Samira Arabgol, Hoo Sang Ko JAISCR, 2013, Vol. 3, No. 3, pp. 251 – 263 DOI 10.2478/jaiscr-2014-0018 [20] D. E. Goldberg, Genetic Algorithms in Search, Waste Management, vol. 29, no. 11, 2009, pp. Optimization, and Machine Learning, Addison- 2874–2879,. Wesley, 1989. [21] S. Jahandideh, S. Jahandideh, E. Barzegari Asad- [22] D. Venkatesan, K. Kannan, and R. Saravanan, ”A B-TREE ALGORITHM COMPLEXITY ANALYSIS TO abadi, M. Askarian, M. M. Movahedi, S. Hos- Genetic Algorithm-based Artificial Neural Net- seini, and M. Jahandideh, ”The Use of Artificial work Model for the Optimization of Machining EVALUATE THE FEASIBILITY OF ITS APPLICATION IN Neural Networks and Multiple Linear Regression Processes,” Neural Computing and Applications, to Predict Rate of Medical Waste Generation,” vol. 18, no. 2, 2009, pp. 135–140. THE UNIVERSITY COURSE TIMETABLING PROBLEM Marco Antonio Cruz Chvez, Alina Martnez Oropeza CIICAp, Universidad Autnoma del Estado de Morelos Av. Universidad 1001. Col. Chamilpa, C.P. 62209. Cuernavaca, Morelos, Mxico [email protected] Abstract This paper presents a comparative analysis of complexity between the B-TREE and the Binary Search Algorithms, both theoretically and experimentally, to evaluate their effi- ciency in finding overlap of classes for students and teachers in the University Course Timetabling Problem (UCTP). According to the theory, B-TREE Search complexity is lower than Binary Search. The performed experimental tests showed the B-TREE Search Algorithm is more efficient than Binary Search, but only using a dataset larger than 75 students per classroom. 1 Introduction sity where there are many constraints to be consid- ered in order to obtain a feasible solution. Resource Scheduling is a very important area To treat the problem of scheduling at a univer- within computer sciences due to its wide applica- sity level, there is a proposed mathematical model tion in different areas, such as manufacturing, the and some benchmarks which are used as a paradigm academic environment, and vehicle routing, among to try to obtain feasible and better solutions. The others. One of the typical problems in this area name of the problem in combinatorial optimiza- is the scheduling in an academic environment [6], tion is the University Course Timetabling Problem where school schedules must be established. Their (UCTP), which has been classified within the set development requires long periods of time due to of NP-Complete problems due to its complexity [1, the number of variables to assess. Also on occasion, 2]. This means that there is no known deterministic unfeasible solutions are obtained because overlaps algorithm bound by a polynomial temporal func- could occur. An overlap is the timeslot in which the tion to solve this kind of problem [3]. Therefore, same event or activity happens simultaneously; in the scientific community has focused its research this context, when scheduling classes, teachers and on the treatment of NP-Complete problems through students cannot give or take different classes during non-deterministic algorithms, which are bound by the same timeslot. polynomial time. These algorithms (better known Designing school schedules is not an easy task, as Heuristics) need to be constantly improved in but its complexity depends on the level. For exam- attempt to obtain better solutions or reduce the ex- ple in an elementary school the complexity is re- ecution time [4]. There are several techniques that duced due to the smaller quantity of teachers, class- have been used to find feasible solutions to this rooms, and the fixed student groups, so they do not problem, but the most common are metaheuristics. have to take into account so many variables. The complexity increases for a high school or a univer- 252 Cruz Chvez M. A., Martnez Oropeza A. To find a good solution to UCTP using any the efficiency of the search algorithms applied to metaheuristic, local search application is necessary. this specific case. The heart of a local search is the application of a The contribution of this work is to find the most neighborhood structure, which generates exchanges efficient search algorithm to detect overlap of events of elements. Iterative local search, guided by an for students and teachers in the UCTP. objective function, can improve the solution of a problem as UCTP, but every swap generated by the The present research is divided as follows. Sec- neighborhood structure through local search could tion one provides a general introduction to the generate an unfeasible solution. To detect a move- problem, and emphasizes the importance of having ment of the neighborhood structure that can gener- an efficient method which can obtain good solu- ate an unfeasible solution in UCTP, it is necessary tions in a reasonable computational time. Section to use a search algorithm. For example, in a UCTP, two presents the formal definition of the Univer- moving a student e to take a class on day i, at a sity Course Timetabling Problem, focusing on the specific