Algorithms

Algorithms - An introduction

Algorithms - An introduction

Algorithms Computer Science

Algorithms

Algorithms - An introduction

The NOTION of an algorithm is basic to all computer programming. Donald E. Knuth

Even you do not care about algorithms, if you program in any programming language you use, produce and read algorithms.

A Programming Language is the language you can express algorithms to the computer can undestand it. So, here I want to make a difference between Programming Languages and Presentation Languages. HTML for me is not a programming language… No worries if you believe it is. 🤓

Of course HTML is a programming language

Defining Algorithms

The Oxford Dictionary define algorithm as:

A set of rules that must be followed when solving a particular problem.

Any computer program is developed to solve a problem. Or to create a problem… no worries 😉

So a program SHOULD follow a Recipe, and that is the algorithm. This is ubiquitous in any computing.

Characteristics

An algorithm is a recipe, but a recipe is not an algorithm. Why? You can undestand a recipe, but a computer cannot! You are much smarter than a computer.

To be an algorithm, a set of rules must have the following features

  1. Finiteness
  2. Definiteness
  3. Input
  4. Output
  5. Effectiveness

1. Finiteness

An algorithm SHOULD always terminate after a finite number of steps.

There are some algorithms that search infinitely for a optimal value. Even this kind of algorithms should have a end, probably an finite number of iteractions or an “stable” value.

2. Definiteness

Each step of an algorithm SHOULD be precisely defined.

You can make assumptions about something you read, but a computer cannot make any assumption. Each step should be precise.

Alt Text

Imagine an algorithm that says:

Divide m by n and assign x with the value

This step is defined? No! You can make questions to this step: Is x an Integer value? If we have m=4 and n=3, the result in x can be x=1 or x=1.333. Depending from the algorithm it can be any value.

The correct step shuold be:

Assign the x (interger) with the value of m divided by n

In an algorithm there is no space for questions in steps.

3. Input

An algorithm has zero or more inputs.

4. Output

An algorithm has one or more outputs.

5. Effectiveness

All steps need to be effective, that means it should be simple enough to be done without any other knowledge

Notation

For writing an algorithm we need:

Sorting a array

How can I sort an array of integers?

GIVEN:
    v    → The given array
    n    → The length of the given array
    x[i] → The value stored in the position i from the array x

START:
  S1: FOR EACH i IN [0, n - 2]:
  S2:   min_index ← i
  S3:   FOR EACH j IN [i + 1, n - 1]:
  S4:     IF v[j] < v[min_index]?
  S5:       min_index ← j
  S6:   IF v[i] != v[min_index]?
  S7:     aux ← v[min_index]
  S8:     v[min_index] ← v[i]
  S9:     v[i] ← aux

About this algorithm we can ask:

  1. Is it finite? Yes, it will iterate over the array.
  2. Is it definite? Yes, every step can be done without any assumption.
  3. What is the input? The array and it’s length.
  4. What is the output? The same array, every change is made inplace
  5. Is it effective? Yes, it will sort any array.

