6.8 KiB
Algorithm
Algorithm is an exact description of how to solve a problem. Algorithms are basically what programming is all about: we tell computers, in very exact ways (with programming languages), how to solve problems -- we write algorithms. But algorithms don't have to be just computer programs, they are simply instruction for solving problems. (Cooking recipes are sometimes given as an example of a non-computer algorithm. You may write an algorithm for how to survive in a jungle, but it has to be exact; if there is an ambiguity or incompleteness, it is not an algorithm.)
Interesting fact: contrary to intuition there are problems that are mathematically proven to be unsolvable by any algorithm, see undecidability, but for most practically encountered problems we can write an algorithm (though for some problems even our best algorithms can be unusably slow).
Algorithms are mostly written as a series of steps (or instructions); these steps may be specific actions (such as adding two numbers or drawing a pixel to the screen) or conditional jumps to other steps ("if condition X holds then jump to step N, otherwise continue"). These jumps can be used to create branches (in programming known as if-then-else) and loops (these two constructs are known as control structures -- they don't express an action but control where we move in the algorithm itself). All in all, any algorithm can be written with only these three constructs:
- sequence: A series of steps, one after another.
- selection (branches, if-then-else): Two branches (sequences of steps) preceded by a condition; the first branch is executed only if the condition holds, the second ("else") branch is executed only if the condition doesn't hold (e.g. "If user password is correct, log the user in, otherwise print out an error.").
- iteration (loops, repetition): Sequence of steps that's repeated as long as certain condition holds (e.g. "As long as end of file is not reached, read and print out next character from the file.").
Note: in a wider sense algorithms may be expressed in other ways than sequences of steps (non-imperative ways, see declarative languages), even mathematical equations are often called algorithms because they imply the steps towards solving a problem. But we'll stick to the common meaning of algorithm given above.
Additional constructs can be introduced to make programming more comfortable, e.g. subroutines/functions (kind of small subprograms that the main program uses for solving the problem) or switch statements (selection but with more than two branches). Loops are also commonly divided into several types: counted loops, loops with condition and the beginning and loops with condition at the end (for, while and do while in C, respectively). Similarly to mathematical equations, algorithms make use of variables, i.e. values which can change that have a specific name (such as x or myVariable).
Flowcharts are a way of visually expressing algorithms, you have probably seen some. Decision trees are special cases of algorithms that have no loops, you have probably seen some too. Even though some languages (mostly educational such as Snap) are visual and similar to flow charts, it is not practical to create big algorithms in this way -- serious programs are written as a text in programming languages.
Example
Let's write a simple algorithm that counts the number of divisors of given number x and check if the number is prime along the way. (Note that we'll do it in a naive, educational way -- it can be done better). Let's start by writing the steps in plain English:
- Read the number x from the input.
- Set the divisor counter to 0.
- Set currently checked number to 1.
- While currently checked number is lower or equal than x: a. If currently checked number divides x, increase divisor counter by 1. b. Increase currently checked number.
- Write out the divisor counter.
- If divisor counter is equal to 2, write out the number is a prime.
Notice that x, divisor counter and currently checked number are variables. Step 4 is a loop (iteration) and steps a and 6 are branches (selection). The flowchart of this algorithm is:
START
|
V
read x
|
V
set divisor count to 0
|
V
set checked number to 1
|
------------>|
| |
| V no
| checked number <= x ? -------
| | |
| | yes |
| V |
| checked number no |
| divides x ? -------- |
| | | |
| | yes | |
| V | |
| increase divisor | |
| count by 1 | |
| | | |
--------------<------------- |
V
divisor count = 2 ? ------
| |
| yes |
V |
print "number is prime" |
| |
|<---------------|
V
END
This algorithm would be written in Python as:
x = int(input("enter a number: "))
divisors = 0
for i in range(1,x + 1):
if x % i == 0:
divisors = divisors + 1
print("divisors: " + str(divisors))
if divisors == 2:
print("It is a prime!")
and in C as:
#include <stdio.h>
int main(void)
{
int x, divisors = 0;
scanf("%d",&x); // read a number
for (int i = 1; i <= x; ++i)
if (x % i == 0)
divisors = divisors + 1;
printf("number of divisors: %d\n",divisors);
if (divisors == 2)
puts("It is a prime!");
return 0;
}
Study of Algorithms
TODO: sorting, searching, classical algorithms (primes, haugh transform, FFT, ...), time/space complexity, genetic programming, design patterns, fizzbuzz