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Look around you Computers and networks are everywhere, enabling an intricate web of complex human activities: education, commerce, entertainment, research, manufacturing, health management, human communication, even war Of the two main technological underpinnings of this amazing proliferation, one is obvious: the breathtaking pace with which advances in microelectronics and chip design have been bringing us faster and faster hardware This book tells the story of the other intellectual enterprise that is crucially fueling the computer revolution: ef cient algorithms It is a fascinating story Gather round and listen close
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01 Books and algorithms
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Two ideas changed the world In 1448 in the German city of Mainz a goldsmith named Johann Gutenberg discovered a way to print books by putting together movable metallic pieces Literacy spread, the Dark Ages ended, the human intellect was liberated, science and technology triumphed, the Industrial Revolution happened Many historians say we owe all this to typography Imagine a world in which only an elite could read these lines! But others insist that the key development was not typography, but algorithms Today we are so used to writing numbers in decimal, that it is easy to forget that Gutenberg would write the number 1448 as MCDXLVIII How do you add two Roman numerals What is MCDXLVIII + DCCCXII (And just try to think about multiplying them) Even a clever man like Gutenberg probably only knew how to add and subtract small numbers using his ngers; for anything more complicated he had to consult an abacus specialist The decimal system, invented in India around AD 600, was a revolution in quantitative reasoning: using only 10 symbols, even very large numbers could be written down compactly, and arithmetic could be done ef ciently on them by following elementary steps Nonetheless these ideas took a long time to spread, hindered by traditional barriers of language, distance, and ignorance The most in uential medium of transmission turned out to be a textbook, written in Arabic in the ninth century by a man who lived in Baghdad Al Khwarizmi laid out the basic methods for adding, multiplying, and dividing numbers even extracting square roots and calculating digits of These procedures were precise, unambiguous, mechanical, ef cient, correct in short, they were algorithms, a term coined to honor the wise man after the decimal system was nally adopted in Europe, many centuries later 9
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Since then, this decimal positional system and its numerical algorithms have played an enormous role in Western civilization They enabled science and technology; they accelerated industry and commerce And when, much later, the computer was nally designed, it explicitly embodied the positional system in its bits and words and arithmetic unit Scientists everywhere then got busy developing more and more complex algorithms for all kinds of problems and inventing novel applications ultimately changing the world
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02 Enter Fibonacci
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Al Khwarizmi s work could not have gained a foothold in the West were it not for the efforts of one man: the 15th century Italian mathematician Leonardo Fibonacci, who saw the potential of the positional system and worked hard to develop it further and propagandize it But today Fibonacci is most widely known for his famous sequence of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, , each the sum of its two immediate predecessors More formally, the Fibonacci numbers F n are generated by the simple rule Fn 1 + Fn 2 if n > 1 Fn = 1 if n = 1 0 if n = 0
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No other sequence of numbers has been studied as extensively, or applied to more elds: biology, demography, art, architecture, music, to name just a few And, together with the powers of 2, it is computer science s favorite sequence In fact, the Fibonacci numbers grow almost as fast as the powers of 2: for example, F 30 is over a million, and F100 is already 21 digits long! In general, F n 20694n (see Exercise 03) But what is the precise value of F100 , or of F200 Fibonacci himself would surely have wanted to know such things To answer, we need an algorithm for computing the nth Fibonacci number An exponential algorithm One idea is to slavishly implement the recursive de nition of F n Here is the resulting algorithm, in the pseudocode notation used throughout this book: function fib1(n) if n = 0: return 0 if n = 1: return 1 return fib1(n 1) + fib1(n 2)
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Whenever we have an algorithm, there are three questions we always ask about it: 1 Is it correct 2 How much time does it take, as a function of n 3 And can we do better 10
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The rst question is moot here, as this algorithm is precisely Fibonacci s de nition of F n But the second demands an answer Let T (n) be the number of computer steps needed to compute fib1(n); what can we say about this function For starters, if n is less than 2, the procedure halts almost immediately, after just a couple of steps Therefore, T (n) 2 for n 1 For larger values of n, there are two recursive invocations of fib1, taking time T (n 1) and T (n 2), respectively, plus three computer steps (checks on the value of n and a nal addition) Therefore, T (n) = T (n 1) + T (n 2) + 3 for n > 1
Compare this to the recurrence relation for F n : we immediately see that T (n) Fn This is very bad news: the running time of the algorithm grows as fast as the Fibonacci numbers! T (n) is exponential in n, which implies that the algorithm is impractically slow except for very small values of n Let s be a little more concrete about just how bad exponential time is To compute F 200 , the fib1 algorithm executes T (200) F 200 2138 elementary computer steps How long this actually takes depends, of course, on the computer used At this time, the fastest computer in the world is the NEC Earth Simulator, which clocks 40 trillion steps per second Even on this machine, fib1(200) would take at least 2 92 seconds This means that, if we start the computation today, it would still be going long after the sun turns into a red giant star But technology is rapidly improving computer speeds have been doubling roughly every 18 months, a phenomenon sometimes called Moore s law With this extraordinary growth, perhaps fib1 will run a lot faster on next year s machines Let s see the running time of fib1(n) is proportional to 20694n (16)n , so it takes 16 times longer to compute F n+1 than Fn And under Moore s law, computers get roughly 16 times faster each year So if we can reasonably compute F100 with this year s technology, then next year we will manage F 101 And the year after, F102 And so on: just one more Fibonacci number every year! Such is the curse of exponential time In short, our naive recursive algorithm is correct but hopelessly inef cient Can we do better A polynomial algorithm Let s try to understand why fib1 is so slow Figure 01 shows the cascade of recursive invocations triggered by a single call to fib1(n) Notice that many computations are repeated! A more sensible scheme would store the intermediate results the values F 0 , F1 , , Fn 1 as soon as they become known function fib2(n) if n = 0 return 0 create an array f[0 n] f[0] = 0, f[1] = 1 for i = 2 n: f[i] = f[i 1] + f[i 2] return f[n] 11
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