_kkt(cnst_list, var_list, 0))
+      overall_error = 1.0;
+    DEBUG2("Iteration %d: Overall_error : %f", iteration, overall_error);
+    if(!dual_updated) {
+      DEBUG1("Could not improve the convergence at iteration %d. Drop it!",iteration);
+      break;
     }
-  
   }
-  
-  //verify the KKT property of each flow
-  xbt_swag_foreach(var, var_list){
-    if(var->bound <= 0 || var->weight <= 0) continue;
-    tmp = 0;
-    tmp = (var->value - var->bound);
 
-    
-    if(tmp != 0.0 ||  var->mu != 0.0){
-      WARN3("The flow (%p) doesn't match the KKT property, value expected (=0) got (lambda=%e) (sum_rho=%e)", var, var->mu, tmp);
-    }
 
+  __check_kkt(cnst_list, var_list, 1);
+
+  if (overall_error <= epsilon_min_error) {
+    DEBUG1("The method converges in %d iterations.", iteration);
+  }
+  if (iteration >= max_iterations) {
+    DEBUG1
+	("Method reach %d iterations, which is the maximum number of iterations allowed.",
+	 iteration);
   }
+/*   INFO1("Method converged after %d iterations", iteration); */
 
-  if(overall_error <= epsilon_min_error){
-    DEBUG1("The method converge in %d iterations.", iteration);
-  }else{
-    WARN1("Method reach %d iterations, which is the maxmimun number of iterations allowed.", iteration);
+  if (XBT_LOG_ISENABLED(surf_lagrange, xbt_log_priority_debug)) {
+    lmm_print(sys);
   }
 }
 
 /*
- * Returns a double value corresponding to the result of a dicotomi proccess with
+ * Returns a double value corresponding to the result of a dichotomy proccess with
  * respect to a given variable/constraint (\mu in the case of a variable or \lambda in
  * case of a constraint) and a initial value init. 
  *
@@ -213,174 +273,227 @@ void lagrange_solve(lmm_system_t sys)
  * @param var_cnst a pointer to a variable or constraint 
  * @param min_erro a minimun error tolerated
  *
- * @return a double correponding to the result of the dicotomial process
+ * @return a double correponding to the result of the dichotomyal process
  */
-double dicotomi(double init, double diff(double, void*), void *var_cnst, double min_error){
+double dichotomy(double init, double diff(double, void *), void *var_cnst,
+		 double min_error)
+{
   double min, max;
   double overall_error;
   double middle;
   double min_diff, max_diff, middle_diff;
-  
+  double diff_0 = 0.0;
   min = max = init;
 
-  if(init == 0){
-    min = max = 1;
+  XBT_IN;
+
+  if (init == 0.0) {
+    min = max = 0.5;
   }
 
   min_diff = max_diff = middle_diff = 0.0;
   overall_error = 1;
 
-  if(diff(0.0, var_cnst) > 0){
-    DEBUG1("====> returning 0.0 (diff = %e)", diff(0.0, var_cnst));
+  if ((diff_0 = diff(1e-16, var_cnst)) >= 0) {
+    CDEBUG1(surf_lagrange_dichotomy, "returning 0.0 (diff = %e)",
+	    diff_0);
+    XBT_OUT;
     return 0.0;
   }
 
-  DEBUG0("====> not detected positive diff in 0");
+  min_diff = diff(min, var_cnst);
+  max_diff = diff(max, var_cnst);
 
-  while(overall_error > min_error){
+  while (overall_error > min_error) {
+    CDEBUG4(surf_lagrange_dichotomy,
+	    "[min, max] = [%1.20f, %1.20f] || diffmin, diffmax = %1.20f, %1.20f", min, max,
+	    min_diff,max_diff);
 
