=Paper=
{{Paper
|id=Vol-1964/PU2
|storemode=property
|title=Proof-of-Concept: Creating "Fuzzy" Sorting Algorithms
|pdfUrl=https://ceur-ws.org/Vol-1964/PU2.pdf
|volume=Vol-1964
|authors=Stephany Coffman-Wolph
|dblpUrl=https://dblp.org/rec/conf/maics/Coffman-Wolph17
}}
==Proof-of-Concept: Creating "Fuzzy" Sorting Algorithms==
<pdf width="1500px">https://ceur-ws.org/Vol-1964/PU2.pdf</pdf>
<pre>
Stephany Coffman-Wolph                                          MAICS 2017                                                pp. 151–156




                 Proof-of-Concept: Creating “Fuzzy” Sorting Algorithms
                                                      Stephany Coffman-Wolph
                                               West Virginia University Institute of Technology
                                             405 Fayette Pike, Montgomery, West Virginia 25136
                                                       sscoffmanwolph@mail.wvu.edu




                                Abstract                                    Kountanis, 2013a]. (In the event that high precision is re-
   Sorting algorithms are common tools for manipulating data                quired, the results from the fuzzy algorithm could be de-
   and used in both standalone circumstances and within larger              fuzzified and run through a non-fuzzy algorithm to complete
   more complex algorithms. Thus, it is highly desirable for                the sorting).
   sorting algorithms to be efficient in terms of storage and com-
   putation. By applying the concept of fuzzy logic (an abstract
   version of Boolean logic) to any well-known algorithm, it
   generates an abstract version (i.e., fuzzy algorithm) that often
                                                                                       Introduction and Background
   results in computational improvements. Although the algo-                The next five sections provide the necessary background
   rithm may produce a less precise result, this is counteracted
   by gaining computational efficiency with minimal acceptable              materials for this paper. The first section discusses the
   trade-offs (e.g., small increase in space requirements, loss of          three-step fuzzy algorithm framework used to convert a tra-
   precision). Using an established three-step framework for                ditional non-fuzzy algorithm into the corresponding equiva-
   fuzzification of an algorithm, the resulting new fuzzy algo-             lent fuzzy algorithm. The second section briefly discusses
   rithm goes beyond a simple conversion of data from raw to                sorting algorithms and their use within the paper. In the
   fuzzy data by converting the operators and concepts within
   the algorithm to their abstract equivalents. The purpose of              third section the author discusses types of data and the fuzzy
   this paper is to demonstrate, as a proof-of-concept, that sort-          sorting algorithms. The fourth section covers the notation
   ing algorithms can be converted into their corresponding                 used within the paper to denote when an element is fuzzy.
   fuzzy sorting algorithms. This paper discusses the prelimi-              In the final section, the author provides a description of what
   nary results of: (1) how to apply the general framework by               it means to be fuzzily sorted.
   developing the corresponding fuzzy algorithms for a variety
   of sort algorithms, (2) the success of applying the framework
   through the development of several fuzzy sort algorithms in-             The Fuzzy Algorithm Framework
   cluding fuzzy shell sort, fuzzy strand sort, and fuzzy bucket            The general broad-spectrum framework for the fuzzification
   sort, and (3) the possible applications and benefits of these
   fuzzy sort algorithms.                                                   of any algorithm is a simple three-step conversion process
                                                                            that converts any algorithm from a traditional non-fuzzy
                                                                            version into a corresponding fuzzy version [Coffman-
                            Motivation                                      Wolph and Kountanis, 2013b]. The three steps of the frame-
                                                                            work are as follows:
Sorting algorithms are used prolifically and have been thor-
oughly researched. This resulted in a wealth of algorithm
                                                                                  1. Decide what can/should be fuzzified and determine
options for sorting. Every sort algorithm produces the same
                                                                                  the category of each piece to be fuzzified
result, but many have unique properties or techniques (e.g.,
                                                                                  2. Fuzzify each piece (identified in step 1) based on the
used for an almost sorted list, reversing a sorted list, or eas-
                                                                                  category
ily adding an additional value). Given the frequent use of
                                                                                       a. Scale/normalize the data, then fuzzify the data
sorting algorithms, being able to sort efficiently (in regards
                                                                                       b. Fuzzify the operators
to both space and time) is highly desirable. If precise sorting
                                                                                       c. Fuzzify the concepts
is not necessary, a fuzzy sort algorithm could be advanta-
                                                                                  3. Defuzzification (if needed/applicable)
geous as fuzzy algorithms take advantage of integer calcu-
                                                                               As can be seen in the framework, there are three main cat-
lations and generally execute fewer steps than the equivalent
                                                                            egories of components within an algorithm that can be fuzz-
traditional non-fuzzy algorithm [Coffman-Wolph and
                                                                            ified or made more abstract: (1) data, (2) operators, and (3)


