# ALGORITHMS JOHNSONBAUGH PDF

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One of the major contentions is the approach to take when there are conflicting constraints. We deem the best approach for optimality is existence of some protocol which suggests a ranking of the constraints. Most approaches have been found to use graph coloring where the classical chromatic number of a graph is found.

This approach finds the smallest number of colors with which the vertices of a graph may be colored so that no two adjacent vertices receive the same color. This is a well known combinatorial optimization problem and is widely encountered in scheduling problems Nieuwoudt Whereas our algorithm schedules examination for a given period, due consideration is given to the uniqueness to the examination scheduling problem; the algorithm attempts to provide a solution that integrates almost all these peculiarities.

Review of Examination Scheduling Algorithms The best algorithms known for finding the chromatic number of a graph have exponential worst-case time complexity in the number of vertices of the graph Rosen Analysis of various examination scheduling algorithms suggests a significant number of these are within a category of only seven types of algorithms. Algorithms 1 , 2 and 3 are based on graph coloring, while algorithms 4 , 5 and 6 are based on other mathematical or graphical techniques.

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Due to the differences in emphases of the various algorithms, there is not an ease of comparison. Instead, we choose to highlight the major strengths of most of these algorithms as the same are considered for our proposed solution.

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The major strengths will be identified with respect to the uniqueness of the examination scheduling problem. We examine first the algorithms that are based on coloring. The maximizing of interexamination gaps spreading of course examinations is a major emphasis of the coloring with the use of heuristics. Their results compared favorably against the actual timetable produced by the current manual system Zhaohui The need not only for maximizing but the ease of spreading of course examinations is a recommended student-centric feature of examination scheduling.

The generic graph coloring algorithm by Malkawi et al has as its objective the achievement of fairness, accuracy and optimal exam time period Malkawi et al The exact methods coloring algorithms also by Nieuwodt, namely the irredundant and the critical coloring algorithms were previously stated to be deemed less efficient than the algorithms with heuristics, and so these were not given detail analysis in this paper in view of the student-centric, growth-oriented examination scheduling algorithm.

The innovative use of three matrices representing event, room and student, and a three-step process of filling events into rooms with a creation of a conflict graph, using max-flow min-cut algorithm to fit allocations into smaller rooms to minimize wastage, and assigning virtual timeslots into actual timeslots, is done by Cheng et al Casey and Thompson employ a greedy randomized adaptive search procedure.

They accomplish a two-phased multi-start method of construction then improvement where the solution is optimized. A number of examination scheduling algorithms exist but there seems to be a very limited number of, if any, algorithm that provides the significant or comprehensive algorithmic solution that considers equitable scheduling course examination in view of lecturers, students, examination administrative personnel and location personnel, and the balancing of each individual and possibly conflicting demands for examination scheduling.

We begin from the basic principle of a non- planar, non-directional, non-bipartite, simple graph G. In our context within a Higher Education Institution HEI , the courses for which there are examinations in a particular period are modeled as vertices of the graph G, and the existence of courses which cannot be done concurrently due primarily to student schedule and workload, is modeled as edges reflecting adjacency.

The coloring of the graph G by the colors C1, C2, …, Cn assigns to each vertex a color Ci so that any vertex has a color different from that of any adjacent vertex Johnsonbaugh , Rosen Notwithstanding, Lewis makes it clear that the examination scheduling problem is indeed NP-Complete and therefore like all other nondeterministic polynomial problems, an exponential growth in time is expected as we endeavor to find an optimal solution.

Whereas our proposal considers complexity analysis, performance analysis and algorithm efficiency algorithm, the major aim of the algorithm was to be student-centric and growth-oriented.

Consequently while we attempt to achieve an optimal examination schedule, the major endeavor was achieving fairness and accuracy amidst efficiency. In order to minimize on the space and time complexity, the degree Epp of each node course is determined and this is matched against the chromatic number Rosen determined within the algorithm. The Examination Scheduling Algorithm in Pseudocode Assuming that we have stored representations of all courses, courses and all course registrations, regist relevant to the period for which an examination schedule is to be generated.

