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complexity Time of an algorithm: =============================== Time complexity is a characterization of the amount of work performed by a particular algorithm in solving a problem as a function of the problem size. We assume that time to complete the algorithm is directly dependent on the amount of work it does. In "work" we include the following: 1. arithmetic operations (+, -, * or /) 2. number of reads or writes from disk 3. mathematical function calls (sine, cosine, logarithm, etc) 4. comparisons, logical operations We also sometimes include: 5. writes to memory 6. reads from memory 7. any other specific operation considered to be time consuming Since all the above operations can take different times on different computers, and in fact different relative times (i.e., an operation that is 2 times faster than another on computer A is not necessarily 2 times faster on computer B, and it may in fact be slower), we assume for purposes of analysis that each operation takes 1 time unit. In analyzing algorithms, note that a straight-line section of code (that is, a sequence of statements that is executed a single time) always executes a constant amount of work. It performs a fixed number of operations. Such a section of code inside a loop is multiplied by the number of loop iterations; so if the work performed by the code inside the loop is K and the loop repeats N times, then the work done is KN, where K is a constant. As we will see below, multiplication by a constant does not affect the complexity. Therefore our chief concern in analyzing the work of an algorithm is examining the loops and determining how many times the code will execute. If we make a mistake in estimating how many operations are performed in straight-line code, only the constant is affected which makes no difference to the order. We need to pay special attention to nested loops, since in general the total number of times the code inside the innermost loop is repeated is the product of the number of executions of all the loops. This, however is not always easy to calculate because the number of times an inner loop executes may depend on the index of outer loops. Do not, for example, assume that because the outer loop executes N times the inner loop will also, giving you N squared repetitions. Consider, for example, an inner loop that goes from 1 to N - square this will give a total of N cubed. On the other hand, the inner loop index could decrease as the outer loop progresses, which could give a result less than N squared. Formally, we have the following definition: Let W(x) be a function that quantifies (counts) the amount of work performed by an algorithm in terms of the size of input data x, such that W(x)>0 for all x>0. Let there be a function of the form Cf(x), where C is some constant. We say that an algorithm is O( f(x) ) if: 1. There is some value x0>0 such that Cf(x) > W(x) for all x>x0. 2. There is no function g(x)<f(x) for all x>x1 (some value x1) such that condition 1 holds for g(x). That is, there is some function f(x) that, when multiplied by some constant C is always bigger than W(x) for large x, and there is no other function smaller than f(x) for which this is true. If these conditions are met, then we can say that the algorithm behaves asymptotically (for arbitrarily large data size) as the function f(x), and abbreviate this by saying it is O( f(x)). To prove directly that a given algorithm is O( f(x) ) we need to compare the behavior of f(x) with W(x) for large x. Comparison generally is done by subtraction: X-Y>0 means X>Y. Therefore, for condition 1 we need to show: 1. Cf(x)-W(x)>0 for all x>x0 To show condition 2, we can often find some constant K>0 and some value x1: 2. f(x)-KW(x)<0 for all x>x1 (In effect this shows that f(x) is of order W(x), and therefore no function smaller than f(x) could meet condition 1.) Note that the constants C, x0, K, x1 above are all chosen by you. The proof does not require that the conditions hold for every combination of constants, all we need is one example of some constants that do meet the conditions. So pick constants that make it easy to show what you need. For example, since Cf(x) is supposed to be bigger than W, a good rule of thumb is to pick C so it is larger than any constant in W. It is often the case that we can use properties of specific functions W(x) to simplify the proof. For example, if W(x) includes a term Ax, then clearly f(x) cannot be smaller than x (so it could not be log(x) or sqrt(x)). So if W(x)=Ax, then we can show condition 1 true by letting C>A and f(x)=x, in which case Cx > Ax for all x>0. So we can say that Ax is O( x ) immediately, because we know it can not be smaller than x.
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Path: Texas >> OKAZAKIT >> 51686 Fall, 2009
Description: Copyright by Taichiro Okazaki 2004 The Dissertation Committee for Taichiro Okazaki Certifies that this is the approved version of the following dissertation: SEISMIC PERFORMANCE OF LINK-TO-COLUMN CONNECTIONS IN STEEL ECCENTRICALLY BRACED FRAMES Co...
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Path: Texas >> BAMFORDW >> 82161 Fall, 2009
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Path: Texas >> RUSSELLR >> 74662 Fall, 2009
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Path: CSU San Bernardino >> CS >> 201 Fall, 2009
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Path: Texas >> MUKADAMA >> 15106 Fall, 2009
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Path: Texas >> KELLERKM >> 71167 Fall, 2009
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Path: Texas >> ENGELAS >> 504835 Fall, 2009
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stanleyk74304.pdf
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Path: Texas >> PROTSENKOD >> 026 Fall, 2009
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Path: Concordia NE >> PHYS >> 110 Fall, 2009
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Path: Texas >> JOHNSONAM >> 71217 Fall, 2009
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complex.txt
Path: CSU San Bernardino >> CS >> 330 Fall, 2009
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Path: Texas >> HAMILTONT >> 84490 Fall, 2009
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Path: Texas >> KOTRLAKA >> 518287 Fall, 2009
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MATH107A46024536.doc
Path: MD University College >> ASIA >> 2092 Fall, 2009
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goncalvesac026.pdf
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burtnerjc90760.pdf
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Description: Copyright by Erin Wells Bonning 2004 The Dissertation Committee for Erin Wells Bonning certifies that this is the approved version of the following dissertation: Computational and Astrophysical Studies of Black Hole Spacetimes Committee: Richard ...
CMIS141AA44024445.doc
Path: MD University College >> ASIA >> 2092 Fall, 2009
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Path: Texas >> KULKARNIS >> 86095 Fall, 2009
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Path: Texas >> MICHALSKYL >> 026 Fall, 2009
Description: Copyright by Linda Oldfather Michalsky 2002 The Dissertation Committee for Linda Oldfather Michalsky Certifies that this is the approved version of the following dissertation: Evaluation of an Interactive Multimedia Program on Calcium and Folate Co...
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Path: Texas >> LODOWSKID >> 97061 Fall, 2009
Description: Copyright by David T. Lodowski 2004 The Dissertation Committee for David Thomas Lodowski Certifies that this is the approved version of the following dissertation: Structural Basis for the Regulation of GRK2 by G Committee: John Tesmer, Supervisor...
raichlend29983.pdf
Path: Texas >> RAICHLEND >> 29983 Fall, 2009
Description: Copyright by David Allan Raichlen 2004 The Dissertation Committee for David Allan Raichlen Certifies that this is the approved version of the following dissertation: The Relationship Between Limb Muscle Mass Distribution and the Mechanics and Energ...
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Path: Texas >> MEHDIABADI >> 026 Fall, 2009
Description: Copyright by Natasha Jum Mehdiabadi 2002 The Dissertation Committee for Natasha Jum Mehdiabadi Certifies that this is the approved version of the following dissertation: ANT SYMBIOSES: COLONY-LEVEL EFFECTS OF ANTAGONISTIC AND MUTUALISTIC INTERACTION...
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Path: Penn State >> ME >> 581 Fall, 2009
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alvaradocg86236.pdf
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Path: Texas >> COWMEADOWR >> 17589 Fall, 2009
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Path: Texas >> CRABTREEJC >> 17037 Fall, 2009
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