Copyright held by the author.




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Proof-of-Concept: Creating “Fuzzy” Sorting Algorithms                                                                  pp. 151–156


concepts. The fuzzification of data is the process of taking            Data
“raw”/non-fuzzy data and converting it into fuzzy data. The             Sorting algorithms are generally able to handle a wide vari-
fuzzification of operators is the process of converting a               ety of data types (e.g., numbers, strings, or combinations).
mathematical, logical, or comparative operator to its fuzzy             The fuzzy sort algorithms will need to have the data “con-
counterpart, which operates on fuzzy sets instead of pure               verted” to the fuzzy equivalent. Fuzzy algorithms generally
numbers. The fuzzification of concepts, the most difficult              take advantage of integer calculations and, usually, have the
of the three, is the conversion of an idea into a similar fuzzy         data in integer format. In the circumstances that require a
idea. Together these three core techniques create a fuzzy               conversion of non-integer numeric data, it is often accom-
algorithm.                                                              plished using scaling. A fuzzy number is not a single value,
   The first and second steps of the framework are needed to            but is represented by a set of values. The size/range of val-
apply the fuzzy algorithm to a variety of problems. The first           ues is problem/data dependent and can also be affected by
step of the framework is to make an in-depth examination of             the level of precision the user desires. For strings, there are
the traditional (non-fuzzy) algorithm and decide which ele-             several possibilities for fuzzifying the data including: con-
ments to fuzzify. It is often helpful to determine what the             verting to a number or considering only the ASCII value
solution will look like in the fuzzy form. This will lay the            equivalent of the first character (or the first few characters).
foundation for the entire fuzzified algorithm. Therefore, it            If there is a combination of data (i.e., records from a data-
can be helpful to work backwards through the algorithm to               base), then a combination of fuzzy data would be required.
determine what elements (data, operators, or concepts) need             The greatest performance advantage is obtained using inte-
to be changed for the solution to be produced. Defuzzifica-             ger values, so the closer the data is to an integer format the
tion, the opposite of fuzzification, is used to convert the re-         better.
sult back to a non-fuzzy result and can be applied to data or
concepts. In some cases, this step might be unnecessary and,            Notation
hence, why the framework denotes this step as optional. De-
ciding to do defuzzification is both problem and usage de-              Throughout this paper, a subscript f will be used to denote
pendent. For example, if the fuzzy algorithm is being used              when a value, variable, operator, or concept is fuzzy. Given
to narrow the solution space and the information will be fed            that not everything is required to be fuzzy within a fuzzy
into a non-fuzzy algorithm, then it is important to develop a           algorithm, the subscript f will help the reader distinguish be-
method of defuzzification.                                              tween portions of the algorithm that remain traditional/non-
   This process has already been successfully applied to the            fuzzy and those that have been converted to a corresponding
following algorithms: (1) a concept-oriented fuzzification              fuzzy version. As mentioned in the previous section, it is
for finding fuzzy patterns [Coffman-Wolph and Kountanis,                important to keep in mind that a single fuzzy value is repre-
2013c], (2) a fuzzification of both data and the operators for          sented by a set of non-fuzzy traditional values. For example,
a fuzzy process particle swarm optimization (FP2SO) [Coff-              5f is a “fuzzy-five” and contains the value 5 and possibly the
man-Wolph and Kountanis, 2013a], (3) a fuzzification of                 values 4 and 6 as well (depending on the definition of 5f in
multiple algorithm components to find strategies for adver-             the context of the problem). In other circumstances, 5f could
sarial situations from game theory [Coffman-Wolph, 2013],               represent all the real number values in the range of 4.5 to
(4) a fuzzification of the simple simplex method for the                6.5.
transportation problem [Coffman-Wolph and Coffman, Jr,
2014], (5) the fuzzification of various linear and nonlinear            Definition of Fuzzily Sorted
search techniques used in optimization algorithms [Coff-                A traditional sort algorithm stops when the algorithm has
man-Wolph and Coffman, Jr, 2016], and (6) the fuzzifica-                guaranteed that all data are sorted in the required order (e.g.,
tion of the Golden Ratio Search [Coffman-Wolph, 2016].                  numerical order, alphabetical order, reverse order, or a com-
                                                                        bination). To proceed with the fuzzification of the sort al-
The Sorting Algorithm                                                   gorithm, the key question to answer is: when is the dataf
Generally, sort algorithms put data “in order”. This paper              from a fuzzy sort algorithm considered sorted sufficiently?
will explore a variety of sort algorithms ranging from the              In general, adding fuzzy logic to an algorithm creates a more
simplistic comparison sorts to slightly more complex distri-            abstract version of this algorithm. Therefore, with a fuzzy
bution sorts. Given the commonality of these sort algo-                 sort, it will be a more abstract version of the sort algorithm
rithms, this paper will focus only on the specific algorithmic          and, thus, the output is abstractly sorted. The definition of
details needed for fuzzification conversion and skip in-depth           “abstractly sorted” will depend on the circumstances of the
discussions of the algorithms themselves. To simplify the               application and may be problem specific and/or sorting al-
discussion, this paper will primarily consider integer data.            gorithm specific (i.e., dependent on the fuzzification pro-
These concepts are easily extended to other data types as               cess). These definitions have one thing in common, that the
discussed in the next section.                                          data will not always be perfectly sorted. One could describe




