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Program to Implement Dijkstra's Algorithm Using Set

/* Program to Implement Dijkstra's Algorithm Using Set This is a C++ Program to find shortest path. Dijkstra's algorithm is very similar to Prim's algorithm for minimum spanning tree. Like Prim's MST, we generate a SPT (shortest path tree) with given source as root. We maintain two sets, one set contains vertices included in shortest path tree, other set includes vertices not yet included in shortest path tree. At every step of the algorithm, we find a vertex which is in the other set (set of not yet included) and has minimum distance from source. Below are the detailed steps used in Dijkstra's algorithm to find the shortest path from a single source vertex to all other vertices in the given graph. Algorithm 1) Create a set sptSet (shortest path tree set) that keeps track of vertices included in shortest path tree, i.e., whose minimum distance from source is calculated and finalized. Initially, this set is empty. 2) Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE. Assign distance value as 0 for the source vertex so that it is picked first. 3) While sptSet doesn't include all vertices ....a) Pick a vertex u which is not there in sptSetand has minimum distance value. ....b) Include u to sptSet. ....c) Update distance value of all adjacent vertices of u. To update the distance values, iterate through all adjacent vertices. For every adjacent vertex v, if sum of distance value of u (from source) and weight of edge u-v, is less than the distance value of v, then update the distance value of v. */ // A C++ program for Dijkstra's single source shortest path algorithm. // The program is for adjacency matrix representation of the graph #include <stdio.h> #include <limits.h> // Number of vertices in the graph #define V 9 // A utility function to find the vertex with minimum distance value, from // the set of vertices not yet included in shortest path tree int minDistance(int dist[], bool sptSet[]) { // Initialize min value int min = INT_MAX, min_index; for (int v = 0; v < V; v++) if (sptSet[v] == false && dist[v] <= min) min = dist[v], min_index = v; return min_index; } // A utility function to print the constructed distance array int printSolution(int dist[], int n) { printf("Vertex Distance from Source\n"); for (int i = 0; i < V; i++) printf("%d \t\t %d\n", i, dist[i]); } // Funtion that implements Dijkstra's single source shortest path algorithm // for a graph represented using adjacency matrix representation void dijkstra(int graph[V][V], int src) { int dist[V]; // The output array. dist[i] will hold the shortest // distance from src to i bool sptSet[V]; // sptSet[i] will true if vertex i is included in shortest // path tree or shortest distance from src to i is finalized // Initialize all distances as INFINITE and stpSet[] as false for (int i = 0; i < V; i++) dist[i] = INT_MAX, sptSet[i] = false; // Distance of source vertex from itself is always 0 dist[src] = 0; // Find shortest path for all vertices for (int count = 0; count < V - 1; count++) { // Pick the minimum distance vertex from the set of vertices not // yet processed. u is always equal to src in first iteration. int u = minDistance(dist, sptSet); // Mark the picked vertex as processed sptSet[u] = true; // Update dist value of the adjacent vertices of the picked vertex. for (int v = 0; v < V; v++) // Update dist[v] only if is not in sptSet, there is an edge from // u to v, and total weight of path from src to v through u is // smaller than current value of dist[v] if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX && dist[u] + graph[u][v] < dist[v]) dist[v] = dist[u] + graph[u][v]; } // print the constructed distance array printSolution(dist, V); } // driver program to test above function int main() { /* Let us create the example graph discussed above */ int graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 }, { 4, 0, 8, 0, 0, 0, 0, 11, 0 }, { 0, 8, 0, 7, 0, 4, 0, 0, 2 }, { 0, 0, 7, 0, 9, 14, 0, 0, 0 }, { 0, 0, 0, 9, 0, 10, 0, 0, 0 }, { 0, 0, 4, 0, 10, 0, 2, 0, 0 }, { 0, 0, 0, 14, 0, 2, 0, 1, 6 }, { 8, 11, 0, 0, 0, 0, 1, 0, 7 }, { 0, 0, 2, 0, 0, 0, 6, 7, 0 } }; dijkstra(graph, 0); return 0; }

#include is a way of including a standard or user-defined file in the program and is mostly written at the beginning of any C/C++ program. This directive is read by the preprocessor and orders it to insert the content of a user-defined or system header file into the following program. These files are mainly imported from an outside source into the current program. The process of importing such files that might be system-defined or user-defined is known as File Inclusion. This type of preprocessor directive tells the compiler to include a file in the source code program.