hour j, in classroom k, (i, j, k), may cre- explanation of the set of constraints, which define ate schedule conflicts (overlap) for the student and the specific case. Section three discusses the theo- other classes that he takes. To determine if there retical complexity managed in the literature for the is overlap, it is necessary to perform a search in all principal search methods. An explanation is given classrooms (k = 1,..., n) to verify that the student e for the methods of both Binary Search and B-TREE is not taking another class in the same timeslot (i, Search. The fourth section presents the temporal j). functions specified to each algorithm. Section five shows the experimental results, and the compara- According to theory, the search algorithms with tive analysis of both methods, taking into account the least theoretical complexity are B-TREE and their theoretical and experimental complexity. Fi- Binary search. The B-TREE complexity, in the nally, the conclusions of this research are presented worst case, is lower than the Binary search com- in section six. plexity. For this reason, in this work, these two search algorithms were tested for UCTP. A set of hard constraints were accounted for, which ensured 2 University Course Timetabling non-overlapping classes for both teachers and stu- Problem dents [5]. The fulfillment of these constraints must be verified; it is necessary to check that students do The University Course Timetabling Problem not take more than one subject per timeslot (i, j), (UCTP) has been classified within the Complexity and in one classroom k. If a student e takes more Theory as NP-Complete [1]. This problem could be than one subject in the same timeslot (i, j), the stu- solved by deterministic or non-deterministic meth- dent is overlapping; therefore, the obtained solution ods. Using a deterministic method, it is only fea- is unfeasible. sible to obtain the optimal solution for a small in- In this research, the two most efficient search al- stance, because if the instance size grows, the com- gorithms, according to their theoretical complexity, putation time of a deterministic algorithm increases were used to conduct an experimental comparison. exponentially. As a result, most researchers have focused on heuristics, which get approximated so- Application of a search method allows for re- lutions in a reasonable time; although, there is no vision of each student e assigned to an event in a guarantee of their optimality. timeslot in a CR classroom. It also allows for revi- sion of possible overlaps with other events the stu- The features of the components that are consid- dent takes, where an event is a class taken by the ered in this problem, such as students, classrooms, student e. According to this reasoning, the meth- facilities, teachers, and classes, when considered in ods applied to UCTP require a very efficient search combination with the features of limited resources, algorithm within their structure due to the iterative time considerations and the available spaces for as- nature which is required for this problem. Accord- signing the events, make it difficult to get feasible ing to theoretical and experimental results, a con- solutions in a reasonable computing time. clusion was reached as to the best option, based on Cruz Chvez M. A., Martnez Oropeza A. B-TREE ALGORITHM COMPLEXITY ANALYSIS TO . 253 To find a good solution to UCTP using any the efficiency of the search algorithms applied to The goal of finding solutions to the UCTP is all the students assigned to attend the event. metaheuristic, local search application is necessary. this specific case. to obtain a scheduling such that both students and H3. The assigned classroom must have the required The heart of a local search is the application of a The contribution of this work is to find the most teachers can attend all their programmed classes facilities to carry out the assigned event. neighborhood structure, which generates exchanges without time difficulties. The UCTP has some spe- efficient search algorithm to detect overlap of events H4. No classroom can have more than one event in of elements. Iterative local search, guided by an for students and teachers in the UCTP. cific features that must be respected to obtaining a objective function, can improve the solution of a feasible solution. These features are known as con- the same timeslot. problem as UCTP, but every swap generated by the The present research is divided as follows. Sec- straints, and are classified into two kinds of con- H5. No teacher attends more than one event in the neighborhood structure through local search could tion one provides a general introduction to the straints. same timeslot. generate an unfeasible solution. To detect a move- problem, and emphasizes the importance of having an efficient method which can obtain good solu- Hard Constraints: They must be fulfilled com- Set of Soft Constraints ment of the neighborhood structure that can gener- pletely, and must not be violated.