For ease of analysis and usage within the algorithm, we begin from a position that both stored representations allow for direct or indexed accessing of the data, rather than sequential access. Notwithstanding, for our proposed algorithm, only the fields defined are necessary to be accessed from the courses and registrations files for a successful generation of an examination schedule.

Generate Exam Schedule using Coloring Algorithm The algorithm is modular for ease of understanding and enhancement. A student-centric examination schedule is generated in view of the desired constraints.

The algorithm exemplifies flexibility with provisions made for simple constraints not known at the time of initial coding or prior to implementation. Our algorithm spans five modules from the construction of a non-directional course adjacency graph through to a per period listing of all courses that may be done concurrently. The adjacency graph is non- directional and contains an array of neighboring secondary courses per primary course for which there is at least one registered student pursuing both the primary course and secondary course.

The clash matrix is not necessary for the success of the algorithm. However, it is desirable to have a record of the number of students pursuing each primary-secondary course combination. The concept of primary and secondary courses is used only to ensure that a course is fixed while adjacency is being constructed or checked for the other courses.

Whereas constraints are user-driven and are expected to be unique to each organization, it has been found that a number of constraints exists in most solutions for the examination scheduling problem. Our solution also provides a facility for simple user- generated constraints. These would be determined by the lecturer s of the courses. The availability is subject to three variables — the resources may be available in different durations of 2 hours or 3 hours the resources may be available for a difference in a number of sessions per day, namely 2 or 3 units the resources availability may vary in periodic life span, namely 2 weeks or 3 weeks A single resource may have variable usage at a point in time.

A single resource may contain sets of 3-hour examinations and 2-hour examinations.

This is normal in scheduling theory as it is unlikely that two diverse jobs will use the resource for the exact duration. The difference with examination scheduling is external to the actual scheduling. There is dialogue that where the resource is being used and a shorter job comes to an end, namely the 2-hour examination, this could cause undue disruption to the other user of the resource, the 3-hour examination.

Optimality is achieved only when all competing tasks have been assigned the resource. No resource is allocated and all wait in an unused state until there is a signed-off assessment of what is acceptable where optimality is not achieved. This concept is not specific to examination scheduling.

However, uniqueness arises in view of the short life span for which optimality must be achieved. The resource may be setup such that two jobs obtain use of the resource at the same date and time.

Another case may exist for which it is desired that two jobs are to be allocated to the resource but they must be allocated at the same date-different times, or different times-same date.

Efficiency is assessed based on minimizing the number of resources, not necessarily the use of the resources. Due to cost considerations in some HEIs, the scarce resources the examination locations are more efficiently used if each is filled to capacity before another resource is considered.

Additionally, there is merit in using the larger resources first. No attempts were made to solve this uniqueness in the proposed algorithm outlined in this paper. The concept of Fairness or Reasonableness exists. The usage of the resource is to be analyzed in view of spreading the courses per student.

This uniqueness may conflict with that of the efficiency where the attempt is to use the minimum number of rooms. One of the major contentions is the approach to take when there are conflicting constraints. We deem the best approach for optimality is existence of some protocol which suggests a ranking of the constraints. Most approaches have been found to use graph coloring where the classical chromatic number of a graph is found. This approach finds the smallest number of colors with which the vertices of a graph may be colored so that no two adjacent vertices receive the same color.

This is a well known combinatorial optimization problem and is widely encountered in scheduling problems Nieuwoudt Whereas our algorithm schedules examination for a given period, due consideration is given to the uniqueness to the examination scheduling problem; the algorithm attempts to provide a solution that integrates almost all these peculiarities.

Review of Examination Scheduling Algorithms The best algorithms known for finding the chromatic number of a graph have exponential worst-case time complexity in the number of vertices of the graph Rosen Analysis of various examination scheduling algorithms suggests a significant number of these are within a category of only seven types of algorithms.