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Stephany Coffman-Wolph                                     MAICS 2017                                                    pp. 151–156


the data as “slightly out of order”. For a fuzzy sort, one               and the un-sorted portion towards the back of the list). Dur-
measure could be an associated percentage which represents               ing the process of converting the algorithm to a fuzzy algo-
the required/desired level of sufficient “sorted-ness”. The              rithm, the concept of the xth element will be fuzzified using
level or percentage can be adjusted as needed for desired                the concept of a fuzzy set [Zadeh, 1965]. For the traditional
precision. Another measure, for a fuzzy sort, could be that              shell sort, the algorithm takes every xth element within the
the results are sorted within a specific number of significant           un-sorted portion of the list and sorts them into the correct
digits or letters/symbols.                                               sequence before merging them into the sorted portion of the
                                                                         list. In the corresponding fuzzified version of this algorithm,
                                                                         the algorithm will take every xfth set of elements from the
             The Fuzzy Sort Algorithm                                    un-sorted portion of the list, sortf these elements, and merge
The following sections discuss the fuzzification of several              these elements into the sortedf portion of the list. The fuzzy
well-known sort algorithms including: selection, insertion,              shell sorting algorithms would likely have an advantage in
bubble, shell, strand, and bucket sort. Each subsection be-              situations where data naturally “clusters”.
low will focus on how the algorithm is converted from the                   Consider the following data points: 88, 92, 98, 100, 7, 5,
traditional algorithm to a fuzzy algorithm using the three-              3, 13, 17, 15, 24, 36, 42, 57, 55, 73, and 77 to illustration the
step framework for conversion. A discussion of the likely                fuzzy shell sort (and the traditional shell sort). When using
advantages and disadvantages will be included for each of                the fuzzy shell sort, the first step is to determine the defini-
the sort algorithms.                                                     tion of the xfth element(s). For this small of a list, the fuzzy
                                                                         algorithm will use a two-element range at each xfth element.
Fuzzy Version of Selection, Insertion, and Bubble                        Thus, the algorithm will consider the xth element and the
Sort                                                                     (x+1)th element to be the xfth element. The fuzzy algorithm
                                                                         begins by selecting the 5fth elements: (88, 92), (5, 3), (24,
Selection, Insertion, and Bubble sorts are three of the most             36), and (73, 77). These elements are sorted, removed from
common comparison-based O(n2) sort algorithms [Sedge-                    the un-sorted portion of the list, and placed into the sorted
wick, 1998; Weiss, 1999]. Each algorithm can be easily                   part of the list: 3, 5, 24, 36, 73, 77, 88, and 92 and the un-
