Merge 07727b8 into 56f5cd6

Author: ulvgard

AUTHORS

@@ -43,6 +43,7 @@ Roman Lebedev <lebedev.ri@gmail.com>
 Shuo Chen <chenshuo@chenshuo.com>
 Steinar H. Gunderson <sgunderson@bigfoot.com>
 Stripe, Inc.
+Tobias Ulvgård <tobias@ulvgard.se>
 Yixuan Qiu <yixuanq@gmail.com>
 Yusuke Suzuki <utatane.tea@gmail.com>
 Zbigniew Skowron <zbychs@gmail.com>

src/complexity.cc

@@ -73,31 +73,41 @@ std::string GetBigOString(BigO complexity) {
 //   - time          : Vector containing the times for the benchmark tests.
 //   - fitting_curve : lambda expression (e.g. [](int64_t n) {return n; };).
 
-// For a deeper explanation on the algorithm logic, look the README file at
-// http://github.com/ismaelJimenez/Minimal-Cpp-Least-Squared-Fit
+// This algorithm works like this:
+//
+// Let the execution times t_i for input sizes n_i be the function 
+// t_i = f(n_i) = c*g(n_i), where c is a coefficient. The coefficient 
+// c is compued by assuming that t_i and f(n_i) are the same function
+// with different coefficients t_i = c_1*g(n_i) and f(n_i) = c_2*g(n_i).
+// Since t_i and f(n_i) are the same function, they only differ in 
+// proportionality: c_2 = c_1 * g(n_i) /  g(n_i)
+//
+// If, t_i and f(n_i) is not the same function, the Root Mean Square
+// (RMS) error will be large and dominated by n.
+// Example t_i = c_1*O(n^2) and  f(n_i) = c_2*O(n*log(n)):
+// ||e|| = sqrt(1/numSamples*sum((t_i - f(n_i))^2))
+// ||e|| = sqrt(1/numSamples*sum((c_1*n_i^2 - c_2*n_i*log(n_i))^2))
+// ||e||^2*numSamples = sum((c_1*n_i^2 - c_2*n_i*log(n_i)^2)
+// which is dominated by the difference (n_i^2 - n_i*log(n_i))^2
 
 LeastSq MinimalLeastSq(const std::vector<int64_t>& n,
                        const std::vector<double>& time,
                        BigOFunc* fitting_curve) {
   double sigma_gn = 0.0;
-  double sigma_gn_squared = 0.0;
   double sigma_time = 0.0;
-  double sigma_time_gn = 0.0;
 
   // Calculate least square fitting parameter
   for (size_t i = 0; i < n.size(); ++i) {
     double gn_i = fitting_curve(n[i]);
     sigma_gn += gn_i;
-    sigma_gn_squared += gn_i * gn_i;
     sigma_time += time[i];
-    sigma_time_gn += time[i] * gn_i;
   }
 
   LeastSq result;
   result.complexity = oLambda;
 
   // Calculate complexity.
-  result.coef = sigma_time_gn / sigma_gn_squared;
+  result.coef = sigma_time / sigma_gn;
 
   // Calculate RMS
   double rms = 0.0;