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Figure 2-2 Comparison of orders of growth. For instance, solving the problem for a matrix of 20 entries will require about a million units of time, but solving the problem for a matrix of 50 entries will require about a million billion units of time. If a unit of time is a millionth of a second, the problem of size 20 will require a second to compute, but the problem of size 50 will require more than 25 years. The ALFRED database is of size 494 1600 = 790,400. Students hoping to graduate need a better algorithm or a different problem! Another example of an intractable problem is the famous traveling salesman problem. This problem is so famous it has its own acronym, TSP. The salesman needs to visit each of several cities, and wants to do so without visiting any city more than once. In the interest of efficiency, the salesman wants to minimize the length of the trip. The salesman must visit each city, but he can visit the cities in any order. Finding the shortest route requires computing the total distance for each permutation of the cities the salesman must visit, and selecting the shortest one. Actually, since a route in one direction is the same distance as the reverse route, only half of the permutations of cities need to be calculated. Since the number of permutations of n objects is equal to n-factorial (n! or n (n 1) (n 2) ... 2 1), the number of routes to test grows as the factorial of the number of cities, divided by 2. So the order of growth for the TSP problem is n-factorial; (n!).
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Q k lg n Classification Constant: run time is fixed, and does not depend upon n. Most instructions are executed once, or only a few times, regardless of the amount of information being processed. Logarithmic: when n increases, so does run time, but much more slowly than n does. When n doubles, lg n increases by a constant, but does not double until n increases to n2. Common in programs which solve large problems by transforming them into smaller problems. Linear: run time varies directly with n. Typically, a small amount of processing is done on each element. When n doubles, run time slightly more than doubles. Common in programs which break a problem down into smaller subproblems, solve them independently, and then combine solutions. Quadratic: when n doubles, runtime increases fourfold. Practical only for small problems; typically the program processes all pairs of input (e.g., in a double nested loop). Exponential: when n doubles, run time squares. This is often the result of a natural, brute force solution. Such problems are not computable in a reasonable time when the problem becomes at all large.
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