converted from the traditional algorithm into a fuzzy equiv-             sorted portion is: 98, 100, 7, 13, 17, 15, 42, 57, and 55. The
alent version. The comparison operators, < and >, would be               next step is to take the 3fth elements which are (98, 100), (17,
converted to <f and >f to handle the fuzzified data, but the             15), and (57, 55). These elements are sorted into the existing
algorithms would otherwise remain unchanged and operate                  list and removed from the unsorted list: 3, 5, 15, 17, 24, 36,
in the same manner. The fuzzy selection, fuzzy insertion,                55, 57, 73, 77, 88, 92, 98, and 100 and the remaining portion
and fuzzy bubble sorts do not experience any computational               of the unsorted list is: 7, 13, and 42. These last three ele-
advantages except the possible use of integer comparisons                ments can be sorted into the sorted portion of the list: 3, 5,
over more complex data types. Additionally, the output of                7, 13, 15, 17, 24, 36, 42, 55, 57, 73, 77, 88, 92, 98, and 100.
these fuzzy sorts would be dependent on the precision level              For each of the xfth elements, the only additional processing
the user wants to achieve when they fuzzify the data and de-             required is a simple swap (as there are only two elements)
cide the exact functionality of the <f and >f. Therefore, the            before sorting and merging. The traditional version of the
author concludes that while these algorithms can be con-                 shell sort algorithm run on the same data set would require
verted to their fuzzy equivalents, there is no advantage to do           5 total passes to get the data in numeric order, which is an
this unless they are being applied as part of a more complex             additional two passes over this fuzzy algorithm version. The
sort algorithm (e.g., the bubble sort as a last step in a more           author projects from preliminary data that this technique
sophisticated sort algorithm).                                           would cut the number of passes in almost half as each pass
                                                                         pulls twice the amount of data.
Fuzzy Version of the Shell Sort
The traditional shell sort is an “in-place” sort and considered          Fuzzy Version of the Strand Sort
an abstract version of an insertion sort [Sedgewick, 1998;               The strand sort is a fairly straightforward comparison-based
Weiss, 1999]. (The comb sort is a very similar “in-place”                sorting algorithm. The algorithm looks for strands of in-or-
sort but uses the bubble sort as an underlying basis and ba-             der data and slowly builds the sorted list [Black, 2008]. The
sically differs only at the implementation level). When the              following data points: 55, 57, 89, 92, 98, 95, 100, 3, 5, 7, 13,
fuzzified version of the shell sort is implemented, the algo-            15, 17, 24, 36, 42, 73, and 77 will be used to demonstrate
rithm will be even further abstracted. Both the fuzzy and tra-           the fuzzy strand sort (and the traditional strand sort). The
ditional versions of the sort use a single list with portions of         traditional algorithm begins by creating a sub-list obtained
the list designated as sorted and un-sorted. (Traditional im-            by comparing each value to the next value and stopping
plementations have the sorted portion at the front of the list




