92 lines
3.9 KiB
Markdown
92 lines
3.9 KiB
Markdown
# BigInt Technical Details
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In this document you can find a quick overview of the technical
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aspects (internal design, memory layout, etc.) of the `BigInt` data structure.
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`BigInt` is a data type for arbitrary precision arithmetic that supports addition,
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subtraction, multiplication, division and modulo operations on signed integers of unlimited size. Internally, it uses
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the `Vector` data structure to represent big numbers using the following layout:
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```
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Number: 2485795518678991171206065
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Internally: [ 171206065, 518678991, 2485795 ]
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/ | \
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/ | \
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digit[0] digit[1] digit[2]
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(LSB) (MSB)
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```
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That is, each element of the vector stores 9 digits in base $10^9$ using
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**little-endian order**. Each such digits can therefore store values from `0` up to
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`999,999,999`.
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This scheme maps to the following structure:
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```c
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typedef struct {
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vector_t *digits;
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bool is_negative;
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} bigint_t;
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```
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where the `digits` array stores the representation in base $10^9$ of the big integer
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and the boolean `is_negative` variable denotes its sign.
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The `BigInt` data structure supports the following methods:
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- `bigint_result_t bigint_from_int(value)`: creates a big integer from a primitive `int` type;
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- `bigint_result_t bigint_from_string(string_num)`: creates a big integer from a C string;
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- `bigint_result_t bigint_to_string(number)`: converts a big integer to a C string;
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- `bigint_result_t bigint_clone(number)`: clones a big integer;
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- `bigint_result_t bigint_compare(x, y)`: compares two big integers, returning either `-1`, `0` or `1` if the first is less than, equal than or greater than the second, respectively;
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- `bigint_result_t bigint_add(x, y)`: adds two big integers together in $\mathcal{O}(n)$;
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- `bigint_result_t bigint_sub(x, y)`: subtracts two big integers in $\mathcal{O}(n)$;
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- `bigint_result_t bigint_prod(x, y)`: multiplies two big integers using Karatsuba's algorithm in $\mathcal{O}(n^{1.585})$;
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- `bigint_result_t bigint_divmod(x, y)`: divides two big integers using _Knuth's Algorithm D_ in $\mathcal{O}(n \times m)$ where $n$ and $m$ are the number of base-10^9
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parts/limbs in the divisor and the quotient, respectively. This method returns both the quotient and the remainder;
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- `bigint_result_t bigint_mod(x, y)`: calls `bigint_divmod`, discards the quotient and yields the remainder;
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- `bigint_result_t bigint_destroy(number)`: deletes the big number;
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- `bigint_result_t bigint_printf(format, ...)`: `printf` wrapper that introduces the `%B` placeholder to print big numbers. It supports variadic parameters.
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As you can see from the previous function signatures, methods that operate on the
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`BigInt` data type return a custom type called `bigint_result_t` which is defined as
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follows:
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```c
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typedef enum {
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BIGINT_OK = 0x0,
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BIGINT_ERR_ALLOCATE,
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BIGINT_ERR_DIV_BY_ZERO,
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BIGINT_ERR_INVALID
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} bigint_status_t;
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typedef struct {
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bigint_t *quotient;
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bigint_t *remainder;
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} div_result_t;
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typedef struct {
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bigint_status_t status;
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uint8_t message[RESULT_MSG_SIZE];
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union {
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bigint_t *number;
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div_result_t division;
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int8_t compare_status;
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char *string_num;
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} value;
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} bigint_result_t;
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```
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Each method that returns such type indicates whether the operation was successful or not
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by setting the `status` field and by providing a descriptive message on the `message`
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field. If the operation was successful (that is, `status == BIGINT_OK`), you can either
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move on with the rest of the program or read the returned value from the sum data type.
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The sum data type (i.e., the `value` union) defines four different variables. Each
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of them has an unique scope as described below:
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- `number`: result of arithmetical, cloning and creating functions;
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- `division`: result of `bigint_divmod`;
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- `compare_status`: result of `bigint_compare`;
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- `string_num`: result of `bigint_to_string`.
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