Running with the pen

GIVEN: 
  v = [5, 100, -6, 98, -111, 0]
  n = 6

  S1: i ← 0                                v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S2: min_index ← 0                        v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S3: j ← 1                                v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S4: v[1] < v[0] ? (100 < 5)    NO        v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S3: j ← 2                                v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S4: v[2] < v[0] ? (-6 < 5)     YES       v = [5, 100, -6, 98, -111, 0]   min_index = 0   aux = 0
  S5: min_index ← 2                        v = [5, 100, -6, 98, -111, 0]   min_index = 2   aux = 0
  S3: j ← 3                                v = [5, 100, -6, 98, -111, 0]   min_index = 2   aux = 0
  S4: v[3] < v[2] ? (98 < -6)    NO        v = [5, 100, -6, 98, -111, 0]   min_index = 2   aux = 0
  S3: j ← 4                                v = [5, 100, -6, 98, -111, 0]   min_index = 2   aux = 0
  S4: v[4] < v[2] ? (-111 < -6)  YES       v = [5, 100, -6, 98, -111, 0]   min_index = 2   aux = 0
  S5: min_index ← 4                        v = [5, 100, -6, 98, -111, 0]   min_index = 4   aux = 0
  S3: j ← 5                                v = [5, 100, -6, 98, -111, 0]   min_index = 4   aux = 0
  S4: v[5] < v[4] ? (0 < -111)   NO        v = [5, 100, -6, 98, -111, 0]   min_index = 4   aux = 0
  S6: v[0] != v[4] ? (5 != -111) YES       v = [5, 100, -6, 98, -111, 0]   min_index = 4   aux = -111
  S7: aux ← -111                           v = [5, 100, -6, 98, -111, 0]   min_index = 4   aux = -111
  S8: v[4] ← 5                             v = [5, 100, -6, 98, 5, 0]      min_index = 4   aux = -111
  S9: v[0] = -111                          v = [-111, 100, -6, 98, 5, 0]   min_index = 4   aux = -111
  S1: i ← 1                                v = [-111, 100, -6, 98, 5, 0]   min_index = 4   aux = -111
  S2: min_index ← 1                        v = [-111, 100, -6, 98, 5, 0]   min_index = 1   aux = -111
  S3: j ← 2                                v = [-111, 100, -6, 98, 5, 0]   min_index = 1   aux = -111
  S4: v[2] < v[1] ? (-6 < 100)   YES       v = [-111, 100, -6, 98, 5, 0]   min_index = 1   aux = -111
  S5: min_index ← 2                        v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S3: j ← 3                                v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S4: v[3] < v[2] ? (98 < -6)    NO        v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S3: j ← 4                                v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S4: v[4] < v[2] ? (5 < -6)     NO        v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S3: j ← 5                                v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S4: v[5] < v[2] ? (0 < -6)     NO        v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S6: v[1] != v[2] ? (100 != -6) YES       v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -111
  S7: aux ← -6                             v = [-111, 100, -6, 98, 5, 0]   min_index = 2   aux = -6
  S8: v[2] ← 100                           v = [-111, 100, 100, 98, 5, 0]  min_index = 2   aux = -6
  S9: v[1] = -6                            v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S1: i ← 2                                v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S2: min_index ← 2                        v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S3: j ← 3                                v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S4: v[3] < v[2] ? (98 < 100)   NO        v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S3: j ← 4                                v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S4: v[4] < v[2] ? (5 < 100)    YES       v = [-111, -6, 100, 98, 5, 0]   min_index = 2   aux = -6
  S5: min_index ← 4                        v = [-111, -6, 100, 98, 5, 0]   min_index = 4   aux = -6
  S3: j ← 5                                v = [-111, -6, 100, 98, 5, 0]   min_index = 4   aux = -6
  S4: v[5] < v[4] ? (0 < 5)      YES       v = [-111, -6, 100, 98, 5, 0]   min_index = 4   aux = -6
  S5: min_index ← 5                        v = [-111, -6, 100, 98, 5, 0]   min_index = 5   aux = -6
  S6: v[2] != v[5] ? (100 != 0)  YES       v = [-111, -6, 100, 98, 5, 0]   min_index = 5   aux = -6
  S7: aux ← 0                              v = [-111, -6, 100, 98, 5, 0]   min_index = 5   aux = 0
  S8: v[5] ← 100                           v = [-111, -6, 100, 98, 5, 100] min_index = 5   aux = 0
  S9: v[2] = 0                             v = [-111, -6, 0, 98, 5, 100]   min_index = 5   aux = 0
  S1: i ← 3                                v = [-111, -6, 0, 98, 5, 100]   min_index = 2   aux = 0
  S2: min_index ← 3                        v = [-111, -6, 0, 98, 5, 100]   min_index = 3   aux = 0
  S3: j ← 4                                v = [-111, -6, 0, 98, 5, 100]   min_index = 3   aux = 0
  S4: v[4] < v[3] ? (5 < 98)     YES       v = [-111, -6, 0, 98, 5, 100]   min_index = 3   aux = 0
  S5: min_index ← 4                        v = [-111, -6, 0, 98, 5, 100]   min_index = 4   aux = 0
  S3: j ← 5                                v = [-111, -6, 0, 98, 5, 100]   min_index = 4   aux = 0
  S4: v[5] < v[4] ? (100 < 5)    NO        v = [-111, -6, 0, 98, 5, 100]   min_index = 4   aux = 0
  S6: v[3] != v[4] ? (98 != 5)   YES       v = [-111, -6, 0, 98, 5, 100]   min_index = 4   aux = 0
  S7: aux ← 5                              v = [-111, -6, 0, 98, 5, 100]   min_index = 5   aux = 5
  S8: v[4] ← 98                            v = [-111, -6, 0, 98, 98, 100]  min_index = 5   aux = 5
  S9: v[3] = 5                             v = [-111, -6, 0, 5, 98, 100]   min_index = 5   aux = 5
  S1: i ← 4                                v = [-111, -6, 0, 5, 98, 100]   min_index = 5   aux = 5
  S2: min_index ← 4                        v = [-111, -6, 0, 5, 98, 100]   min_index = 5   aux = 5
  S3: j ← 5                                v = [-111, -6, 0, 5, 98, 100]   min_index = 5   aux = 5
  S4: v[5] < v[4] ? (100 < 98)   NO        v = [-111, -6, 0, 5, 98, 100]   min_index = 5   aux = 5

As we can see, given the input [5, 100, -6, 98, -111, 0], it returns the output [-111, -6, 0, 5, 98, 100]. So in the output there is no value v[i] and v[j] where i < j and v[i] > v[j], that means the array is sorted.

Algorithms

Algorithms - An introduction
Originally published January 09, 2020