BITS IN A BINARY NUMBER

Code 39 Extended Drawer In JavaUsing Barcode generation for Java Control to generate, create Code 39 Extended image in Java applications.

HOWMANY BITS ARE

Barcode Drawer In JavaUsing Barcode generator for Java Control to generate, create barcode image in Java applications.

The number of bits required represent numbers is logarithmic

Barcode Decoder In JavaUsing Barcode recognizer for Java Control to read, scan read, scan image in Java applications.

REQUIRED TO REPRESENT

Drawing Code 3 Of 9 In C#Using Barcode maker for VS .NET Control to generate, create Code-39 image in .NET framework applications.

CONSECUTIVE INTEGERS

Code 39 Full ASCII Maker In .NETUsing Barcode maker for ASP.NET Control to generate, create Code 39 Extended image in ASP.NET applications.

A 16-bit s h o r t integer represents the 65,536 integers in the range -32,768 to 32,767 In general, B bits are sufficient to represent 2B different integers Thus the number of bits B required to represent N consecutive integers satisfies the equation 2B 2 N Hence we obtain B 2 log N, so the minimum num ber of bits is [log ~ 1(Here [XI is the ceiling function and represents the smallest integer that is at least as large as X The corresponding floor funcrepresents the largest integer that is at least as small as X) tion

Code-39 Encoder In VS .NETUsing Barcode drawer for Visual Studio .NET Control to generate, create Code39 image in VS .NET applications.

REPEATED DOUBLING STARTING X = 1 , HOW MANY FROM

Code 39 Extended Creator In VB.NETUsing Barcode generator for .NET Control to generate, create Code39 image in .NET framework applications.

IS AT LEAST AS LARGE AS

Bar Code Encoder In JavaUsing Barcode creator for Java Control to generate, create barcode image in Java applications.

TIMES SHOULD X BE DOUBLED BEFORE IT

UPCA Generator In JavaUsing Barcode drawer for Java Control to generate, create UPC Symbol image in Java applications.

The ~ o ~ a r i t h m - r n

Generate Code 3 Of 9 In JavaUsing Barcode generator for Java Control to generate, create Code 3/9 image in Java applications.

Suppose that we start with $1 and double it every year How long would it take to save a million dollars In this case, after 1 yr we would have $2; after 2 yr, $4; after 3 yr, $8, and so on In general, after K years we would have 2K dollars, so we want to find the smallest K satisfying 2 2 N This is the same K equation as before, so K = [log ~ 1After 20 yr, we would have more than a million dollars The repeated doubling principle holds that, starting from 1, we can repeatedly double only [log times until we reach N

Encoding Data Matrix 2d Barcode In JavaUsing Barcode printer for Java Control to generate, create Data Matrix image in Java applications.

The repeated doubling principle holds that, starting at 1, we can repeatedly double only logarithmically many times until we reach N

EAN-13 Creation In JavaUsing Barcode generation for Java Control to generate, create EAN13 image in Java applications.

REPEATED HALVING STARTING FROM X = N , IF N IS REPEATEDLY HALVED, HOW M N AY ITERATIONS MUST BE APPLIED TO MAKE N SMALLER THAN O R EQUAL

British Royal Mail 4-State Customer Code Maker In JavaUsing Barcode generation for Java Control to generate, create British Royal Mail 4-State Customer Code image in Java applications.

The repeated halving principle holds that, starting at N, we can halve only logarithmically many times This process is used to obtain logarithmic routines for searching The Nth harmonic number is the sum of the reciprocals of the first N positive integersThe growth rate of the harmonic number is logarithmic

Drawing Bar Code In Visual C#Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in VS .NET applications.

If the division rounds up to the nearest integer (or is real, not integer, division), we have the same problem as with repeated doubling, except that we are going in the opposite direction Once again the answer is [log N1 iterations If the division rounds down, the answer is Llog N ] We can show the difference by starting with X = 3 Two divisions are necessary, unless the division rounds down, in which case only one is needed Many of the algorithms examined in this text contain logarithms, introduced because of the repeated halving principle, which holds that, starting at N, we can halve only logarithmically many times In other words, an algo1)) rithm is O(1og N) if it takes constant (0( time to cut the problem size by a constant fraction (usually 112) This condition follows directly from the fact that there will be O(1og N) iterations of the loop Any constant fraction will do because the fraction is reflected in the base of the logarithm, and Theorem 64 tells us that the base does not matter All the remaining occurrences of logarithms are introduced (either directly or indirectly) by applying Theorem 65 This theorem concerns the Nth harmonic number, which is the sum of the reciprocals of the first N positive integers, and states that the Nth harmonic number, H,, satisfies HN = O(1og N) The proof uses calculus, but you do not need to understand the proof to use the theorem

Code 128B Recognizer In VS .NETUsing Barcode scanner for .NET Control to read, scan read, scan image in VS .NET applications.

Let H N = I In N + 0577

Code 128A Encoder In .NET FrameworkUsing Barcode drawer for ASP.NET Control to generate, create Code128 image in ASP.NET applications.

1 /i Then HN = @(logN ) A more precise estimate is

Printing Bar Code In .NETUsing Barcode generation for ASP.NET Control to generate, create barcode image in ASP.NET applications.

Theorem 65

Painting DataMatrix In .NETUsing Barcode creation for ASP.NET Control to generate, create Data Matrix ECC200 image in ASP.NET applications.

-~Gthrn

Encoding ECC200 In Visual Studio .NETUsing Barcode encoder for .NET Control to generate, create ECC200 image in VS .NET applications.

Analysis

Generating GTIN - 12 In VS .NETUsing Barcode generator for .NET Control to generate, create UPC-A Supplement 5 image in Visual Studio .NET applications.

Proof

The intuition qf the proof is that a discrete ~ l r i 1r~ i t ~ l approximated by , l to show the (continuous) integral The proof uses n cor~st~~rctioi~ that the - 111sum H , can be bounded above and belo\\ 01 J - \tYrh uppropriate Y limits Details are left as Exercise 618