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LectureNotes22 - ECS 120 Lesson 22 Time Complexity Oliver...

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ECS 120 Lesson 22 – Time Complexity Oliver Kreylos Monday, May 21th, 2001 Until now, we investigated the computability of problems – the question whether a certain problem can be solved by some algorithm. We defined a problem to be decidable if and only if there exists a Turing Machine that halts on every input word and that accepts exactly those words that are instances of the problem. In doing so, we neglected the fact that, even if a problem is decidable, it might take a Turing Machine a very long time to halt on some input words. Some decidable problems require any Turing Machine deciding them to take so many steps that solving them is impractical for but the most simple instances. From now on, we will focus our attention on decidable problems and analyze the running-time of Turing Machines solving them – that is, the number of computation steps a Turing Machine has to perform to accept or reject an input word of some length. Our goal is to be able to estimate how long a computation will take before actually performing it. This means, we need tools to estimate running-time from the description of an algorithm alone. 1 Running-time Analysis of Algorithms As a running example, let us consider the language L = 0 n 1 n n 0 . We have seen that this language is context-free, so it must be decidable. A Turing Machine deciding this language is given by the following algorithm: Algorithm L = 0 n 1 n n 0 On input w , 1. Mark the beginning of the tape. 2. Scan until the end of input and reject if any zero appears after any one, or if any other characters but zeros and ones appear in the input word. 1
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3. Return the tape head to the leftmost position. 4. Scan to the right until the first non-crossed-out character is encoun- tered. If it is a blank, accept. If it is a one, reject. 5. Cross out the current character (a zero). 6. Scan to the right until the first character that is neither crossed out nor a zero is encountered. If it is a blank, reject. 7. Cross out the current character (a one). 8. Repeat from step 3. If given an algorithm like the one above, we are now interested in how long a Turing Machine implementing this algorithm would run, or more precisely, how many computation steps it would perform before halting. It is apparent that the number of steps a Turing Machine performs depends on the specific input word. Therefore, every deciding Turing Machine M defines a running- time function rt M : Σ * N , that returns rt M ( w ), the number of steps M performs on input word w . 1.1 Input Size The running-time function rt M as defined above is too difficult to compute for practical purposes – in general, the only way to evaluate it for a word w is to actually run machine M on word w . This defeats the purpose of running- time analysis: We want to be able to estimate how long a computation will take before actually starting it. The first simplification is to not evaluate the running-time for a specific input word, but for a class of input words of the same size . The exact definition of size depends on the particular problem:
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