The Search-o-Sort Theory
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe authors proposed the Search-o-Sort and its further expansion for the theoretical provenance was done by its application on k-ary, Interpolation, and Jump Search. This paper can be published in the journal after minor revisions:
1) Abstract: in this section, the authors stated the characteristics of the Search-o-Sort theory. You need to analyze the work you have performed.
2) Even though the term "Search-o-Sort" is a new proposition for visualizing Sorting in terms of Searching, the authors should provide the details of searching and sorting statistically. You need to extend the introduction section by using at least 10 papers in the field.
3) Please explain the theory for extrapolation sort.
4) Can we use the theory for non-linear interpolation sort?
Author Response
General Comment: The authors proposed the Search-o-Sort and its further expansion for the theoretical provenance was done by its application on k-ary, Interpolation, and Jump Search. This paper can be published in the journal after minor revisions.
Response: Thank you, Professor for finding our work suitable for publication. In the revised version, we have tried to the best of our ability to address all the following comments. For brevity, we have highlighted the modifications in blue.
Comment #1: Abstract: in this section, the authors stated the characteristics of the Search-o-Sort theory. You need to analyze the work you have performed.
Response: We have rewritten the abstract of the manuscript focusing on the analysis and findings of the manuscript. Alongside the possible applications of the proposed Search-o-Sort theory have been analysed.
Comment #2: Even though the term "Search-o-Sort" is a new proposition for visualizing Sorting in terms of Searching, the authors should provide the details of searching and sorting statistically. You need to extend the introduction section by using at least 10 papers in the field.
Response: Professor, as suggested by another reviewer, we have segregated the Introduction Section into (1.1) General Context, (1.2) Motivation, (1.3) Literature Review, (1.4) Contribution and Scope, and (1.5) Document Structure. In the Literature Review subsection, we have identified few potential works for comparing, and establishment of Search-o-Sort theory.
Comment #3: Please explain the theory for extrapolation sort.
Response: Professor, to the best of our knowledge, there exists no explicit mention of extrapolation search (and therefore sort) in literature, but as suggested, we have tried to extend the conceptualization of Interpolation Search with extended bounds. Please refer to the Subsection 2.5. Further, we have tried to prove that the proposed Search-o-Sort theory holds well for the Extrapolation Search as well.
Comment #4: Can we use the theory for non-linear interpolation sort?
Response: We acknowledge this concern, and have addressed the same in the revised version of the manuscript. Please refer to the portion marked in blue between line numbered 279 – 283.
Thank you once again Professor for the suggestions and comments.
Reviewer 2 Report
Comments and Suggestions for AuthorsArticle:
Summary: This paper proposes Search-o-Sort, a theoretical framework that reconceptualizes sorting as a searching process. Validated through classical methods like Linear and Binary Search, and extended to advanced techniques such as Interpolation and Jump Search, the study highlights new possibilities for improving sorting efficiency. The findings suggest potential refinements to the established O(x log (x)) complexity bound, inviting further exploration into tighter theoretical limits.
Comments:
- The research topic is appropriate and aligns well with the scope of the journal. However, the authors must significantly improve the presentation of the manuscript. In particular, the Introduction section requires further development. The authors should present the current advancements in sorting theory and clearly explain how their contribution advances research in this area. I recommend reorganizing the Introduction using the following structure: (1.1) General Context, (1.2) Motivation, (1.3) Literature Review, (1.4) Contribution and Scope, and (1.5) Document Structure.
- Please revise Figure 1 to improve its visual quality and consistency. I recommend recreating it using vector graphics, as the current version displays inconsistent letter sizes and font types compared to those used in the main text. Ensuring uniform typography and scalable quality will enhance the overall presentation of the figure.
- Revise the Journal format regarding numbering all equations, if needed.
- Algorithms 1 and 2 require additional detail, as they form the core of the proposed validation. The current descriptions are not sufficiently clear for readers to fully understand the implementation and logic. I recommend expanding these algorithms with step-by-step explanations, clarifying input parameters, intermediate steps, and expected outputs. Adding brief contextual notes to highlight their role in the overall methodology will also improve readability and comprehension.
Once all these comments must be addressed, this paper will be ready for being published in this research journal.
