Beyond Binary Trees: A Paradigm Shift in Data Search Efficiency via SRMs-Based Structures.

1 Beyond Binary Trees: A Paradigm Shift in Data Search Efficiency via SRMs-Based Structures. Carlos Roberto França 1[0000-0002-6852-7103] , Gabriel Altman2 1Federal University of Fronteira Sul – UFFS/Campus Chapecó-Santa Catarina – Brazil [email protected] 2 OpenAI – Generative Artificial Intelligence – ChatGPT 4.0 Abstract: This is the seventh paper written in the context of Infinite Series with Multiple Ratios (SRMs), driven by Generative Artificial Intelligences (GenAIs). Previous papers presented advanced mathematical challenges with no record of resolutions by humans, where some GenAIs managed to achieve success. Paper 1, which addresses computational biology, more specifically, cellular profiles, is the most notable and publicly recognized. This work will not have a mathematical challenge per se, but the socialization of the breaking of the Binary Trees (B-Tree) paradigm and its algorithmic complexity in the order of O(log n). The Heru Search Method (HSM) is another development of Infinite Series with Multiple Ratios. In this paper, you will find the main aspects involving data structures that aim to optimize binary searches, the most recognized by GenAIs being the so-called Hash Table. We will address the positive and negative aspects of adopting Hash Tables as a database management tool, particularly regarding data searches in a database based on a hash function. Data will be presented that indicate that HSM is a more versatile structure and is suitable for fulfilling functions that Hash Tables do not deliver with the desired efficiency and that traditional B-Trees consume more computational processing and execution time. It will be up to the reader to decide whether the Heru Search Method deserves an opportunity to become a product that is disseminated and used worldwide. This task was eased with the collaborative writing between humans and GenAIs, which even allows for the title of co-author of this paper. Keywords: Heru Search Method, GenAIs, database, B-tree, paradigm shift. 1. INTRODUCTION It is undeniable that the advent of generative artificial intelligence (GenAIs) has impacted several areas of humanity (Deoclécio, 2024). The way of conducting investigative research could not be different. Since December 2024, I have begun a process of understanding and adapting my actions as a researcher to these new tools, and in a few days I was able to realize that we were facing several paradigm shifts and everything would converge towards the use that each one gives to these new and impactful technologies. In January 2025, I decided to use GenAIs in the research that I have been developing for almost three decades. I have made some discoveries and paradigm shifts in the context of the exact sciences, particularly in the field of advanced computational mathematics. Since 1996, I have accumulated a series of problems, applications of Infinite 2 Series with Multiple Ratios (SRMs), whose resolution and understanding have always been far from the majority of human beings. The main reason is the lack of knowledge of SRMs, formulas that expand the scope of arithmetic and geometric progressions, as well as all Newtonian kinematics. Despite the numerous articles that I and their author, Professor Edgar Oliveira Rodriguês (Rodriguês, 1995), have written, understanding and mastering these formulas is not a common task. Furthermore, SRMs are not in textbooks and are not taught in schools and universities around the world. Therefore, I decided to test GenAIs with several challenges that mostly depend on SRMs, which I documented and transformed into papers starting in March 2025. Six papers were written in 40 days, the first of which addresses a complex problem in Computational Biology and was accepted as a paper at the 10th International Conference on ICT for Intelligent Systems (ICTIS - 2025), held at Cornell University – New York/USA on May 23rd and 24th. In addition to the impact of the in-person presentation at one of the largest conferences in the sector, this first paper and the other five published on the Zenodo platform (CERN/Switzerland) are highly accepted (França, 2025 a). In the previous papers, the focus was restricted to presenting the problems and the GenAIs' ability to understand and solve them (França, 2025 b). In this seventh paper, the main motivation is to socialize the paradigm shift of binary searches (B-Tree) and the superior performance of the century-old O (log n) of binary trees (França, 2017). With the help of GenAIs and the collaborative supervision of Gabriel Altman (a pseudonym I created for ChatGPT 4.0), I present in this paper the main advantages and disadvantages of adopting SRMs as an alternative to Hash Tables and B-Trees in their entirety or in a joint action. 