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C++ Program to Check Whether a Vertex Cover of Size k Exists

/* C++ Program to Check Whether a Vertex Cover of Size k Exists This is a C++ Program To Check Whether A Vertex Cover Of Size K exists For A Given Graph The problem takes E edges as input and outputs whehter vertex cover of size K of the graph exists or not. Vertex Cover of a Graph is, a set of vertices S, such that for every edge connecting A to B in graph, either A or B (or both) are present in S. Possible Vertex Covers are: S = {1,2} or {1,3} or {1,4} or {2,3} or {2,4} or {3,4} or {1,2,3} or {1,2,4} or {2,3,4} or {1,3,4} or {1,2,3,4}. Here minimal vertex cover size is 2. So a vertex cover of size 1 does not exist, rest all sizes exist. Some of the possible vertex covers are: {1,2,4,7} or {1,3,5,6} and many more. Here minimum possible vertex cover size is 3, so size 1 and 2 vertex covers won't exist. Size of vertex cover, is the cardinality of the vertex cover. Generate all possible subsets of size K, and check whether it covers all edges or not. 1. Derive an adjacency matrix from the graph. 2. Using Gosper's Hack, generate all subsets. 3. Mark all edges, connected with the vertices in generated subset. 4. Increment the count on visiting new edge. 5. If the final count is equal to edges, then vertex cover is possible. 6. Else, generate new subset, until all subsets are exhausted. 7. If no vertex cover, exist then we do not have a vertex cover of size K. */ #include<bits/stdc++.h> using namespace std; bool graph[1001][1001]; bool vis[1001][1001]; int i,j,k,v; int n,e,x,y; bool isVertexCover(int sz) { int c,r,cnt=0; int set = (1 << sz) - 1; int limit = (1 << n); while (set < limit) { memset(vis,0,sizeof(vis)); cnt = 0; for (j = 1, v = 1 ; j < limit ; j = j << 1, v++) { if (set & j) { for (k = 0 ; k <= n-1 ; k++) { if (graph[v][k] && !vis[v][k]) { vis[v][k] = 1; vis[k][v] = 1; cnt++; } } } } if (cnt == e) return true; c = set & -set; r = set + c; set = (((r^set) >> 2) / c) | r; } return false; } int main() { cout<<"Enter number of vertices:"; cin>>n; cout<<"\nEnter number of Edges:"; cin>>e; for(i=0;i<e;i++) { cout<<"Enter the end-points of edge "<<i+1<<":"; cin>>x>>y; x--; y--; graph[x][y]=1; graph[y][x]=1; } cout<<"Enter the size of Vertex Cover to check for (should be less than number of vertices) :"; cin>>k; if(isVertexCover(k)) cout<<"Vertex Cover of size"<<" "<<k<<" exist.\n"; else cout<<"Vertex Cover of size"<<" "<<k<<" does not exist.\n"; return 0; }

C supports nesting of loops in C. Nesting of loops is the feature in C that allows the looping of statements inside another loop. Any number of loops can be defined inside another loop, i.e., there is no restriction for defining any number of loops. The nesting level can be defined at n times. You can define any type of loop inside another loop; for example, you can define 'while' loop inside a 'for' loop. A loop inside another loop is called a nested loop. The depth of nested loop depends on the complexity of a problem. We can have any number of nested loops as required. Consider a nested loop where the outer loop runs n times and consists of another loop inside it. The inner loop runs m times. Then, the total number of times the inner loop runs during the program execution is n*m.

The cout is a predefined object of ostream class. It is connected with the standard output device, which is usually a display screen. The cout is used in conjunction with stream insertion operator (<<) to display the output on a console. On most program environments, the standard output by default is the screen, and the C++ stream object defined to access it is cout. The "c" in cout refers to "character" and "out" means "output". Hence cout means "character output". The cout object is used along with the insertion operator << in order to display a stream of characters.

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 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.

