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binary trie A data structure in which a left branch represents 0 and a right branch represents 1 The path to a node indicates its representation (p 440) compression The act of reducing the number of bits required for data representation, which actually has two phases: the encoding phase (compression) and the decoding phase (uncompression) (p 440) cross-reference generator A program that lists identifiers and their line numbers It is a common application because it is similar to creating an index (p 461) full tree A tree whose nodes either are leaves or have two children (P 441) Huffman's algorithm An algorithm that constructs an optimal prefix code by repeatedly merging the two minimum weight trees (p 442) prefix code Code in which no character code is a prefix of another character code This condition is guaranteed in a trie if the characters are only in leaves A prefix code can be decoded unambiguously (p 441)
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1 When working with character I/O, you often need to use an i n t to store the characters because of the additional EOF symbol There are several other tricky coding issues For instance, when reading an input stream a second time, you must clear the EOF error
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2 Using too much memory to store the compression table is a common mistake Doing so limits the amount of compression that can be achieved
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The compression program and cross-reference generator is available
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hzipcpp Contains the source for the Huffman coding compression and uncompression program Xrefcpp Contains the source for the cross-reference generator
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In Short
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131 Show the Huffman tree that results from the following distribution of punctuation characters and digits: colon (loo), space (605), newline ( 1 OO), comma (705), 0 (43 I), 1 (242), 2 ( 176), 3 (59), 4 (185), 5 (250), 6 (174), 7 (199), 8 (205), and 9 (217) 132 Most systems come with a compression program Compress several types of files to determine the typical compression rate on your system How large do the files have to be to make compression worthwhile Compare their performance with the Huffman coding program ( h z i p ) provided in the online source code 133 What happens if a file compressed with Huffman's algorithm is used to transmit data over a phone line and a single bit is accidentally lost What can be done in this situation
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134 Prove the correctness of Huffman's algorithm by expanding the following steps a Show that no node has only one child b Show that the two least frequent characters must be the two deepest nodes in the tree c Show that the characters in any two nodes at the same depth can be swapped without affecting optimality d Use induction: As trees are merged, consider the new character set to be the characters in the tree roots
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135 Under what circumstances could a Huffman tree of ASCII characters generate a 2-bit code for some character Under what circumstances could it generate a 20-bit code 136 Show that, if the symbols have already been sorted by frequency, Huffman's algorithm can be implemented in linear time 137 Huffman's algorithm occasionally generates compressed files that are larger than the original Prove that all compression algorithms must have this property (ie, no matter what compression algorithm you design, some input files must always exist for which the algorithm generates compressed files that are larger than the originals)
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138 In the cross-reference generator, store the line numbers in a v e c t o r instead of a 1 i s t and compare performance 139 If a word occurs twice on a line, the cross-reference generator will list it twice Modify the algorithm so that duplicates are only listed once 1310 Modify the algorithm so that, if a word appears on consecutive lines, a range is indicated For example,
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Programming Projects 1311 Modify the C h a r c o u n t e r class to use an array of i n t s instead of a map, and explain whether this modification is likely to affect the program's running time Incorporate the change and measure the running time on a large data file for both compression and uncompression 1312 Storing the character counts in the encoding table gives the uncompression algorithm the ability to perform extra consistency checks Add code that verifies that the result of the uncompression has the same character counts as the encoding table claimed 1313 Describe and implement a method of storing the encoding table that uses less space than the trivial method of storing character counts
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1314 Add the robust error checks for the compression program suggested at the end of Section 1313 1315 Analyze empirically the performance of the compression program and determine whether its speed can be significantly improved If so, make the required changes 1316 Split the Tokenizer class into three classes: an abstract base class that handles the common functionality and two separate derived classes (one that handles the tokenization for the balanced symbol program, and another that handles the tokenization for the crossreference generator)
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Generate an index for a book The input file consists of a set of index entries Each line consists of the string IX:, followed by an index entry name enclosed in braces and then by a page number enclosed in braces Each ! in an index entry name represents a sublevel A I ( represents the start of a range and a I ) represents the end of the range Occasionally, this range will be the same page In that case, output only a single page number Otherwise, do not collapse or expand ranges on your own As an example, Figure 1333 shows sample input and Figure 1334 shows the corresponding output
IX: IX: IX: IX: IX: IX : IX: IX: IX: IX: {series ( 1 {2 ) {series!geometricl ( } {4) {Euler's constant} {4) {~eries!geometricl)} { 4 3 {~eries!arithmetic((} 14) {~eries!arithmeticl)} {5) {~eries!harmonicl 1 ( 153 {Euler'sconstant) {5) {~eries!harmonic( ) } {5} {Series 1 ) 1 {5}