Initializing the Model

The model for a program using arithmetic coding needs to have three pieces of information for each symbol: the low end of its range, the high end of its range, and the scale of the entire alphabet’s range (this is the same for all symbols in the alphabet). Since the top of a given symbol’s range is the bottom of the next, we only need to keep track of N + 1 numbers for N symbols in the alphabet.

An example of how the range information would be created is shown for symbols A, B, C, D, and E. These symbols have the counts 5, 2, 3, 3, and 8 respectively. The numbers can be arranged in any order along the range, if done consistently.

 Range : 21 E : 13 D : 10 C : 7 B : 5 A : 0

In this case, the alphabet has five symbols, and the number of counts to keep track of is six. The array is formed by creating the cumulative total at each symbol, starting at zero and working up to the range.

Given a structure like this, it is simple to derive the three items of information that define the symbol for the arithmetic encoder. For symbol x in the array, the low count can be found at totals[ x ], the high count at totals[ x + 1 ], and the range of scale at totals[ N ], N being the number of symbols in the alphabet.

The routines that do compression create this array. In this program, the array is named totals[], and it has 258 elements. The number of symbols in the alphabet is 257, the normal 256 plus one for the end-of-stream symbol.

One additional constraint is placed on these counts. Two things determine the highest possible count: the number of bits available in the high and low registers and the number of bits in the code values. Since floating-point calculations are performed in fixed-length registers, we have to minimize the amount of precision in our calculations so errors do not occur.

As it happens, there are two restrictions on the number of bits used in arithmetic coding: (1) the number of bits in the frequency values must be at least two less than the number of bits in the high and low registers; and (2) the number of bits in the frequency values plus the bits used in either the high and low registers must not exceed the number of bits used in arithmetic calculations during the coding process.

During calculations on the arithmetic registers, the code in this chapter uses unsigned long values, which give 32 bits to work with. Since our high and low registers and our frequency counts are limited to 16 bits in unsigned short ints, we meet restriction 2 implicitly. Restriction 1, however, requires more modifications. Since we are only using 16-bit registers for our high and low values, we have to restrict the highest cumulative counts in the totals[] array to no more than 14 bits, or 16,384.

The code to make sure that our total count of symbols is less than 16,384 is in the build_model routine called on initialization of the model. It takes the process a step further, scaling the counts down so they all fit in a single byte. This is done so that the count data stored in the output file takes a minimal amount of space.

During the compression phase of the program, the build_model() routine is called to perform all the necessary chores associated with the order-0 modeling used in this program. The four lines of code from build_model() are shown here:

```count_bytes( input, counts );
scale_counts( counts, scaled_counts );
output_counts( output, scaled_counts );
build_totals( scaled_counts );
```

As you can see above, the build_model routine does no work on its own. It calls a series of four worker routines to handle the data for it. The first routine is count_bytes(). It does just that, counting all the occurrences of each symbol in the file and maintaining the total in an array, like so:

```input_marker = ftell( input );
while ( ( c = getc( input )) != EOF )
counts[ c ]++;
fseek( input, input_marker, SEEK_SET );
```