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:

Check whether eofbit is set. Returns true if the eofbit error state flag is set for the stream. This flag is set by all standard input operations when the End-of-File is reached in the sequence associated with the stream. Note that the value returned by this function depends on the last operation performed on the stream (and not on the next). Operations that attempt to read at the End-of-File fail, and thus both the eofbit and the failbit end up set. This function can be used to check whether the failure is due to reaching the End-of-File or to some other reason.

Consider a situation, when we have two persons with the same name, jhon, in the same class. Whenever we need to differentiate them definitely we would have to use some additional information along with their name, like either the area, if they live in different area or their mother's or father's name, etc. Same situation can arise in your C++ applications. For example, you might be writing some code that has a function called xyz() and there is another library available which is also having same function xyz(). Now the compiler has no way of knowing which version of xyz() function you are referring to within your code.

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 while loop, condition is evaluated first and if it returns true then the statements inside while loop execute, this happens repeatedly until the condition returns false. When condition returns false, the control comes out of loop and jumps to the next statement in the program after while loop. The important point to note when using while loop is that we need to use increment or decrement statement inside while loop so that the loop variable gets changed on each iteration, and at some point condition returns false. This way we can end the execution of while loop otherwise the loop would execute indefinitely. A while loop that never stops is said to be the infinite while loop, when we give the condition in such a way so that it never returns false, then the loops becomes infinite and repeats itself indefinitely.

The cin object is used to accept input from the standard input device i.e. keyboard. It is defined in the iostream header file. C++ cin statement is the instance of the class istream and is used to read input from the standard input device which is usually a keyboard. The extraction operator(>>) is used along with the object cin for reading inputs. The extraction operator extracts the data from the object cin which is entered using the keyboard. The "c" in cin refers to "character" and "in" means "input". Hence cin means "character input". The cin object is used along with the extraction operator >> in order to receive a stream of characters.

Fill block of memory. Sets the first num bytes of the block of memory pointed by ptr to the specified value (interpreted as an unsigned char). This function converts the value of a character to unsigned character and copies it into each of first num character of the object pointed by the given str[]. If the num is larger than string size, it will be undefined.

The bitwise operators are the operators used to perform the operations on the data at the bit-level. When we perform the bitwise operations, then it is also known as bit-level programming. It consists of two digits, either 0 or 1. It is mainly used in numerical computations to make the calculations faster. We have different types of bitwise operators in the C++ programming language. The following is the list of the bitwise operators: Bitwise AND operator is denoted by the single ampersand sign (&). Two integer operands are written on both sides of the (&) operator. If the corresponding bits of both the operands are 1, then the output of the bitwise AND operation is 1; otherwise, the output would be 0. This is one of the most commonly used logical bitwise operators. It is represented by a single ampersand sign (&). Two integer expressions are written on each side of the (&) operator.

The sizeof() is an operator that evaluates the size of data type, constants, variable. It is a compile-time operator as it returns the size of any variable or a constant at the compilation time. The size, which is calculated by the sizeof() operator, is the amount of RAM occupied in the computer. The sizeof is a keyword, but it is a compile-time operator that determines the size, in bytes, of a variable or data type. The sizeof operator can be used to get the size of classes, structures, unions and any other user defined data type. The data_type can be the data type of the data, variables, constants, unions, structures, or any other user-defined data type.

The if...else statement executes two different codes depending upon whether the test expression is true or false. Sometimes, a choice has to be made from more than 2 possibilities. The if...else ladder allows you to check between multiple test expressions and execute different statements. In C/C++ if-else-if ladder helps user decide from among multiple options. The C/C++ if statements are executed from the top down. As soon as one of the conditions controlling the if is true, the statement associated with that if is executed, and the rest of the C else-if ladder is bypassed. If none of the conditions is true, then the final else statement will be executed.

#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.

'Insert element' into the vector. Deleting last element of the vector. Size of the vector and Display by index. "Dislplay by iterator". Clear the vector. Enter value to be inserted. Display

This C++ Program code example deals with all two-d transformation such as translation, 'scaling', 'rotation', 'reflection', 'shearing' in homogeneous coordinates. Transformation