Author Response
General Comment: This paper proposes Search-o-Sort, a theoretical framework that reconceptualizes sorting as a searching process. Validated through classical methods like Linear and Binary Search, and extended to advanced techniques such as Interpolation and Jump Search, the study highlights new possibilities for improving sorting efficiency. The findings suggest potential refinements to the established O(x log (x)) complexity bound, inviting further exploration into tighter theoretical limits.
Response: Thank you, Professor for finding our work suitable for publication. In the revised version, we have tried to the best of our ability to address all the following comments. For brevity, we have highlighted the modifications in blue.
Comment #1: The research topic is appropriate and aligns well with the scope of the journal. However, the authors must significantly improve the presentation of the manuscript. In particular, the Introduction section requires further development. The authors should present the current advancements in sorting theory and clearly explain how their contribution advances research in this area. I recommend reorganizing the Introduction using the following structure: (1.1) General Context, (1.2) Motivation, (1.3) Literature Review, (1.4) Contribution and Scope, and (1.5) Document Structure.
Response: Professor, we have segregated the Introduction Section into (1.1) General Context, (1.2) Motivation, (1.3) Literature Review, (1.4) Contribution and Scope, and (1.5) Document Structure. In the Literature Review subsection, we have identified few potential works for comparing, and establishment of Search-o-Sort theory. Please refer to the Section 1 of the revised manuscript.
Comment #2: Please revise Figure 1 to improve its visual quality and consistency. I recommend recreating it using vector graphics, as the current version displays inconsistent letter sizes and font types compared to those used in the main text. Ensuring uniform typography and scalable quality will enhance the overall presentation of the figure.
Response: Professor, we have updated all the figures (Fig. 1, 2, and 3) as vector images. Currently, the font style (Palatino Linotype) and size (18) matches with that of the main text. We believe this enhances the overall presentation of the manuscript.
Comment #3: Revise the Journal format regarding numbering all equations, if needed.
Response: Professor, we have checked the journal format regarding numbering of equations. We have tried to give numbers to all the important equations, as was suggested in the guidelines.
Comment #4: Algorithms 1 and 2 require additional detail, as they form the core of the proposed validation. The current descriptions are not sufficiently clear for readers to fully understand the implementation and logic. I recommend expanding these algorithms with step-by-step explanations, clarifying input parameters, intermediate steps, and expected outputs. Adding brief contextual notes to highlight their role in the overall methodology will also improve readability and comprehension.
Response: We have updated all the algorithms (Alg. 1, 2, 3, and 4) with comments (wherever needed) to improve readability and understandability of the manuscript for a wider reader group (if accepted for publication).
Thank you once again Professor for the suggestions and comments.
Reviewer 3 Report
Comments and Suggestions for AuthorsThese comments are from a person with a strong appreciation for mathematics but limited background in algorithms.
My understanding is that the lowest possible computational complexity (in terms of time) for sorting $n$ objects without any known structure is $n \log_2 n$. The abstract of this paper suggests that while the $n \log_2 n$ bound remains valid, some algorithm may achieve a runtime of $n \log_2 n - a_n$ for some lower-order term $a_n$. However, the paper’s presentation is somewhat elusive, and I am not entirely confident in my interpretation. This suggests that the exposition could be improved. It is unclear whether the authors have discovered a more efficient algorithm or are aiming to demonstrate that certain existing algorithms are already this efficient.
The mathematical interpretations of ${\cal O}(n)$ and ${\cal \Theta}(n)$ are not yet fully standardized in the community. This lack of clarity is significant, as the paper’s results hinge on precise definitions. I do not understand why $g(n) = {\cal O}(f(n))$ would imply $f(n) \geq g(n)$. My understanding is that this only implies the existence of a constant $C > 0$ such that $g(n) < C f(n)$ for sufficiently large $n$ (assuming all quantities are positive). This kind of ambiguity recurs throughout the paper. For instance, in the expression $\int_{i=1}^n \phi(i)$, is this meant to denote an integral? If so, how should one interpret $\phi(1.4)$? If it is meant to be a sum, why not write it as such?
On line 14 of page 4, the paper states: “since $\max - \min = {\cal O}(n)$.” Is this intended as a general truth or as an assumption? In Figure 2(b), it appears that the size of the ball should exceed that of the hole. Section 2.2 would benefit from an introductory sentence at the beginning and a concluding remark at the end. This applies similarly to other sections.