Algorithms 1 , 2 and 3 are based on graph coloring, while algorithms 4 , 5 and 6 are based on other mathematical or graphical techniques. Due to the differences in emphases of the various algorithms, there is not an ease of comparison. Instead, we choose to highlight the major strengths of most of these algorithms as the same are considered for our proposed solution.

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The major strengths will be identified with respect to the uniqueness of the examination scheduling problem. We examine first the algorithms that are based on coloring. The maximizing of interexamination gaps spreading of course examinations is a major emphasis of the coloring with the use of heuristics. Their results compared favorably against the actual timetable produced by the current manual system Zhaohui The need not only for maximizing but the ease of spreading of course examinations is a recommended student-centric feature of examination scheduling.

The generic graph coloring algorithm by Malkawi et al has as its objective the achievement of fairness, accuracy and optimal exam time period Malkawi et al The exact methods coloring algorithms also by Nieuwodt, namely the irredundant and the critical coloring algorithms were previously stated to be deemed less efficient than the algorithms with heuristics, and so these were not given detail analysis in this paper in view of the student-centric, growth-oriented examination scheduling algorithm.

The innovative use of three matrices representing event, room and student, and a three-step process of filling events into rooms with a creation of a conflict graph, using max-flow min-cut algorithm to fit allocations into smaller rooms to minimize wastage, and assigning virtual timeslots into actual timeslots, is done by Cheng et al Casey and Thompson employ a greedy randomized adaptive search procedure.

They accomplish a two-phased multi-start method of construction then improvement where the solution is optimized. A number of examination scheduling algorithms exist but there seems to be a very limited number of, if any, algorithm that provides the significant or comprehensive algorithmic solution that considers equitable scheduling course examination in view of lecturers, students, examination administrative personnel and location personnel, and the balancing of each individual and possibly conflicting demands for examination scheduling.

## 0math 61 Discrete Mathematics Johnsonbaugh 7e

We begin from the basic principle of a non- planar, non-directional, non-bipartite, simple graph G. In our context within a Higher Education Institution HEI , the courses for which there are examinations in a particular period are modeled as vertices of the graph G, and the existence of courses which cannot be done concurrently due primarily to student schedule and workload, is modeled as edges reflecting adjacency.

The coloring of the graph G by the colors C1, C2, …, Cn assigns to each vertex a color Ci so that any vertex has a color different from that of any adjacent vertex Johnsonbaugh , Rosen A full discussion of the big oh, omega, and theta notations for the growth of functions Section 4. Having all of these notations available makes it possible to make precise statements about the growth of functions and the time and space required by algorithms.

An introduction to number theory Chapter 5. This chapter includes classical results e.

## Algorithms

The major application is the RSA public-key cryptosystem Section 5. The calculations required by the RSA public-key cryptosystem can be performed using the algorithms previously developed in the chapter. Combinations, permutations, discrete probability, and the Pigeonhole Principle Chapter 6. Two optional sections Sections 6.

Recurrence relations and their use in the analysis of algorithms Chapter 7. Graphs, including coverage of graph models of parallel computers, the knights tour, Hamiltonian cycles, graph isomorphisms, and planar graphs Chapter 8.

Theorem 8. Trees, including binary trees, tree traversals, minimal spanning trees, decision trees, the minimum time for sorting, and tree isomorphisms Chapter 9. Networks, the maximal ow algorithm, and matching Chapter A treatment of Boolean algebras that emphasizes the relation of Boolean algebras to combinatorial circuits Chapter An approach to automata emphasizing modeling and applications Chapter The SR ip-op circuit is discussed in Example Fractals, including the von Koch snowake, are described by special kinds of grammars Example The difference with examination scheduling is external to the actual scheduling.

Using paper and pencil.

AS OF: Separate sections Problem-Solving Corners show students how to attack and solve problems and how to do proofs. Whereas there is an examination scheduling system within the institution, students continue to experience unfavorable examination schedule including five exams within three days. More examples and exercises are included to highlight common errors for example, the subsection Some Common Errors that precedes the Section 2.

Since r is true.