In a given graph, a tour that commences and concludes at the same vertex, while also visiting all other vertices just once, is known as a Hamiltonian circuit. The complexity of determining the existence of such a circuit in a graph categorizes it as an NP-complete problem.
Deciphering Hamiltonian Circuits Using Backtracking
John developed an algorithm using the backtracking method to identify all Hamiltonian circuits present within a graph. The algorithm accepts an adjacency matrix as its input and subsequently triggers a specific user-defined function, presenting an array that depicts the sequence of vertices for every Hamiltonian circuit detected.
To standardize and prevent redundancy, John decided to initiate all circuits at the vertex labeled “0”. Furthermore, the terminal vertex is always earlier in sequence than the second vertex. This approach averts the repetition of circuits which are just cyclic permutations or simple reversals.
```c
#include <stdlib.h>
static unsigned int circuit_contains(const unsigned int *circuit, unsigned int c, unsigned int v) {
unsigned int i;
unsigned int contains = 0;
for (i = 0; i < c && !contains; i++) {
contains = circuit[i] == v;
}
return contains;
}
typedef void (*circuitfn)(const unsigned int *, size_t);
static void hamiltonian_circuits_recursive(unsigned int **adjmat, size_t n, unsigned int *circuit, unsigned int c, circuitfn fun) {
if (c == n) {
fun(circuit, n);
} else {
unsigned int v;
for (v = 1; v < n; v++) {
if (!circuit_contains(circuit, c, v) && adjmat[circuit[c - 1]][v] == 1 && (c < n - 1 || (adjmat[0][v] == 1 && v < circuit[1]))) {
circuit[c] = v;
hamiltonian_circuits_recursive(adjmat, n, circuit, c + 1, fun);
}
}
}
}
void hamiltonian_circuits(unsigned int **adjmat, size_t n, circuitfn fun) {
unsigned int *circuit;
circuit = malloc(n * sizeof(unsigned int));
if (circuit == NULL) {
return;
}
circuit[0] = 0;
hamiltonian_circuits_recursive(adjmat, n, circuit, 1, fun);
free(circuit);
}
```John also showcased a demonstrative program that efficiently identifies all Hamiltonian circuits in a complete graph with 5 vertices. Keeping in mind the constraints regarding cyclic permutations and direction, a complete graph of order “n” will have exactly (n – 1)!/2 Hamiltonian circuits.
```c
#include <stdio.h>
#include <stdlib.h>
static void print_circuit(const unsigned int *circuit, size_t len) {
unsigned int v;
for (v = 0; v < len; v++) {
printf("%d ", circuit[v]);
}
putchar('\n');
}
int main(void) {
unsigned int **adjmat;
const size_t n = 5;
unsigned int i, j;
adjmat = malloc(n * sizeof(unsigned int *));
for (i = 0; i < n; i++) {
adjmat[i] = malloc(n * sizeof(unsigned int));
for (j = 0; j < n; j++) {
adjmat[i][j] = 1;
}
}
hamiltonian_circuits(adjmat, n, print_circuit);
for (i = 0; i < n; i++) {
free(adjmat[i]);
}
free(adjmat);
return 0;
}
```Upon execution, the program produces a precise output. As an example, for a graph with 5 vertices, 12 unique circuits are identified:
```
0 2 3 4 1
0 2 4 3 1
...
0 4 3 1 2
0 4 3 2 1
```By observing the results, John concluded that the algorithm was efficient and provided a clear representation of the Hamiltonian circuits in the graph.
Conclusion
In wrapping up, understanding and identifying Hamiltonian circuits is pivotal in a myriad of computational problems, spanning from traveling salesman problems to molecular chemistry structures. John’s backtracking algorithm stands out as an astute solution, addressing this intricate NP-complete problem. By accepting an adjacency matrix as its input, the algorithm robustly navigates through the intricacies of the graph, unveiling all possible Hamiltonian circuits. The incorporation of standardization techniques, such as fixing the initial vertex and constraining the terminal one, ensures a comprehensive yet non-redundant result.
The demonstrative program further accentuates the algorithm’s efficacy, particularly for complete graphs. By offering a tangible representation of the Hamiltonian circuits in a graph with 5 vertices, it elucidates the algorithm’s scalability and potential applications in larger, more complex graphs. Such algorithms, when applied aptly, could revolutionize the optimization landscape, offering solutions to age-old problems and perhaps paving the way for novel computational paradigms. As we continue to advance in the realm of computer science, such methods remind us of the intricate beauty embedded within graphs and the boundless possibilities they present.