Equations (1) and (2) appear to contradict each other. Using the symbol $\approx$ in equation (2) seems unwise, as the goal of the paper is to provide precise analysis. Why not use ${\cal O}(n)$ or ${\cal \Theta}(n)$, or otherwise describe the relationship more precisely? Furthermore, is a highly accurate version of Stirling’s formula necessary here or elsewhere in the paper?
On page 1, lines 20–21, the subscript $i$ is overused.
On page 2, line 40, please provide a reference for the “well known” claim, and clarify what is meant by “in logarithmic time.”
I do not have the expertise to evaluate the entire paper as a specialist in this field. However, the paper does not appear to be intended solely for experts in the area. Therefore, it would be best to make the exposition accessible to readers with a solid mathematical foundation.
Author Response
General Comment: These comments are from a person with a strong appreciation for mathematics but limited background in algorithms.
Response: Thank you, Professor for aggreging to review our manuscript. In the revised version, we have tried to the best of our ability to address all the following comments. For brevity, we have highlighted the modifications in blue.
Comment #1: My understanding is that the lowest possible computational complexity (in terms of time) for sorting $n$ objects without any known structure is $n \log_2 n$. The abstract of this paper suggests that while the $n \log_2 n$ bound remains valid, some algorithm may achieve a runtime of $n \log_2 n - a_n$ for some lower-order term $a_n$. However, the paper’s presentation is somewhat elusive, and I am not entirely confident in my interpretation. This suggests that the exposition could be improved. It is unclear whether the authors have discovered a more efficient algorithm or are aiming to demonstrate that certain existing algorithms are already this efficient.
Response: Professor, we have segregated the Introduction Section into (1.1) General Context, (1.2) Motivation, (1.3) Literature Review, (1.4) Contribution and Scope, and (1.5) Document Structure. In Contribution and Scope Subsection, we have clarified on the contributions of the work. We have mentioned that Search-o-Sort is a framework which can accommodate any number of algorithms given the corollary holds.
Comment #2: The mathematical interpretations of ${\cal O}(n)$ and ${\cal \Theta}(n)$ are not yet fully standardized in the community. This lack of clarity is significant, as the paper’s results hinge on precise definitions. I do not understand why $g(n) = {\cal O}(f(n))$ would imply $f(n) \geq g(n)$. My understanding is that this only implies the existence of a constant $C > 0$ such that $g(n) < C f(n)$ for sufficiently large $n$ (assuming all quantities are positive). This kind of ambiguity recurs throughout the paper. For instance, in the expression $\int_{i=1}^n \phi(i)$, is this meant to denote an integral? If so, how should one interpret $\phi(1.4)$? If it is meant to be a sum, why not write it as such?
Response: Professor, we would respectfully disagree to this concern. ${\cal O}(n)$ and ${\cal \Theta}(n)$ stands for the Big-Oh, and Big-Theta notation, which are well known asymptotic notations in the algorithmic community. Infact, $g(n) = {\cal O}(f(n))$ implying $f(n) \geq g(n)$ derives from the fundamental definition of Bog-Oh notation. Further, regarding the ambiguity arising out of the usage of integral over summation have been clarified in the revised version of the manuscript. Please refer to the portions highlighted in blue in Subsection 2.1.
Comment #3: On line 14 of page 4, the paper states: “since $\max - \min = {\cal O}(n)$.” Is this intended as a general truth or as an assumption?
Response: Professor, $\max - \min = {\cal O}(n)$ is a general truth arising from the Min-Max Algorithm, and have been adapted for the realization of the Search-o-Sort theory.
Comment #4: In Figure 2(b), it appears that the size of the ball should exceed that of the hole. Section 2.2 would benefit from an introductory sentence at the beginning and a concluding remark at the end. This applies similarly to other sections.
Response: We apologize for any confusion caused. We have added the necessary clarifications for a general audience (if the work is accepted) to the caption of Figure 2.
Comment #5: Equations (1) and (2) appear to contradict each other. Using the symbol $\approx$ in equation (2) seems unwise, as the goal of the paper is to provide precise analysis. Why not use ${\cal O}(n)$ or ${\cal \Theta}(n)$, or otherwise describe the relationship more precisely? Furthermore, is a highly accurate version of Stirling’s formula necessary here or elsewhere in the paper?
