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where 8 is the opening angle between the two leptons and where the electron velocity Ve == 1 in our approximation. Here, we have used the trace theorems of Section 6.4; see also (12.25) and (12.26). Substituting (12.21) into (12.20), the transition rate becomes

-[CIA] files" << endl;

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string option = argv[ 1 I ; for( int i = 2 ; i < argc; i++ i i string nextFile = argv[ i 1 ; if( option == "-c" 1 Compressor::compress( nextFile ) ; else if( option == "-u" 1 Compressor::uncompress(nextFile ) ; else i cerr < < "Illegal option; usage: " << argv[OI << ' -[cu] files" < < endl; return 1; 1 1 return 0;

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Finally, main is shown in Figure 1327 If invoked with the -c argument, it compresses; with the -u argument it uncompresses

The program as written serves its main purpose of illustrating the basics of the Huffman coding algorithm It achieves some compression, even on moderately sized files For instance, it obtains roughly 36 percent compression when run on its own source file, hzip cpp However, the program could be improved in several ways

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1 The error checking is extremely limited A production program must verify that files open correctly and must check all reads and all writes It must rigorously ensure that the file being decompressed is actually a compressed file (One way to have it do so is to write extra information in the encoding table) The internal routines should have more checks

(3.11)

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table For large files this lack is of little consequence, but for smaller files a large encoding table can be unacceptable because the encoding table takes up space itself 3 A robust program checks the size of the resulting compressed file and aborts if the size is larger than the original 4 In many places we made little attempt to optimize for speed During compression, input files are read twice, which can be slow on some systems Further, memoization could be used to avoid repeated searching of the tree for codes, and maps could be replaced with a more efficient data structure when the key is simply a charOften vectors are copied and returned, perhaps resulting in a performance hit Further improvements to the program are left as an exercise for you to do

In this section, we design a program called a cross-reference generator that scans a C++ source file, sorts the identifiers, and outputs all the identifiers, along with the line numbers on which they occur One compiler application is to list, for each function, the names of all other functions that it directly calls However, this is a general problem that occurs in many other contexts For instance, it can be used to generalize the creation of an index for a book Another use, spell checking, is described in Exercise 1318 As a spelling checker detects misspelled words in a document, those words are gathered, along with the lines on which they occur This process avoids repeatedly printing out the same misspelled word and indicates where the errors are

(2'17 )

A cross-reference generator lists identifiers and their line numbersIt is a common application because it'is similar to creating an index

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