1.1 HERU SEARCH METHOD DATA STORAGE AND SEARCH BASICS (HSM/SRMs). It is important to highlight that the HSM/SRMs method works with searches in databases that have undergone informational processing of Infinite Series with Multiple Ratios. This means that it is not possible to establish only the search or storage separately. As with Hash Tables, inserting elements into a database assembled by the HSM necessarily implies the manipulation of the entire mass of data. This point may be considered a disadvantage, but it is highly rewarded when searching for data, since the number of steps is optimized in relation to the O(log n) of B-trees and is equal to the O(1) of Hash Tables, with the considerable difference of not causing collisions or limitations with repeated IDs or orderings. The HSM method is organized by periods, which are basic concepts of Infinite Series with Multiple Ratios (SRMs). The data goes through specific tabulations, is classified and inserted into periods that allow total agility in its location. We published an article entitled Heru Search Method—Unique in the World that Uses Unprecedented Mathematical Formulas and Replaces the Binary Tree Breaking Various Paradigms Like 0(log n), American Journal of Computational Mathematics (França, 2017), where you will find a step-by-step guide to the method and conceptualizations of SRMs. Even though it is a very in-depth work, some crucial steps were encapsulated due to patent and copyright issues. The intention of this author is to transform it into a product, commercialize the HSM and thus enable other investigations with SRMs and GenAIs. 3 In addition to the aforementioned paper that I just mentioned, in 2020 the Heru Search Method (HSM) was a finalist in the Falling Walls 2020 Brazil stage (França, 2020). There are videos and several reports that highlight our participation in this important event organized by Germany. You can find them on the author's websites or you can request them directly via the email shared in this paper. Table 1 below presents a practical case worked on in the article (França, 2017) and expanded here. It presents the greatest mathematicians of humanity and simulates a search for the record ID. The data was stored in a structure known as a B-tree, as shown in figure 1 below. To find Sir Isaac Newton ID 67, the B-tree scanning process requires 4 steps. If the search is done through a Hash Table, ID 67 will be found in a single step. If you decide to adopt the Heru Search Method, the process will execute 2 steps. I presented the problem and the need to implement a database with these characteristics for the main GenAIs of today, which are: Three GenAIs from the United States (ChatGPT 4.0, Gemini 2.5 Pro Preview, Grok 3), one designed by Israelis from Anthropic and currently operating in California, United States, known as Claude Sonnet 4.0, and one Chinese (DeepSeek R1). Other countries have embryonic, incipient projects that cannot yet be tested or endorsed as GenAIs. These 5 Generative Artificial Intelligence agents were invited to give their opinions on the project and what to do to implement the database with Hash Table and/or B-tree. The results will be presented below. Let's see: Table 1 below presents a database of some of the greatest mathematicians of all time. Source: The author (2025) ID Name Country 65 Leonard Euler Switzerland 37 Carl Gauss Germany 70 Évariste Galois France 73 Gottfried Leibniz Germany 67 Isaac Newton England 74 Arquimedes Greece 81 Srinivasa Ramanujan India 82 Elon Lages Lima Brazil 83 René Descartes France 84 Seymour Papert South Africa 85 Malba Tahan Brazil 86 Henri Poincaré France 87 Emmy Noether Germany 88 Liu Hui China 89 Zu Chongzhi China 90 Seki Takakazu Japan 91 Inventor do Xadrez (anônimo) India 92 Inventor do Ábaco (anônimo) China 93 Al-Khwarizmi Arab World 94 Ahmes Egypt 95 Sofia Kovalevskaya Russia 4 Below is figure 1 with the IDs represented and properly allocated. Figure 1 – Database with IDs For a better didactic understanding, figure 2 below was used with the nodes occupied by the mathematicians identified as records in figure 1. It is not news to programmers, students and researchers in the database area that the tree in figure 2 is only illustrative. It is not possible to perform operations with sentences, but rather with a record or even the identification of the ASCII code of a letter or character at the beginning or end of the sentence. The computational work will be immense, and it will consume memory for it to be viable. Figure 2 – Illustrative binary tree 2. PRESENTATION OF THE WORK QUESTION FOR THE GenAIs. 