In computer programming, we use the if statement to run a block code only when a certain condition is met. An if statement can be followed by an optional else statement, which executes when the boolean expression is false. There are three forms of if...else statements in C++: • if statement, • if...else statement, • if...else if...else statement, The if statement evaluates the condition inside the parentheses ( ). If the condition evaluates to true, the code inside the body of if is executed. If the condition evaluates to false, the code inside the body of if is skipped.

An array is a collection of data items, all of the same type, accessed using a common name. A one-dimensional array is like a list; A two dimensional array is like a table; The C++ language places no limits on the number of dimensions in an array, though specific implementations may. Some texts refer to one-dimensional arrays as vectors, two-dimensional arrays as matrices, and use the general term arrays when the number of dimensions is unspecified or unimportant. (2D) array in C++ programming is also known as matrix. A matrix can be represented as a table of rows and columns. In C/C++, we can define multi dimensional arrays in simple words as array of arrays. Data in multi dimensional arrays are stored in tabular form (in row major order).

A program shall contain a global function named main, which is the designated start of the program in hosted environment. main() function is the entry point of any C++ program. It is the point at which execution of program is started. When a C++ program is executed, the execution control goes directly to the main() function. Every C++ program have a main() function.

In computer programming, loops are used to repeat a block of code. For example, when you are displaying number from 1 to 100 you may want set the value of a variable to 1 and display it 100 times, increasing its value by 1 on each loop iteration. When you know exactly how many times you want to loop through a block of code, use the for loop instead of a while loop. A for loop is a repetition control structure that allows you to efficiently write a loop that needs to execute a specific number of times.

In the C++ Programming Language, the #define directive allows the definition of macros within your source code. These macro definitions allow constant values to be declared for use throughout your code. Macro definitions are not variables and cannot be changed by your program code like variables. You generally use this syntax when creating constants that represent numbers, strings or expressions. The syntax for creating a constant using #define in the C++ is: #define token value

Return distance between iterators. Calculates the number of elements between first and last. If it is a random-access iterator, the function uses operator- to calculate this. Otherwise, the function uses the increase operator (operator++) repeatedly. The distance() function in C++ helps find the distance between two iterators. In other words, we can use this function to calculate the number of elements present between the two iterators. This function is available in the <iterator> header file. InputIterator shall be at least an input iterator. Function returns the number of elements between first and last.

Logical Operators are used to compare and connect two or more expressions or variables, such that the value of the expression is completely dependent on the original expression or value or variable. We use logical operators to check whether an expression is true or false. If the expression is true, it returns 1 whereas if the expression is false, it returns 0. Assume variable A holds 1 and variable B holds 0:

An array is defined as the collection of similar type of data items stored at contiguous memory locations. Arrays are the derived data type in C++ programming language which can store the primitive type of data such as int, char, double, float, etc. It also has the capability to store the collection of derived data types, such as pointers, structure, etc. The array is the simplest data structure where each data element can be randomly accessed by using its index number. C++ array is beneficial if you have to store similar elements. For example, if we want to store the marks of a student in 6 subjects, then we don't need to define different variables for the marks in the different subject. Instead of that, we can define an array which can store the marks in each subject at the contiguous memory locations.

C++ program displays the maximum number of edge disjoint paths present between two vertices. 'Maximum number' of edge disjoint paths refers to the maximum flow or shortest