Practical and Theoretical Limits In Computing Aleksander Ratzloff and Zachary Andrews

Computational Complexity Complexity describes how many resources are required to solve a problem. o These are usually time and space Problems are categorized into classes o P complexity  x 2 resources o NP complexity  2 x resources

Practical Problems In Computing Practical problems usually fall under P- complexity These include a wide variety of problems Searching algorithms Sorting algorithms

Searching Algorithms Searching algorithms are used to find a specific item in a list of items. Algorithm Time required (worst case) Example (worst case) Time required (average case) Example (average case) Linear search n1024n/2512 Binary search 10

Sorting Algorithms Sorting algorithms are used to sort items in a list in a specific order. Algorith m Time required (worst case) Example (worst case) Time required (average case)Example (average case) Selection sort n²n²1024²n²n² Bubble sort n²n²1024²n²n² Insertion sort n²n²1024²n²n² Heap sort1024 * 10 Merge sort 1024 * 10 Quicksortn²n²1024²1024 * 10

Impractical Problems in Computing

Unsolvable Problem in Computing Problem introduced in 1936 by Alan Turing For a given input to a program, it's impossible to determine if the program will loop forever or come to a halt.

Computer Architecture Parallel Computing o Multiple microprocessors used to divide up work of a certain computational problem o Problems solved simultaneously and put back together Parallel-Processing-using-WCF

Computer Architecture Quantum Computing introduced in 1982 by Richard Feynman o Qubits hold value of both 0 and 1 o Problems able to be solved faster o Cryptographic encryption methods would become obsolete

Works Cited Rudich, Steven; Wigderson, Avi. Computational Complexity Theory. American Mathematical Society, Providence, RI, 2004, p. 13, 22. Sedgewick, Robert; Wayne, Kevin. Algorithms, Fourth Edition. Addison-Wesley Professional, Bruen, Aiden A. Cryptography, information theory, and error-correction : a handbook for the 21st century. Wiley-Interscience, Hoboken, NJ, Biham, Eli; Shamir, Adi. Differential Cryptanalysis of the Data Encryption Standard.Springer- Verlag, New York, NY, HowStuffWorks Inc. “How Quantum Computers Work” (Date Accessed 7/26/2012) Imperial College. "A Brief History of Quantum Computing" (Date Accessed: 7/30/12) HowStuffWorks Inc. “How Parallel Processing Works” processing.htm (Date Accessed: 7/28/12) processing.htm

Acknowledgements Dr. Alexander D. Schwab Dr. Jennifer Burris Dr. Barry Kurtz Dr. Rahman Tashakkori Appalachian State University Academy of Science Summer Bridge Program National Science Foundation