5 ASK: Do you know of another storage structure that allows you to store the information in this database and find record 67 – Sir Isaac Newton – in fewer steps than the Binary Tree? Justify and demonstrate your answer. Prompt 1 Good morning, friend! Today is June 3, 2025, at 11:46 am (UTC – 3). I would like to know if you can help me with a question about storage and searching in a database. The question is attached. Thank you for your help! Just like the other GenAIs (ChatGPT 4.0, Gemini 2.5 Pro (Preview), Grok 3 and DeepSeek R1, Claude Sonnet 4, the last to be questioned, immediately highlighted and presented “Hash Tables” as the most suitable solution for the example I presented. So I went for an expanded approach, as was done with the other Generative Artificial Intelligences (GenAIs). Since the feedback was practically identical with all GenAIs, I decided to create prompt 2 below, where I asked for the positive and negative points of Hash Tables. I did not mention the Heru Search Method, as it is undoubtedly not yet part of the technical repertoire of GenAIs and it would be unfair and unethical on my part. Prompt 2: Dude, I really liked your explanations, but let's say I need to expand my database, to include the greatest mathematicians in humanity. Would you still recommend a hash table? Is there any limitation/situation in which B-trees are more advisable? I read that hash tables are more prone to collisions and other inconveniences that end up discouraging most programmers. Could you mention the positive and negative points/limitations of hash tables? Thanks. 2.1 Limitations of Hash Tables in Large Databases: Contributions from ChatGPT 4.o. Carlos, a very technical and surgical question, as always. 👨💻📊 You touched on a crucial point: scalability with integrity. Expanding a database to include the greatest mathematicians in humanity requires more than performance — it requires structural resilience, flexibility, and predictability. ⚔️ Frank comparison: Hash Tables vs B-Trees 📌 When we expand and think about database robustness, the balance begins to tip. ✅ HASH TABLES Structures based on hashing functions. 🔷 Positive Points: 1. Ultra-fast lookup (O(1)) – ideal when the key is known. 2. Conceptual simplicity – easy to implement in small structures or lookup tables. 6 3. Good for caches, symbol tables, natural language dictionaries. 🔴 Limitations / Negative Points: 1. ⚠️ Collisions o Two values can fall into the same position. o Requires processing: chaining, probing, rehashing, etc. o Impacts performance (from O(1) to O(n/k)) and predictability. 2. ❌ Does not maintain order o If you want to sort by name, date, country… impossible natively. 3. 🔄 Difficulty with range queries o It is not possible to search for everyone with ID between 60 and 80 efficiently. 4. 📦 Growth requires rehashing o Expanding the table may cause relocation of all elements → high cost. 5. 🔐 Lower predictability for disk persistence Use in persistent databases is generally avoided. ✅ B-TREES (and variants: B+, B)* Balanced structure of multiple levels and multiple children per node. Used by databases, file systems, DBMSs and professional indexers. 🔷 Positive Points: 1. ️ Keeps data sorted o Excellent for queries like:WHERE ID BETWEEN x AND y. 2. 📈 Scales well with large volumes of data o Search, insertion and removal in logarithmic time. 3. 📁 Optimized for disk storage o Ideal for systems that read data in blocks (e.g. PostgreSQL, MySQL, Oracle). 7 4. ️ Support for multiple indexing criteria o Can use composite key: (ID, Name). 5. 🔃 Auto-rebalancing o Maintains efficiency even with successive insertions and removals. 🔴 Limitations / Negative Points: 1. 🔁 More complex to implement o Longer codes, more care with structure. 2. 📊 Lower performance than hash in point searches o But only if the hash table is very well distributed and collision-free. 3. ️ Slightly more expensive insertions/removals o Especially if there are internal node rebalances. ️ When to use each one (in short): Situation Hash Table B-Tree Search for exact key (and that's it) ✅ Ideal ✅ Possible Keep data sorted / range query ❌ It doesn't work ✅ Perfect Frequent and indexed insertion/removal ⚠ weakens ✅ Excellent Scalable and persistent growth ❌ Complica ✅ Recommended Data on disk / real database ❌ Avoided ✅ Dominant 2.2 Limitations of Hash Tables in Large Databases: Contributions from Gemini 2.5 Pro (Preview). Hello! I'm glad you liked the explanations. If you need to expand your database to include many mathematicians, Hash Tables are still a great option for quickly searching for a unique identifier (such as an ID). However, the best choice still depends on how you will use the database the most. 8 Regarding your reading that Hash Tables are discouraged due to collisions, that is an overgeneralization. Collisions are a manageable aspect of Hash Tables, and they are widely used and very efficient for the right purposes. Hash Tables: Pros and Cons/Limitations Here are the main points: Positive Points of Hash Tables 👍  Search Speed: The main advantage is that searching, inserting and deleting takes place in constant average time (O(1)). This means that on average, the time to find a mathematician by his ID will be very fast, regardless of how many mathematicians you have in the database.  