1 / 7

Understanding Intractable Algorithms: Challenges and Insights from Computer Science

This discussion explores the complexities of intractable algorithms in computer science, emphasizing how certain problems become unsolvable within reasonable timeframes as their scale increases. Focusing on practical examples like the Traveling Salesperson Problem and the Towers of Hanoi, the text illustrates the exponential growth of possibilities and the limitations of computing. As algorithms scale, the computation time can explode, raising questions about whether advancements in hardware can truly alleviate these challenges. Analyzing these topics sheds light on the fundamental limits of computing.

gunnar
Télécharger la présentation

Understanding Intractable Algorithms: Challenges and Insights from Computer Science

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Today’s Topics Computer Science Program Execution Time: Intractable Algorithms Upcoming Parallel Computing Great Ideas, Chapter 14 Reading Great Ideas, Chapter 13

  2. On the Limits of Computing • Intractable Algorithms • Computer "crawls" or seems to come to halt for large N • Large problems essentially unsolved • May never be able to compute answer for some obvious questions • Chess • Note: here N is number of moves looking ahead • We have an Algorithm! • Layers of look-ahead: If I do this, then he does this, .... • Problem Solved (?!) • Can Represent Possibilities by Tree • Assume 10 Possibilities Each Move • t = A * 10^N • Exponential ! ! !

  3. Exponential Algorithms • Recognizing Exponential Growth • Things get BIG very rapidly • Numbers seem to EXPLODE • KEY: at each added step, work multiplies rather than adds • Exponential = Intractable • Traveling Salesperson Example • Visit N Cities in Optimal Order • Optimize for minimum: • Time • Distance • Cost • N factorial (N!) Possibilities • N! is (very) roughly N N • Sterling’s approximation: N! = sqrt(2*Pi*N)*(N/e)N • Typical of some very practical problems

  4. Traveling Salesperson Examples • 3 cities 2! = 2 possible routes (1 of interest) • abc • acb • 4 cities 3! = 6 possible routes (3 of interest) • abcd • abdc • acbd • acdb • adbc • adcb • (Only half usually of interest because just reverse of another path)

  5. 5 cities 4! = 24 possible routes abcde abced abdce abdec abecd abedc acbde acbed acdbe acdeb acebd acedb (12 of interest) adbce adbec adcbe adceb adebc adecb aebcd aebdc aecbd aecdb aedbc aedcb Traveling Salesperson Examples

  6. Towers of Hanoi N t t = 0.00549 * 2N 5 .17 sec (for a very old PC) 10 5.62 sec 15 3.00 min 20 1.6 hour 25 2.13 day 30 68.23 day 35 5.98 year What would a faster computer 40 191.3 year do for these numbers? 45 6120 year 50 196 K year 55 6.27 M year 60 201 M year 65 6.42 G year 70 205 G year

  7. Intractable Algorithms • Other Games • More hardware not the answer! • Predicting Yesterday's Weather • Actual Examples for Time Complexity

More Related