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The idea behind this proposal is born from the analysis of phenomena for which there are events with a greater frequency of being present, but for which the factor that determines these magnitude is not known. Being concrete, ***LRPS*** originates from my own attempt to replicate the composition of music through genetic algorithms with randomness, being clear the fact that depending on the style of music, genre, era, etc., certain composition patterns are more common; for example: 4/4 time signature, long or short notes, simple or complex harmonies, etc. Popular music tends to have with more frequency the 4/4 signature, and only major-minor chords, slow voice melodies, relaxed rhythms, among others; It is then that in order to try to replicate the composition of popular music, we could consider these characteristics as the most common, but without neglecting the fact that it is possible to have more variety of decision.

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In the model it has been decided that Cookies n’ cream flavor is the favorite, so its selection should be more frequent, followed by strawberry, chocolate, etc. The important issue is that there is no a tangible number that allows determining this importance bias with respect to the other flavors, it is simply known that flavors are preferred more as they are higher on the top. The logarithmic scale can provide a growth ratio based on its exponential mathematical nature:

A loose representation of what is the difference between linear and logarithmic scales looks like is shown in the image above; if we take it as an example, we could understand this decision algorithm as placing a point randomly in any of the two scales, where the space between two divisions represents an object to select. For the case of the linear scale, the probability of finding the object between **0 and 1** is the same as that of the object between **1 and 2,** or **2 and 3.** On the other hand, with the logarithmic scale, the probability of falling into the object between **0 and 1** is very different from that of the object between **5 and 6.**

Using real data for the example of choosing an ice cream flavor, we would have the following scales comparison:

This is what the ***LRPS*** algorithm scale looks compared to a roulette algorithm. If we throw a random point in this space; we would obtain the decisions for both types of algorithms, which would be:

- **(🍫) Chocolate for a roulette algorithm.**

- **(🍓) Strawberry: for the LRPS algorithm.**

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That is, given the same case of building the scale on a Cartesian axis, with ***n*** number of spaces (objects) and therefore ***n*** number of division points, the distance between each point ***i*** and the ***origin*** (coordinate (0, 0)) is given by the formula:

If it is wanted to find the coordinate of point 1, for the example of five ice cream flavors (with n = 5), the coordinate would be:

So, to find all the points needed to build the logarithmic scale of five objects:

**Note: The use of the Cartesian axis is unnecessary for the algorithm, since it is only necessary to know the distance between the origin and any other point on the scale; so only the distance formula will be necessary: dᵢ = nlogₙ₊₁(i+1).**

The pseudocode of the selection algorithm is shown below:

A list of objects is received as input from which the value of ***n*** is obtained as the number of objects in the selection. A couple of stopping cases occur when the list is empty or contains just one element, otherwise proceeds with the rest of the algorithm, where a random decimal number between 0 and ***n*** is obtained, and a loop for 1 to ***n*** is started.

**1. Having 5 objects and 100 repetitions:**

**2. Having 5 objects and 1,000 repetitions:**

**3. Having 5 objects and 1,000,000 repetitions:**

**4. Having 10 objects and 100,000 repetitions:**

**5. Having 20 objects and 10,000,000 repetitions:**

**It can be verified that the shape of a logarithm curve is obtained.**

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In the event that for a specific problem it is desired that two or more objects have the same selection probability, it can be considered –for this implementation– that these objects are included in a data structure that contains them as **ONE** single object, so that if the algorithm chooses them, it will throw the objects as a whole, later and by means of a roulette-type decision (such as the linear scale) one of these can be chosen.

Write to me at **arhcoder@gmail.com** for any query, or if this contribution was useful to you in any application. Also look for me on social media as **@arhcoder.**