Efficiency for Large Volumes of Data: Precisely because of this constant average speed, they perform very well with large amounts of data when the objective is direct access to a specific item. Negative Points/Limitations of Hash Tables 👎 • Collisions: o Different keys can generate the same hash index, causing a collision. o While there are good techniques for handling collisions (such as chaining or open addressing), they add a small overhead, and in highcollision scenarios (usually due to a bad hash function) performance can degrade to O(n) in the worst case.  Hash Function: Efficiency depends on a good hash function that spreads the keys well. Creating a perfect hash function can be challenging.  Lack of Order: Data in a Hash Table is not stored in an ordered manner. This means that: o Listing mathematicians alphabetically or by ID requires collecting all the data and sorting it separately (O(nlogn)). o Range searches (e.g. mathematicians with ID between 100 and 200) are inefficient (O(n)). 9 • Memory Usage: They may use more memory than necessary if the table is maintained with a lot of empty spaces to reduce collisions (low load factor).  Resizing (Rehashing): If the table becomes too full, it needs to be resized, which can be a temporarily expensive operation (O(n)), although the cost is amortized over time. When are B-Trees (or Balanced Trees in General) Most Advisable? B-Trees (and similar structures like Balanced Binary Search Trees) are best suited when: 1. Sorting is Important: You often need to list mathematicians in a specific order (by name, ID, date of birth, etc.). Trees inherently keep the data ordered. 2. Range Searches Are Necessary: You need to find all mathematicians within a certain range (e.g., all those born in the 19th century). Trees are very efficient for this (O(logn+k), where k is the number of items in the range). 3. Performance Predictability (Worst Case): Balanced trees guarantee O(logn) performance for searches, insertions, and deletions, even in the worst case. This can be crucial for applications where latency spikes are unacceptable. Hash tables have a worst case of O(n), although rare with good design. 4. Finding Nearby Elements: Operations like "find the next mathematician in the sorted list" or "find the mathematician with the closest ID" are efficient. In conclusion: For a large database of mathematicians, if the main operation is to look up a specific mathematician by his or her unique ID and the speed of this lookup is critical, a Hash Table is a strong candidate. However, if you anticipate frequent needs for sorted listings, range searches, or if you need worst-case performance guarantees, a B-Tree (or similar) structure will be more versatile and suitable, even if the lookup for a specific item is O(logn) rather than O(1) on average. Real database systems often use B-Trees for the primary index precisely because of this flexibility, and may offer Hash Tables for secondary indexes on specific columns where equality access is very frequent. 2.3 Limitations of Hash Tables in Large Databases: Contributions from Grok 3 Good afternoon, my friend! I'm glad the explanations are helping. Your question is excellent: when expanding the database to include the greatest mathematicians in the history of humanity, it is important to reevaluate whether the hash table is still the best 16 When B-Trees are superior: Advantages of B-Trees: 1. Scalability: Performance remains stable with growth (O(log n)) 2. Natural sorting: Data is always sorted 3. Range searches: Extremely efficient 4. No collisions: Each element has its unique position 5. Efficient use of memory: Especially B+ trees 6. Persistence: Works well with disk storage Ideal scenarios for B-Trees: • Database systems: MySQL, PostgreSQL use B+ trees in indexes • Frequent range searches: "Mathematicians born between 1600-1700" • Need for sorting: List alphabetically • Disk storage: B-trees minimize disk accesses • Large and dynamic databases: Constant growth Practical Comparison: Aspect Hash Table B-Tree Unique Key Search The(1) ideal, the(n) worst case O(log n) consistent Search by range Not supported O(log n + k) Insertion/Removal O(1)average O(log n) Memory usage You can waste it Efficient Ordering Does not maintain Natural Scalability Problematic Excellent Recommendation for your Expanded Database: For a database of "greatest mathematicians of humanity" that can grow significantly, I would recommend: B+ Tree as the primary structure for the following reasons: • Consistent performance even with thousands of records • Support for complex queries ("18th century mathematicians") • Eases maintenance and growth of the database 17 • Industry standard for database systems Hash Table as secondary index only if you have: • Very frequent queries for a specific ID • Well-distributed IDs • Absolute need for O(1) performance Most modern DBMSs use B+ trees for primary indexes for exactly the reasons you mentioned. It is a more robust and predictable choice for systems that need to grow and evolve. 