-    min_diff = diff(min, var_cnst);
-    max_diff = diff(max, var_cnst);
-
-    DEBUG2("DICOTOMI ===> min = %e , max = %e", min, max);
-    DEBUG2("DICOTOMI ===> diffmin = %e , diffmax = %e", min_diff, max_diff);
-
-    if( min_diff > 0 && max_diff > 0 ){
-      if(min == max){
+    if (min_diff > 0 && max_diff > 0) {
+      if (min == max) {
+	CDEBUG0(surf_lagrange_dichotomy, "Decreasing min");
 	min = min / 2.0;
-      }else{
+	min_diff = diff(min, var_cnst);
+      } else {
+	CDEBUG0(surf_lagrange_dichotomy, "Decreasing max");
 	max = min;
+	max_diff = min_diff;
+
       }
-    }else if( min_diff < 0 && max_diff < 0 ){
-      if(min == max){
+    } else if (min_diff < 0 && max_diff < 0) {
+      if (min == max) {
+	CDEBUG0(surf_lagrange_dichotomy, "Increasing max");
 	max = max * 2.0;
-      }else{
+	max_diff = diff(max, var_cnst);
+      } else {
+	CDEBUG0(surf_lagrange_dichotomy, "Increasing min");
 	min = max;
+	min_diff = max_diff;
       }
-    }else if( min_diff < 0 && max_diff > 0 ){
-      middle = (max + min)/2.0;
-      middle_diff = diff(middle, var_cnst);
+    } else if (min_diff < 0 && max_diff > 0) {
+      middle = (max + min) / 2.0;
+      CDEBUG1(surf_lagrange_dichotomy, "Trying (max+min)/2 : %1.20f",middle);
 
-      if(max != 0.0 && min != 0.0){
-	overall_error = fabs(min - max)/max;
+      if((min==middle) || (max==middle)) {
+	DEBUG0("Cannot improve the convergence!");
+	break;
       }
+      middle_diff = diff(middle, var_cnst);
 
-      if( middle_diff < 0 ){
+      if (middle_diff < 0) {
+	CDEBUG0(surf_lagrange_dichotomy, "Increasing min");
 	min = middle;
-      }else if( middle_diff > 0 ){
+	min_diff = middle_diff;
+      } else if (middle_diff > 0) {
+	CDEBUG0(surf_lagrange_dichotomy, "Decreasing max");
 	max = middle;
-      }else{
-	WARN0("Found an optimal solution with 0 error!");
+	max_diff = middle_diff;
+      } else {
 	overall_error = 0;
-	return middle;
       }
-
-    }else if(min_diff == 0){
-      return min;
-    }else if(max_diff == 0){
-      return max;
-    }else if(min_diff > 0 && max_diff < 0){
-      WARN0("The impossible happened, partial_diff(min) > 0 && partial_diff(max) < 0");
+    } else if (min_diff == 0) {
+      max=min;
+      overall_error = 0;
+    } else if (max_diff == 0) {
+      min=max;
+      overall_error = 0;
+    } else if (min_diff > 0 && max_diff < 0) {
+      CWARN0(surf_lagrange_dichotomy,
+	     "The impossible happened, partial_diff(min) > 0 && partial_diff(max) < 0");
+      abort();
+    } else {
+      CWARN2(surf_lagrange_dichotomy,
+	     "diffmin (%1.20f) or diffmax (%1.20f) are something I don't know, taking no action.",
+	     min_diff, max_diff);
+      abort();
     }
   }
 
-
-  DEBUG1("====> returning %e", (min+max)/2.0);
-  return ((min+max)/2.0);
+  CDEBUG1(surf_lagrange_dichotomy, "returning %e", (min + max) / 2.0);
+  XBT_OUT;
+  return ((min + max) / 2.0);
 }
 
 /*
  *
  */
-double partial_diff_mu(double mu, void *param_var){
-  double mu_partial=0.0;
-  double sigma_mu=0.0;
-  lmm_variable_t var = (lmm_variable_t)param_var;
+double partial_diff_mu(double mu, void *param_var)
+{
+  double mu_partial = 0.0;
+  double sigma_mu = 0.0;
+  lmm_variable_t var = (lmm_variable_t) param_var;
   int i;
-
+  XBT_IN;
   //compute sigma_i
-  for(i=0; i<var->cnsts_number; i++)
+  for (i = 0; i < var->cnsts_number; i++)
     sigma_mu += (var->cnsts[i].constraint)->lambda;
-  
+
   //compute sigma_i + mu_i
   sigma_mu += mu;
-  
+
   //use auxiliar function passing (sigma_i + mu_i)
-  mu_partial = diff_aux(var, sigma_mu) ;
- 
+  mu_partial = diff_aux(var, sigma_mu);
+
   //add the RTT limit
   mu_partial += var->bound;
 
