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Understanding Intractable Algorithms and Exponential Growth in Computer Science

This chapter explores the concept of intractable algorithms, focusing on problems that become unsolvable or impractical as input sizes grow, particularly exponential algorithms. It discusses examples like the Traveling Salesperson Problem and analyses how computational time escalates with increased complexity, offering insights into cases such as chess moves and the Towers of Hanoi. Through these examples, the chapter emphasizes the limitations of computation and the challenges posed by large-scale problems, elucidating the rapid growth of possibilities in algorithmic challenges.

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Understanding Intractable Algorithms and Exponential Growth in Computer Science

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  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 * 10N • 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

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