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Understanding Complexity - Urban Traffic VS Highway Traffic

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Sometimes, algorithmic complexity, when approached academically, or mathematically, might go astray its literal meaning. Indeed, algorithmic complexity is described using theoretic mathematical expressions, and referred to basically by using Big O notation. As a matter of fact, algorithmic complexity has a picturesque meaning.   Sometimes, a bad algorithm is the one that has a pointless high complexity, and any experienced programmer can easily find room for optimisation. Reducing complexity can sometimes mean that an unnecessary recalculation is repeatedly done. Sometimes, a value should be calculated only by using a formula rather than using (nested) loops etc. A real life situation is traffic in the city versus traffic in the highway. In the city, the distance between home and downtown is 10 km, but it takes me 45 minutes to reach my destination, taking into account traffic lights, intersections, the tramway ... Whereas, the distance between home and a former jo...
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Booth's Algorithm - C Implementation

#include <stdio.h> #define WORD 4 #define VERBOSE 1 //0 /* * 02 DEC 2011 (CSC2304) * Implementation of the Booth's Algorithm. */ void twosComplementAddition(char[], char[]); void rightShift(char[], char); void addition(char[], char[]); char* twosComplementMultiplication(char M[], char Q[]) { char C; char *A = (char*) malloc(sizeof(char)*(2 * WORD + 1)); char processedQ[WORD+ 1]; char Q0, Q_1 = '0'; int i, j; strcpy(A, "0000"); if (VERBOSE) { printf("\n A | Q | M |"); printf("\n %s | %s | %s | Initial", A, Q, M); printf("\n-------------------------------------------------------------"); } for (i = 0, j = 1; i < WORD; i++, j++) { Q0 = Q[WORD - 1]; if (VERBOSE) { printf("\n %s | %s | %s | Cycle %d", A, Q, M, j); } if (Q0 == '0' && Q_1 == '1')...
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N-ary Tree Data Structure API in Java

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TreeDS4j is a Java (6+) API that allows handling n-ary tree data structure in such a flexible way. Indeed, the user is subject to enter structured and coherent data, and the latter would be stored in an hierarchy, accordingly. The link below contains two samples that show how to take advantage of the API. https://sourceforge.net/projects/treeds4j/
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Decreasing Complexity Means Increasing Performance

What is wrong the piece of C Language code below? for ( i = 1 ; i <= length(str) ; i ++ ) { // Do some stuff } There is a recalculation of the length of the string at every iteration, which is a major waste of time. Indeed, at the first iteration, we already know how many times we need to iterate; when it's time for the second iteration, a recalculation of the already known value is done once again ... we keep doing this n times. Hence, complexity here is unnecessarily high. Detailing the code above in a more explicit manner, it may be translated as follows: int n = 0; while (str[i] != '\0') { n ++; } for ( i = 1 ; i <= n ; i ++ ) { // Do some stuff n = 0; while (str[i] != '\0') { n ++; } } At a glance, not only the code becomes redundant and longer, this fragment's time complexity is obviously O(n²) . The easy optimisation is to add a variable n , which will hold the length of the string, and, consequently...
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Indirect Recursion - A Natural Phenomenon

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Indirect recursion could be noticed in seeds and trees: a seed ends up to be a tree, and the latter produces many seeds, which themselves become trees and so forth. An interesting case was beautifully described in the Qur'an, chapter 2, verse 261, a plain description that is easy  to represent: " The example of those who spend their wealth in the way of Allah is like a seed [of grain] which grows seven spikes ; in each spike is a hundred grains. And Allah multiplies [His reward] for whom He wills. And Allah is all-Encompassing and Knowing ". ( Source ). Regardless of the religious aspects, one can notice the exponential growth of seeds, which is appreciated in agriculture, but seriously deprecated in Computer Science. If we represent this phenomenon (seeds and spikes) as a Java algorithm: void seed(x) { System.out.println("Seed " + x); for (i = 0; i < 7; i ++) { spike(i); } } void spike(y) { System.out.println("Sp...
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Triple Nested Loops - Two Linear Loops and One Logarithmic with a non-Constant Base - The Monster

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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...
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Pays/territoire : Rabat, Maroc
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Logarithmic Summations and Discrete Loops - Ceiling and Floor Functions

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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...
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