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Proof-of-Concept: Creating “Fuzzy” Sorting Algorithms                                                                 pp. 151–156


when the next is less than the previous. The traditional al-             and above. The numbers could be sorted as follows using
gorithm would find the following strand: 55, 57, 89, 92, 98.             the fuzzy algorithm (and fuzzy buckets):
The remaining elements are: 95, 100, 3, 5, 7, 13, 15, 17, 24,                 1. 0, 7, 15, 21
36, 42, 73, and 77. However, the fuzzy version of the strand                  2. 24, 36, 50
sort performs fuzzy “less than” comparisons (e.g., <f means                   3. 42, 57, 46, 73
x <f y if x < y + 5 ) and, thus, could create the following sub-              4. 88, 81, 92, 100
list: 55, 57, 89, 92, 98, 95, 100 with the remaining elements
being: 3, 5, 7, 13, 15, 17, 24, 36, 42, 73, and 77. The tradi-           Using the same four fuzzy buckets/bins, the fuzzy algorithm
tional algorithm would need to take an additional step to get            could also have produced the following:
to the same place (merging the 95, 100 into the sub-list).                   1. 0, 7, 15, 21
From that point onward both the traditional and fuzzy ver-                   2. 24, 36, 42, 46
sions of strand sort would produce the same strands. The                     3. 50, 57, 73, 81
final sorted list for the traditional strand sort is always pre-             4. 88, 92, 100
cisely in-order. However, the fuzzy strand sort could have
a few data points slightly out of order (as can be seen with             Observe that the output for the traditional algorithm would
the 98 and 95 above). The fuzzy strand algorithm required                require further processing on each bin to complete the algo-
fewer overall steps and could be the better sort algorithm               rithm. Generally, in these kinds of algorithms, either each
when the slightly out of order data are within an acceptable             bin/bucket is subdivided into more bin/buckets or the con-
level of precision. Given the nature and organization of the             tents of each bucket are sorted using a basic sort algorithm.
data that works well for the strand sort, the fuzzy strand al-           In the first output of the fuzzy algorithm, the data in each
gorithm could take the same or fewer steps than the tradi-               bin/bucket except number four is almost in-order (and bin
tional version of the algorithm. However, the fuzzy strand               number four needs the 88 and 81 flipped). (Merging these
algorithm would never take more steps then the traditional               bins/buckets into a final sorted list would be an easy process
version. The fuzzy strand algorithm output might be slightly             and eliminate the issue that the 46 is in bin number three
out of order, but the max amount is predictable and con-                 while the 50 is located in bin number two). The second out-
trolled by the specific implementation of the fuzzy set size.            put of the fuzzy algorithm, the fuzzily sorted data is com-
                                                                         pletely in order and does not require further processing.
Fuzzy Version of the Bucket Sort                                         Both outputs of the fuzzy algorithm are better in-order than
A fuzzy sort most closely resembles the traditional distribu-            the output produced by the traditional algorithm
tion based sorts or “bucket” sorts, but differs because of the              In this example, it is important to note that that there is
underlying definition of a fuzzy value. In a traditional                 both overlap between the different buckets and different
bucket sort, an item sorts into one category/bucket [Sedge-              sizes for each bucket/bin. Changing the overlap amounts
wick, 1998; Weiss, 1999]. In a fuzzy sort since each (fuzzy)             and size of bins could change the outcome of the algorithm
value is represented by a set, an item can sort into multiple            and it would be important to choose these dependent on the
categories/buckets. Additionally, each category/bucket is a              data being sorted. In other words, the bin size and overlap
(fuzzy) value and it too is represented by a set, thus, the cat-         are both problem and data specific. If the user wants/needs
egories/buckets may overlap and do not have to be distrib-               a high level of precision on the fuzzy sort, then the overlap
uted evenly.                                                             and bin size would need to be chosen carefully and the al-
   Given the following set of numbers: 24, 36, 50, 42, 57,               gorithm would probably need to continue by further sorting
0, 88, 73, 92, 46, 7, 81, 15, 21, 100. Both the traditional and          each bin. However, if each bucket/bin has a small quantity
fuzzy bucket sort algorithms begin by sorting each number                of data, in many cases it will not be necessary to continue
into a bucket/bin. For this example, consider that there are             sorting.
four bucket/bins for the traditional bucket sort: (1) 0-25, (2)             The total number of steps taken by the fuzzy bucket/bin
26-50, (3) 51-75, (4) 75-100. The numbers will be sorted as              sort is controlled in the same way as the traditional algo-
follows using the traditional algorithm:                                 rithm – basically divide into to bucket/bins until each bin
     1. 24, 0, 7, 15, 21                                                 has a certain amount (and then use another sorting technique
     2. 36, 50, 42, 46                                                   to complete each bin). With the fuzzy bucket sort, it would
     3. 57                                                               be possible to eliminate the final step of using another sort-
     4. 88, 73, 92, 81, 100                                              ing technique to completely sort each bin especially when
                                                                         the slightly out of order numbers are within an acceptable
For the fuzzy algorithm, the four fuzzy buckets/bins are de-             lack of precision. The output of a fuzzy bucket/bin algo-
fined as: (1) 25 and below, (2) 20-50, (3) 45-82, and (4) 70             rithm might be slightly out of order, but can be partial con-
                                                                         trolled by the specific implementation of the fuzzy
                                                                         bucket/bin sizes and the overlap allowed among bins. The