+  XBT_OUT;
   return mu_partial;
 }
 
 /*
  *
  */
-double partial_diff_lambda(double lambda, void *param_cnst){
+double partial_diff_lambda(double lambda, void *param_cnst)
+{
 
   int i;
   xbt_swag_t elem_list = NULL;
   lmm_element_t elem = NULL;
   lmm_variable_t var = NULL;
-  lmm_constraint_t cnst= (lmm_constraint_t) param_cnst;
-  double lambda_partial=0.0;
-  double sigma_mu=0.0;
+  lmm_constraint_t cnst = (lmm_constraint_t) param_cnst;
+  double lambda_partial = 0.0;
+  double sigma_i = 0.0;
 
+  XBT_IN;
   elem_list = &(cnst->element_set);
 
-  DEBUG1("Computting diff of cnst (%p)", cnst);
-  
+  CDEBUG1(surf_lagrange_dichotomy,"Computting diff of cnst (%p)", cnst);
+
   xbt_swag_foreach(elem, elem_list) {
     var = elem->variable;
-    if(var->weight<=0) continue;
-    
+    if (var->weight <= 0)
+      continue;
+
     //initilize de sumation variable
-    sigma_mu = 0.0;
+    sigma_i = 0.0;
 
     //compute sigma_i of variable var
-    for(i=0; i<var->cnsts_number; i++){
-      sigma_mu += (var->cnsts[i].constraint)->lambda;
+    for (i = 0; i < var->cnsts_number; i++) {
+      sigma_i += (var->cnsts[i].constraint)->lambda;
     }
-	
+
     //add mu_i if this flow has a RTT constraint associated
-    if(var->bound > 0) sigma_mu += var->mu;
+    if (var->bound > 0)
+      sigma_i += var->mu;
 
     //replace value of cnst->lambda by the value of parameter lambda
-    sigma_mu = (sigma_mu - cnst->lambda) + lambda;
-    
+    sigma_i = (sigma_i - cnst->lambda) + lambda;
+
     //use the auxiliar function passing (\sigma_i + \mu_i)
-    lambda_partial += diff_aux(var, sigma_mu);
+    lambda_partial += diff_aux(var, sigma_i);
   }
 
+
   lambda_partial += cnst->bound;
 
+  XBT_OUT;
   return lambda_partial;
 }
 
 
-double diff_aux(lmm_variable_t var, double x){
-  double tmp_fp, tmp_fpi, tmp_fpip, result;
+double diff_aux(lmm_variable_t var, double x)
+{
+  double tmp_fpi, result;
 
-  xbt_assert0(var->func_fp, "Initialize the protocol functions first create variables before.");
+  XBT_IN2("(var (%p), x (%1.20f))", var, x);
+  xbt_assert0(var->func_fpi,
+	      "Initialize the protocol functions first create variables before.");
 
-  tmp_fp = var->func_fp(var, x);
   tmp_fpi = var->func_fpi(var, x);
-  tmp_fpip = var->func_fpip(var, x);
-
-  result = tmp_fpip*(var->func_fp(var, tmp_fpi));
-  
-  result = result - tmp_fpi;
-
-  result = result - (tmp_fpip * x);
+  result = - tmp_fpi;
 
+  XBT_OUT;
   return result;
-}  
- 
+}
 
 
+/**************** Vegas and Reno functions *************************/
+/*
+ * NOTE for Reno: all functions consider the network
+ * coeficient (alpha) equal to 1.
+ */
 
+/*
+ * For Vegas fpi: $\frac{\alpha D_f}{x}$
+ */
+double func_vegas_fpi(lmm_variable_t var, double x){
+  xbt_assert0(x>0.0,"Don't call me with stupid values!");
+  return VEGAS_SCALING*var->df/x;
+}
 
+/*
+ * For Reno fpi: $\sqrt{\frac{1}{{D_f}^2 x} - \frac{2}{3{D_f}^2}}$
+ */
+double func_reno_fpi(lmm_variable_t var, double x){
+  double res_fpi; 
 
+  xbt_assert0(var->df>0.0,"Don't call me with stupid values!");
+  xbt_assert0(x>0.0,"Don't call me with stupid values!");
 
+  res_fpi = 1/(var->df*var->df*x) - 2/(3*var->df*var->df);
+  if(res_fpi<=0.0) return 0.0;
+  xbt_assert0(res_fpi>0.0,"Don't call me with stupid values!");
+  return sqrt(res_fpi);
+}