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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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A factorial order of growth is even more extreme than an exponential order of growth. For example, there are about 3.6 million permutations of 10 cities, but more than 2 trillion billion permutations of 20. If the computer can compute the distance for a million permutations a second, the TSP problem will take 1.8 seconds for 10 cities, but tens of thousands of years for 20 cities. Figure 2-2 shows the rates of growth for lg n, n, n(lg n), n2, 2n, and n! Table 2.1 summarizes some different orders of growth, and the characteristics of associated algorithms. ALGORITHMS AS TECHNOLOGY It s pretty exciting to buy a new computer with twice, four times, or even ten times the clock rate of the old computer. Many people think of computer hardware speed as the measure of technological advance. Having discussed algorithms and their performance, consider whether a better algorithm on a slower computer might be better than a slower algorithm on a faster computer. As an example, consider a sorting task. Suppose you need to sort a million numbers (social security numbers, for example). You have the choice of using your current computer with a merge sort program, or of buying a new computer, which is 10 times faster, but which uses an insertion sort. The insertion sort on the new computer will require on the order of (10 6)2, or a million million cycles, while the merge sort will require on the order of 10 6(lg 10 6), or 10 6(20), or 20 million cycles. Even when it runs on your old computer, the merge sort will still run four orders of magnitude faster than the insertion sort on the new machine. If it takes 20 seconds to run the merge sort on your old machine, it will take over 27 hours to run the insertion sort on the new machine! Algorithm design should be considered important technology. A better algorithm can make the difference between being able to solve the problem or not, and a better algorithm can make a much greater difference than any near-term improvement in hardware speed. FORMAL MODELS OF COMPUTATION The theory of computing has advanced by adopting formal models of computation whose properties can be explored mathematically. The most influential model was proposed by the mathematician Alan Turing in 1936. Turing used the human as the model computing agent. He imagined a human, in a certain state of mind, looking at a symbol on paper. The human reacts to the symbol on paper by 1 erasing the symbol, or erasing the symbol and writing a new symbol, or neither, 2 perhaps changing his or her state of mind as a result of contemplating the symbol, and then 3 contemplating another symbol on the paper, next to the first. This model of computation captures the ability to accept input (from the paper), store information in memory (also on the paper), take different actions depending on the input and the computing agent s state of mind, and produce output (also on the paper). Turing recast this drastically simple model of computation into mathematical form, and derived some very fundamental discoveries about the nature of computation. In particular, Turing proved that some important problems cannot be solved with any algorithm. He proved not that these problems have no known solution; he proved that these problems cannot ever have a solution. For instance, he proved that one will never be able to write one program that will be able to determine whether any other arbitrary program will execute to a proper completion, or crash. Hmmm... that s too bad... it would be nice to have a program to check our work and tell us whether or not our new program will ever crash. The mathematical conception of Turing s model of computation is called a Turing machine or TM. A TM is usually described as a machine reading a tape.
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The tape contains symbols or blanks, and the tape can be infinitely long. The machine can read one symbol at a time, the symbol positioned under the read/write head of the TM. The machine can also erase the symbol, or write a new symbol, and it can then position the tape one cell to the left or right. The machine itself can be in one of a finite number of states, and reading a symbol can cause the state of the TM to change. A special state is the halting state, which is the state of the machine when it terminates normally.
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