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Stephany Coffman-Wolph                                    MAICS 2017                                                     pp. 151–156


fuzzy bucket/bin algorithm is also more adaptable and, from             sort algorithm and, thus, showed the best promise of the al-
the preliminary data, more accurate when sorting data with              gorithms presented in this paper.
fewer bins than needed by the traditional algorithm.

  Possible Applications for Fuzzy Sort Algo-                                                   Future Work
                    rithms                                              Using the knowledge gained in this proof-of-concept paper,
As mentioned earlier, sorting is most commonly used to ma-              the author plans to further investigate the fuzzification of
nipulate data especially for display to a user or within a              sort algorithms by focusing attention on the bucket-like sorts
search algorithm, but sorting algorithms are found within a             and the “in-place” sorts. One next step would be to com-
diverse range of interesting and complex algorithms. There              plete an implementation of these algorithms (both fuzzy and
are several operation research examples that implement a                non-fuzzy) and run benchmark data on these algorithms to
sort algorithm as an essential element of a larger algorithm            determine if the promise shown above holds for various
including shortest processing time rule, load-balancing                 cases. Additionally, the author needs to: (1) verify that fuzz-
problem, and event-driven simulation [Hillier and Lieber-               ifying the sort algorithm does not have adverse effects to
man, 1990; Lasden, 1970; Luenberger, 1973]. In discrete                 sorting the data and (2) determine the best and worst cases
event simulation, a sorting algorithm is used to order events           for the fuzzy sorts. For adverse effects, the author will thor-
but a fuzzy sort algorithm would be an unworkable solution              oughly investigate the negative or dangerous effects fuzzifi-
because the events are required to be in exactly time order             cation could have on an algorithm and determine if there ex-
(and, as seen earlier, a fuzzy sort could have a few items out          ist any countermeasures. The author also plans to use the
of order). However, there are circumstances where that                  knowledge gleamed from this experience to further refine
functionality of the fuzzy sort algorithm would not have det-           details of the three-step framework. This research encom-
rimental or negative effects on operation research algo-                passes the need to determine which algorithm characteristics
rithms. For example, in queuing problems, sorting is used               imply that there is a smooth transition from non-fuzzy to
to determine the ordering of operations and the operations              fuzzy using the framework and which algorithm character-
list is constantly being re-ordered. Given that a precise or-           istics imply that a transition is infeasible or impractical.
dering is not necessary and the nearly continuously resorting
of the data, a faster fuzzy sort algorithm would be helpful to
the overall runtime of the algorithm. Another example is
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