On page 2, line 40, please provide a reference for the “well known” claim, and clarify what is meant by “in logarithmic time.”
Response: Thank you Professor, for highlighting the typographic error in equation 1, as per the approximation of Euler gamma function, T(n) = log(n!), which can be further simplified to nlogn (the reference is cited). We refrain from using any asymptotic notation to establish the Search-o-Sort theory as a summation of individual computational complexities. The accurate versions of Stirling’s’ formula is used to keep the work apropos the reference cited for the proof [2], where they used an accurate version of the Stirling’s formula by Cristinel Mortici. Please refer to the highlighted portions in page 2 of the manuscript.
Comment #6: On page 1, lines 20–21, the subscript $i$ is overused.
Response: Thank you, Professor, for the suggestion but removing the subscript will cause inconsistency in notation for the readers. Thus, we were unable to address this suggestion.
Comment #6: I do not have the expertise to evaluate the entire paper as a specialist in this field. However, the paper does not appear to be intended solely for experts in the area. Therefore, it would be best to make the exposition accessible to readers with a solid mathematical foundation.
Response: Professor, we understand your concern towards the accessibility of the work for wider group of readers, but the work builds on the theoretical foundations of algorithmic design and analysis, which can’t be overridden otherwise.
Thank you once again Professor for the suggestions and comments.
Round 2
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors fail to respond to my comments seriously. While my comments are not crucial to the significance of the paper itself, it is best to fix all obvious issues, even if they are minor.
One comment is about the use of big O and big Theta. Their usage may be standard in one specialized community, but not in general. The fact that I feel uncomfortable is itself solid evidence.
I do not see that max - min = O(n): if we have a set of number {1, 2^2, 3^3, \ldots n^n}. Then max-min = n^n - 1. If by max, you mean something else, then you are still at fault for not defining it properly.
In short, the authors should have taken my comments with greater care. Go over one by one, either correct the error or explain it away.
Author Response
General Comment: The authors fail to respond to my comments seriously. While my comments are not crucial to the significance of the paper itself, it is best to fix all obvious issues, even if they are minor.
Response: Thank you, Professor, for your suggestions and pointing out the minor issues remaining in the revised version of the manuscript. The modifications have been marked in magenta.
Comment #1: One comment is about the use of big O and big Theta. Their usage may be standard in one specialized community, but not in general. The fact that I feel uncomfortable is itself solid evidence.
Response: Thank you, Professor, for raising this concern. Being from the computer science domain, we are seriously unaware of the standardisation of big Oh and big Theta notations in the general-mathematical domain. To address this concern, we have appended an additional section to the manuscript (please refer to Appendix A). We have tried to define the five prevalent asymptotic notations in algorithmic analysis, with examples relevant to this work.
Comment #2: I do not see that max - min = O(n): if we have a set of number {1, 2^2, 3^3, \ldots n^n}. Then max-min = n^n - 1. If by max, you mean something else, then you are still at fault for not defining it properly.
Response: Thank you, Professor, for the suggestion. In general, for algorithmic analysis, a common assumption is a linear distribution of data indices {0, 1, ... n-1} or {1, 2, ..., n}. Since the context in which \max-\min = \mathcal{O}(n) is defined, is for data indices and has been generally assumed to be linearly distributed, rather than exponentially distributed (or any higher Ackermann function, like tetration), we applied the existing Min-Max algorithm that further guarantees max - min = O(n) in our case. We have mentioned the linear distribution assumption of the data indices and have also formalised the Min-Max algorithm in Appendix B.
Comment #3: In short, the authors should have taken my comments with greater care. Go over one by one, either correct the error or explain it away.
Response: Professor, we respect your comments and suggestions and are sorry as you feel this way. We would like to emphasise that we tried to address all the reviewer comments with equal seriousness. Since we are not from a pure mathematical domain, our comments might seem a little weak in the context of the previous round of revision.
Thank you once again, Professor, for the suggestions and comments. We believe you will find the revised version of the manuscript pertained to your suggestions and comments. Please let us know if any further modifications are needed to improve the technical absorption and readability by a wider audience (if our work is found suitable for publication).
