Ada Programming/Algorithms/Chapter 6
Chapter 6: Dynamic Programming
editFibonacci numbers
editThe following codes are implementations of the Fibonacci-Numbers examples.
Simple Implementation
edit...
To calculate Fibonacci numbers negative values are not needed so we define an integer type which starts at 0. With the integer type defined you can calculate up until Fib (87)
. Fib (88)
will result in an Constraint_Error
.
type
Integer_Typeis
range
0 .. 999_999_999_999_999_999;
You might notice that there is not equivalence for the assert (n >= 0)
from the original example. Ada will test the correctness of the parameter before the function is called.
function
Fib (n : Integer_Type)return
Integer_Typeis
begin
if
n = 0then
return
0;elsif
n = 1then
return
1;else
return
Fib (n - 1) + Fib (n - 2);end
if
;end
Fib; ...
Cached Implementation
edit...
For this implementation we need a special cache type can also store a -1 as "not calculated" marker
type
Cache_Typeis
range
-1 .. 999_999_999_999_999_999;
The actual type for calculating the fibonacci numbers continues to start at 0. As it is a subtype
of the cache type Ada will automatically convert between the two. (the conversion is - of course - checked for validity)
subtype
Integer_Typeis
Cache_Typerange
0 .. Cache_Type'Last;
In order to know how large the cache need to be we first read the actual value from the command line.
Value : constant
Integer_Type :=
Integer_Type'Value (Ada.Command_Line.Argument (1));
The Cache array starts with element 2 since Fib (0) and Fib (1) are constants and ends with the value we want to calculate.
type
Cache_Arrayis
array
(Integer_Typerange
2 .. Value)of
Cache_Type;
The Cache is initialized to the first valid value of the cache type — this is -1
.
F : Cache_Array := (others
=> Cache_Type'First);
What follows is the actual algorithm.
function
Fib (N : Integer_Type)return
Integer_Typeis
begin
if
N = 0or
else
N = 1then
return
N;elsif
F (N) /= Cache_Type'Firstthen
return
F (N);else
F (N) := Fib (N - 1) + Fib (N - 2);return
F (N);end
if
;end
Fib; ...
This implementation is faithful to the original from the Algorithms book. However, in Ada you would normally do it a little different:
when you use a slightly larger array which also stores the elements 0 and 1 and initializes them to the correct values
type
Cache_Arrayis
array
(Integer_Typerange
0 .. Value)of
Cache_Type; F : Cache_Array := (0 => 0, 1 => 1,others
=> Cache_Type'First);
and then you can remove the first if
path.
if
N = 0or
else
N = 1then
return
N; elsif
F (N) /= Cache_Type'Firstthen
This will save about 45% of the execution-time (measured on Linux i686) while needing only two more elements in the cache array.
Memory Optimized Implementation
editThis version looks just like the original in WikiCode.
type
Integer_Typeis
range
0 .. 999_999_999_999_999_999;function
Fib (N : Integer_Type)return
Integer_Typeis
U : Integer_Type := 0; V : Integer_Type := 1;begin
for
Iin
2 .. Nloop
Calculate_Next :declare
T :constant
Integer_Type := U + V;begin
U := V; V := T;end
Calculate_Next;end
loop
;return
V;end
Fib;
No 64 bit integers
editYour Ada compiler does not support 64 bit integer numbers? Then you could try to use decimal numbers instead. Using decimal numbers results in a slower program (takes about three times as long) but the result will be the same.
The following example shows you how to define a suitable decimal type. Do experiment with the digits
and range
parameters until you get the optimum out of your Ada compiler.
type
Integer_Typeis
delta
1.0digits
18range
0.0 .. 999_999_999_999_999_999.0;
You should know that floating point numbers are unsuitable for the calculation of fibonacci numbers. They will not report an error condition when the number calculated becomes too large — instead they will lose in precision which makes the result meaningless.