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Full-Text Articles in Theory and Algorithms
Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke
Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke
Honors Theses
In this thesis, Pascal's Triangle modulo n will be explored for n prime and n a prime power. Using the results from the case when n is prime, a novel proof of Lucas' Theorem is given. Additionally, using both the results from the exploration of Pascal's Triangle here, as well as previous results, an efficient algorithm for computation of binomial coefficients modulo n (a choose b mod n) is described, and its time complexity is analyzed and compared to naive methods. In particular, the efficient algorithm runs in O(n log(a)) time (as opposed to …
Iterated Local Search Algorithms For Bike Route Generation, Aidan Pieper
Iterated Local Search Algorithms For Bike Route Generation, Aidan Pieper
Honors Theses
Planning routes for recreational cyclists is challenging because they prefer longer more scenic routes, not the shortest one. This problem can be modeled as an instance of the Arc Orienteering Problem (AOP), a known NP-Hard optimization problem. Because no known algorithms exist to solve this optimization problem efficiently, we solve the AOP using heuristic algorithms which trade accuracy for speed. We implement and evaluate two different Iterated Local Search (ILS) heuristic algorithms using an open source routing engine called GraphHopper and the OpenStreetMap data set. We propose ILS variants which our experimental results show can produce better routes at the …
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Honors Theses
This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.