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``````package bfs.c.simple;

import java.util.Arrays;
import java.util.PriorityQueue;
import java.util.Queue;

/**
* For a faster implementation, see @see ../fast/BFS.java (using adjacency Lists)
*/
public class BFS {
/**
* This returns nothing (yet), it is meant to be a template for whatever you want to do with it,
* e.g. finding the shortest path in a unweighted graph.
* This has a runtime of O(|V|^2) (|V| = number of Nodes), for a faster implementation see @see ../fast/BFS.java (using adjacency Lists)
*
* @param graph an adjacency-matrix-representation of the graph where (x,y) is true if the the there is an edge between nodes x and y.
* @param start the node to start from.
* @return an array containing the shortest distances from the given start node to each other node
*/
public static int[] bfs(boolean[][] graph, int start) {
//A Queue to manage the nodes that have yet to be visited
Queue<Integer> queue = new PriorityQueue<>();
//Adding the node to start from
//A boolean array indicating whether we have already visited a node
boolean[] visited = new boolean[graph.length];
//(The start node is already visited)
visited[start] = true;
// Keeping the distances (might not be necessary depending on your use case)
int[] distances = new int[graph.length]; // No need to set initial values since every node is visted exactly once
//While there are nodes left to visit...
while (!queue.isEmpty()) {
System.out.println("Visited nodes: " + Arrays.toString(visited));
System.out.println("Distances: " + Arrays.toString(distances));
int node = queue.remove();
System.out.println("Removing node " + node + " from the queue...");
//...for all neighboring nodes that haven't been visited yet....
for (int i = 1; i < graph[node].length; i++) {
if (graph[node][i] && !visited[i]) {
// Do whatever you want to do with the node here.
// Visit it, set the distance and add it to the queue
visited[i] = true;
distances[i] = distances[node] + 1;
System.out.println("Visiting node " + i + ", setting its distance to " + distances[i] + " and adding it to the queue");

}
}
}
System.out.println("No more nodes in the queue. Distances: " + Arrays.toString(distances));
return distances;
}
}
``````

## About the algorithm and language used in this code snippet:

The Breadth-first search algorithm is an algorithm used to solve the shortest path problem in a graph without edge weights (i.e. a graph where all nodes are the same “distance” from each other, and they are either connected or not). This means that given a number of nodes and the edges between them, the Breadth-first search algorithm is finds the shortest path from the specified start node to all other nodes. Nodes are sometimes referred to as vertices (plural of vertex) - here, we’ll call them nodes.

## Description of the Algorithm

The basic principle behind the Breadth-first search algorithm is to take the current node (the start node in the beginning) and then add all of its neighbors that we haven’t visited yet to a queue. Continue this with the next node in the queue (in a queue that is the “oldest” node). Before we add a node to the queue, we set its distance to the distance of the current node plus 1 (since all edges are weighted equally), with the distance to the start node being 0. This is repeated until there are no more nodes in the queue (all nodes are visited).

In more detail, this leads to the following Steps:

1. Initialize the distance to the starting node as 0. The distances to all other node do not need to be initialized since every node is visited exactly once.
2. Set all nodes to “unvisited”
3. Add the first node to the queue and label it visited.
4. While there are nodes in the queue:

1. Take a node out of the queue
2. For all nodes next to it that we haven’t visited yet, add them to the queue, set their distance to the distance to the current node plus 1, and set them as “visited”

In the end, the distances to all nodes will be correct.

## Example of the Algorithm

Consider the following graph:

The steps the algorithm performs on this graph if given node 0 as a starting point, in order, are:

1. Visiting node 0
2. Visited nodes: [true, false, false, false, false, false], Distances: [0, 0, 0, 0, 0, 0]

1. Removing node 0 from the queue…
2. Visiting node 1, setting its distance to 1 and adding it to the queue
3. Visiting node 2, setting its distance to 1 and adding it to the queue
3. Visited nodes: [true, true, true, false, false, false], Distances: [0, 1, 1, 0, 0, 0]

1. Removing node 1 from the queue…
2. Visiting node 3, setting its distance to 2 and adding it to the queue
3. Visiting node 4, setting its distance to 2 and adding it to the queue
4. Visited nodes: [true, true, true, true, true, false], Distances: [0, 1, 1, 2, 2, 0]

1. Removing node 2 from the queue…
5. Visited nodes: [true, true, true, true, true, false], Distances: [0, 1, 1, 2, 2, 0]

1. Removing node 3 from the queue…
2. Visiting node 5, setting its distance to 3 and adding it to the queue
6. Visited nodes: [true, true, true, true, true, true], Distances: [0, 1, 1, 2, 2, 3]

1. Removing node 4 from the queue…
7. Visited nodes: [true, true, true, true, true, true], Distances: [0, 1, 1, 2, 2, 3]

1. Removing node 5 from the queue…
8. No more nodes in the queue. Final distances: [0, 1, 1, 2, 2, 3]

## Runtime Complexity of the Algorithm

The runtime complexity of Breadth-first search is O(|E| + |V|) (|V| = number of Nodes, |E| = number of Edges) if adjacency-lists are used. If a we simply search all nodes to find connected nodes in each step, and use a matrix to look up whether two nodes are adjacent, the runtime complexity increases to O(|V|^2).

Depending on the graph this might not matter, since the number of edges can be as big as |V|^2 if all nodes are connected with each other.

## Space Complexity of the Algorithm

The space complexity of Breadth-first search depends on how it is implemented as well and is equal to the runtime complexity.

Breadth-First Search Algorithm in other languages:

# C

C is a compiled language used for many purposes, although it can be primarily found in systems where importance is important. This is because C offers a lot of low-level support for optimization, at the cost of not having some of the convenient abstractions that other languages offer. C is therefore primarily found in situations where available computation power is low such as embedded systems, or situations where required computation power is high, such as simulation or deep learning.

## Getting to “Hello World” in C

The most important things first - here’s how you can run your first line of code in C.

2. If you’re on Windows, you can also install GCC, even though it might cause problems. You also have other options outlined e.g. here.
3. Open a terminal, make sure the `gcc` command is working (or the according command for whichever compiler you’re using), and that the command your’re going to be using is referring to the version you just installed by running `gcc --version`. If you’re getting a “command not found” error (or similar), try restarting your command line, and, if that doesn’t help, your computer. If the issue persists, here are some helpful forum questions for each platform:

4. As soon as that’s working, copy the following snippet into a file named HelloWorld.c:

``````#include<stdio.h>
int main() {
printf("Hello World\n");
return 0;
}``````
5. Change directory by typing `cd path/to/HelloWorld`, then run `gcc HelloWorld.c` to compile the file (which creates the bytecode), then run `./a.out`. This should print “Hello World” to your Terminal.

That’s it! People who know multiple programming languages will notice that the entry barrier in C is a little lower than Java even though it is lower-level, while the entry barrier to Python is lower even though it is higher-level. My personal observation is that low-level and high-level languages tend to have low barriers of entry, whereas mid-level languages have higher barriers.

## Fundamentals in C

To understand algorithms and technologies implemented in C, one first needs to understand what basic programming concepts look like in this particular language. Each of the following snippets should be compiled and run using the commands mentioned above.

### Variables and Arithmetic

Variables in C are statically typed, meaning the content of a variable needs to be specified when writing the code. The datatype for whole numbers, for example is `int`. Numbers with decimal places are typed `float` or `double` depending on the required precision. The type for text ist `String`.

``````#include<stdio.h>

int main() {
int number = 5;
double decimalNumber = 3.25;
double result = number * decimalNumber;
char callout [] = "The number is ";
// In this instance, the values are concatenated rather than added because one of them is a String.
printf("%s", callout);
printf("%f", result);
printf("\n");
return 0;
}``````

### Arrays

Arrays in C are real arrays (as opposed to e.g. Python where they’re implemented as lists). The implications of that are that the size needs to be set when they are created and cannot be changed, but also that they are more efficient in C than they are in Python. Also, contrary to Java, C does not check array bounds. If you access an index that doesn’t exist, the program will read whatever is in the memory at that location (which will probably be gibberish).

``````int integers[5];
integers[3] = 12; // Assigning values to positions in the array
printf("%d\n", integers[0]); // will be 0
printf("%d\n", integers[3]); // will be 12
printf("%d\n", integers[6]); // will print something random that happened to be at that location in memory
return 0;``````

### Conditions

Just like most programming languages, C can do `if-else` statements. Additionally, C can also do `switch-case` statements.

``````int value = 5;
if(value == 5){
printf("%s\n", "Value is 5");
} else if(value < 5){
printf("%s\n", "Value is less than 5");
} else {
printf("%s\n", "Value is something else");
}

switch (value){
case 1:
printf("%s\n", "Value is 1");
break; // Don't go further down the cases
case 2:
printf("%s\n", "Value is 2");
break; // Don't go further down the cases
case 3:
printf("%s\n", "Value is 3");
break; // Don't go further down the cases
case 4:
printf("%s\n", "Value is 4");
break; // Don't go further down the cases
case 5:
printf("%s\n", "Value is 5");
break; // Don't go further down the cases
default:
printf("%s\n", "Value is something else");
}``````

The above C code will print “Value is 5” twice.

### Loops

C supports `for`, `while` as well as `do while` loops. `break` and `continue` statements are also supported. The below example illustrates the differences:

``````int value = 2;
for (int i = 0; i < value; i++) {
printf("%d\n", i);
}
while (value > 0) {
printf("%d\n", value);
value--;
}
do {
printf("%d\n", value);
value--;
} while (value > 0);``````

This will print the following to the terminal:

``````0
1
2
1
0``````

Note the last `0`: it is printed because in the `do-while`-loop, compared to the `while`-loop. the code block is executed at least once before the condition is checked.

### Functions

Functions in C can be declared similar to Java, but require less boilerplate since they don’t need to be part of classes or objects. Here is a minimal example of a function:

``````#include<stdio.h>

int addNumbers(int numberOne, int numberTwo) {
return numberOne + numberTwo;
}

int main() {
C requires the use of curly brackets (`{}`) to surround code blocks in conditions, loops, functions etc.; It also requires semicolons at then end of statements. While this can lead to some annoying syntax errors, it also means the use of whitespace for preferred formatting (e.g. indentation of code pieces) does not affect the code. Note how the Syntax of C is very similar to Java. The Syntax of Java, and many other languages that came after and/or were derived from C copy many aspects of its Syntax.