vb.net print barcode zebra Figure 15.62 Graph isomorphisms in Java

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Figure 15.62 Graph isomorphisms
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15.19 15.20
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The adjacency matrix and the adjacency list are shown in Figure 15.63 on page 318. a. The breadth-first search visits ABDECHFIGKLJMONPQ; its spanning tree is shown on the left in Figure 15.64 on page 318. b. The depth-first search visits ABCFEIHDKLMJGNPQO; its spanning tree is shown on the right in Figure 15.64 on page 318.
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GRAPHS
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Figure 15.63 Adjacency matrix and adjacency list
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Figure 15.64 Breadth-first search and depth-first search
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APPENDIX
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This appendix summarizes mathematical topics used in the study of data structures. THE FLOOR AND CEILING FUNCTIONS The floor and ceiling functions return the two nearest integers of a given real number. The floor of x, denoted by x , is the greatest integer that is not greater than x. The ceiling of x, denoted by x , is the smallest integer that is not smaller than x. Here are the main properties of these two functions. (The symbol stands for the set of all integers.) Theorem A.1 Properties of the Floor and Ceiling Functions 1. x = max{m | m x}, and x = min{n | n x}. 2. x x < x + 1, and x 1 < x x . 3. x 1 < x x x < x + 1. 4. If n and n x < n + 1, then n = x . If n and n 1 < x < n, then n = x . 5. If x , then x = x = x . 6. If x , then x < x < x . 7. x x and x = x 8. x + 1 = x and x + 1 x LOGARITHMS The logarithm with base b of a positive number x is the exponent y on b for which b y = x. For example, the logarithm of 1000 base 10 is 3 because 103 = 1000. This is written log10 1000 = 3. The logarithm with base 2 is called the binary logarithm and is written lg x = log 2 x. For example, lg 8 = 3. As a mathematical function, the logarithm is the inverse of the exponential function with the same base: by = x y = log b x For example, 3 = lg 8 because 23 = 8.
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ESSENTIAL MATHEMATICS
[APPENDIX
Theorem A.2 Laws of Logarithms 1. log b (b y) = y 2. b logb x = x 3. log b uv = log b u + log b v 4. log b u/v = log b u log b v 5. log b u v = v log b u 6. log b x = (log c x)/(log c b) = (log b c) (log c x) 7. For a positive integer n, lg n + 1 = lgn + 1.
EXAMPLE A.1 Applying the Laws of Logarithms
log 2 256 = log 2 (28) = 8 log 2 1000 = (log 10 1000)/(log 10 2) = 3/0.30103 = 9.966 log 2 1,000,000,000,000 = log 2 10004 = 4(log 2 1000) = 4(9.966) = 39.86 (ln n)/(lg n) = (log e n)/(log 2 n) = log e 2 = ln 2 = 0.693147, for any n > 1
ASYMPTOTIC COMPLEXITY CLASSES In computer science, algorithms are classified by their complexity functions. These are functions that describe an algorithm s running time relative to the size of the problem. For example, the Bubble Sort belongs to the complexity class (n2). This means that if the Bubble Sort takes T milliseconds to sort an array of n elements, then it will take about 4T milliseconds to sort an array of 2n elements because (2n)2 = 4n2. The symbol () is one of five symbols used to describe complexity functions. They all can be defined in terms of the ratios of f(n) and g(n), where f(n) is the algorithm s timing function and g(n) is a characterizing function such as lg n or n2. For a given function g(n), the five asymptotic complexity classes are O(g(n)) = { f(n) S | f(n)/g(n) is bounded } (g(n)) = { f(n) S | g(n)/f(n) is bounded } (g(n)) = { f(n) S | f(n)/g(n) is bounded and g(n)/f(n) is bounded } } o(g(n)) = { f(n) S | f(n)/g(n) 0 as n (g(n)) = { f(n) S | g(n)/f(n) 0 as n } These definitions assume that f(n) and g(n) that are positive ascending functions. As sets of functions, o(g) O(g) (g) (g) (g) = O(g) (g) EXAMPLE A.2 Asymptotic Growth Classes
For every k > 0, nk = o(2n ), because nk/2n 0. For every k > 0, (lgn)k = o(n), because (lgn)k/n 0. For every base b > 1, log b n = (lgn), because log b n/lg n = log b 2. The factorial numbers n! = (2n ), because 2n/n! 0.
APPENDIX]
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