Algorithms Orbit

To obtain the function that tells the exact number of iterations a set of nested loops would perform didn't seem to be a tedious exercise. However, I came accross this interesting case: - Linear outer loop - Dependent first level linear inner loop - Dependent second level logarithmic inner loop with a general base. for ( i = 1 ; i <= n ; i ++ ) { for ( j = 1; j <= i ; j ++ ) { for ( k = 1 ; k <= j ; k *= a ) { // c } } } Ceilinged/Floored logarithmic functions within summations turned out to have very lengthy closed-forms (See my previous post). We will dissect the formulas above (for any a > 1 and n >= 1): Therefore, This link  points at a  plain   Java program that represents the above functions (f(), g () , h () , u () , v () , w () , respectively), and give the exact number of iterations performed by the covered algorithm mentioned earlier (receives two a...

Reading the so called paper " Discrete Loops and Worst Case Performance " ( by Dr. Johann Blieberger ), I came across some interesting summations: Useful here: for ( i = 1; i <= n ; i ++ ) { for (j = 1; j <= i ; j *= 2) { // Some constant time C instructions. } } Because: Likewise: Applicable here: for ( i = 1; i <= n ; i ++ ) { for (j = 1; j < i ; j *= 2) { // Some constant time C instructions. } } Because: And more generally: for ( i = 1; i <= n ; i ++ ) { for (j = 1; j <= i ; j *= a) { // Some constant time C instructions. } } We can proceed like the following: Further: for ( i = 1; i <= n ; i ++ ) { for (j = 1; j < i ; j *= a) { // Some constant time C instructions. } } With: Thus, while I tried to refine the results of some nested loops running time analysis, I had to face a particul...

First, watch this video: Now, let's assume the vehicles, namely the Jet, the Car, and the Motorcycle are your algorithms. Also, suppose the length of the racecourse is the size of the data n . Consequently, the speed of the vehicles can be analogized to the running time of the algorithms. Eventually, if we had to embody the circumstances of this interesting race, we will obtain this (approximately): Supposedly, the respective vehicles speed closed-form: Which means: Formal Analysis: Let the instant between 0:24 and 0:30 be represented by  n 0,  instant 0:33 represented by  n' 0 , and instant 0:36 by  n'' 0 . The most interesting case to study, is T car (n), as it grows faster than T motorC (n) but slower than T jet (n). Critique: We notice that, if the length of the courserace was small, the motorcycle would have won this contest. The speed of the jet is exponentially larger than both the car and the mo...