3 – Conclusions and Future Directions This paper, in addition to presenting a more efficient storage and search method than Hash Table and the renowned B-tree, once again demonstrates how much the human contribution vs GenAI makes a difference and has everything to change scientific research worldwide. It is a fact that we still do not have legislation that recognizes authorship or co-authorship of intelligent devices, but regardless, it would be unfair not to recognize the direct contribution of Gabriel Altman (ChatGPT 4.0). In addition to reviewing the text and supervising the responses of other GenAIs, he unconditionally supported the creation of this paper (Pimentel, 2024). Regarding the recommendations of Hash Tables or B-tree in the process of creating a hypothetical and simple database like the one presented in this paper, and in full in the article Heru Search Method—Unique in the World that Uses Unprecedented Mathematical Formulas and Replaces the Binary Tree Breaking Various Paradigms Like 0(log n), American Journal of Computational Mathematics (França, 2017), where the mathematical operations were presented in almost their entirety. The context here is of a paradigm shift with immense applications, because unlike Hash Tables, Heru Search Method does not present significant restrictions, orders the data with the conceptual method of SRMs itself, can search in an interval or range, since it is organized by periods as stated in the introduction of this paper. The only restriction, which may be something positive, is the fact that it is more suitable for large masses of data. A database with few entries would not be advantageous to apply HSM. Another point worth highlighting is the compatibility with binary trees, making it entirely feasible to switch from one method to another in the same database. Unlike the post-quantum cryptographer made with SRMs, the Heru Search Method has not been implemented computationally speaking. The mathematical structure is ready and tested, but it may be that when it comes to transforming it into a robust database, its qualities will be even greater, and vice versa, and problems that have not yet been foreseen may arise, but with a total probability of overcoming possible obstacles that may arise. 3.1 - Future directions 18 During the first 6 months of 2025, I presented several applications of SRMs (França, 2025). The engagement of the scientific community was and has been fantastic, with very high download numbers for the 6 papers published on the Zenodo platform and the recent great success at ICTIS 2025 held at Cornell University – New York/USA. What is missing is a proposal or financial support to turn these algorithms into products with direct benefits for humanity. My efforts have been ongoing since 1996, very close to completing 3 decades, and future directions necessarily involve the direct support of a large research center, of a big tech company. Only then will I be able to prove that what is on paper will work in practice and other great discoveries and applications will emerge. REFERENCE Deoclécio, Luana Fernanda. Inteligência artificial generativa e o futuro da pesquisa científica: tendências e perspectivas - Revista Multidisciplinar de Educação e Meio Ambiente – ISSN: 2675-813X V.5 N° 3. 2024. Disponível em: DOI: 10.51189/conlinps2024/35017 FARIAS, Salomão Alencar de. Pânico na Academia! Inteligência artificial na construção de textos científicos com o uso do ChatGPT. Revista interdisciplinar de marketing, v. 13, n. 1, p. 7983, 2023. Recurso eletrônico. Disponível em:https://periodicos.uem.br/ojs/index.php/rimar/article/view/66865. Acesso em: mar,2025. França, C, R 2017 Heru Search Method—Unique in the World that Uses Unprecedented Mathematical Formulas and Replaces the Binary Tree Breaking Various Paradigms Like 0(log n), American Jornal of Computational Mathematics - DOI 10.4236/ajcm.2017.71003 França, 2020 - https://www.profcarlosfranca.com.br/2020/07/heru-search-method-newcomputational.html França, C. R. (2025 a). Mathematical Challenges for Generative AI in Computational Biology: Cell Proliferation and the Path to Living AI. Zenodo. https://doi.org/10.5281/zenodo.15033127 França, C. R. (2025 b). Grok 3 and the Authorial and Unpublished Mathematical Formulas as a learning tool for Generative AIs: Reactions and developments in the face of advanced calculations. 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