/// <summary>
/// performs a Branch and Bound search of the state space of partial tours
/// stops when time limit expires and uses BSSF as solution
/// </summary>
/// <returns>results array for GUI that contains three ints: cost of solution, time spent to find solution, number of solutions found during search (not counting initial BSSF estimate)</returns>
public string[] bBSolveProblem()
{
// Overall the worst time and space complexity would be O(n!) and
// the time complexity would be O(n!) where n is the amount of
// cities possible for our route. We know, however, that the branch
// and bound method will do much better as it prunes off states from the
// queue. If we consider the pruning doing a really good job,
// then time complexity tends to be O(n^3) and space complexity tends to be
// O(n^3). This is because ideally be prune often enough that the final
// solution has the most space.
string[] results = new string[3];
Stopwatch timer = new Stopwatch();
int count = 0;
timer.Start();
string[] greedy_r = greedySolveProblem();
// Gready takes O(n^2) time and O(n) space.
State root = new State(Cities.Length);
List <State> queue = new List <State>();
// O(n^2) time and O(n^2) space for n cities in a State S.
for (int i = 0; i < Cities.Length; i++)
{
for (int j = 0; j < Cities.Length; j++)
{
if (i == j)
{
root.matrix[i, j] = double.PositiveInfinity;
}
else
{
root.matrix[i, j] = Cities[i].costToGetTo(Cities[j]);
}
}
}
root.find_lb(); // Takes a most O(n^3) time.
root.route.Add(Cities[0]);
double root_lb = root.lower_bound;
int max_states = 1;
int updates = 0;
int states_created = 0;
int pruned = 0;
int from = 0;
State at_node = root;
do
{
// The do-while loop would cost at most O(n!) becuase each node would
// be considered. But the branch and bound method reduces this significantly.
double current_count = bssf.costOfRoute();
foreach (int to in at_node.get_children(from))
{
// At most there will be n - 1 children to get. Therefore,
// this loop will cost O(n) for each state calculated.
if (to == 0 && at_node.route.Count == Cities.Length)
{
if (root_lb == at_node.lower_bound)
{
count++;
updates++;
bssf = new TSPSolution(at_node.route);
timer.Stop();
results[COST] = costOfBssf().ToString();
results[TIME] = timer.Elapsed.ToString();
results[COUNT] = count.ToString();
return(results);
}
TSPSolution t_bssf = new TSPSolution(at_node.route);
if (t_bssf.costOfRoute() < current_count)
{
count++;
updates++;
bssf = t_bssf;
}
}
else if (to != 0)
{
State temp = new State(Cities.Length);
states_created++;
temp.lower_bound = at_node.matrix[from, to];
// Copying takes O(n^2) time.
Array.Copy(at_node.matrix, temp.matrix, Cities.Length * Cities.Length);
List <Tuple <int, int> > zeros = new List <Tuple <int, int> >();
for (int j = 0; j < Cities.Length; j++)
{
// This loop checks if we are turing a zero into infinity
// and then changes a whole row and column into infinity
// (See TSP branch and bound for details).
if (temp.matrix[from, j] == 0 && j != to)
{
zeros.Add(new Tuple <int, int>(from, j));
}
if (temp.matrix[j, to] == 0 && j != from)
{
zeros.Add(new Tuple <int, int>(j, to));
}
temp.matrix[from, j] = double.PositiveInfinity;
temp.matrix[j, to] = double.PositiveInfinity;
}
if (temp.matrix[from, to] == 0)
{
zeros.Add(new Tuple <int, int>(from, to));
}
temp.matrix[to, from] = double.PositiveInfinity;
// Used to use find_lb() but at wost check_zeros is O(n^2)
// instead of O(n^3).
temp.check_zeros(zeros);
//temp.lb_find()
temp.lower_bound += at_node.lower_bound;
if (temp.lower_bound < current_count)
{
temp.number = to;
temp.depth = at_node.depth + 1;
temp.do_priority();
temp.route = new ArrayList(at_node.route);
temp.route.Add(Cities[to]);
queue.Add(temp);
}
else
{
pruned++;
}
}
}
// This sorting time takes O(nlog(n)) time.
// The queue space compexity is complicated because we don't
// really know how many States will be on the queue. The worst case
// is O(n!) but we usually do not get to this value. (See table in report)
queue = queue.OrderByDescending(x => x.priority).ToList();
if (max_states < queue.Count)
{
max_states = queue.Count;
}
pruned--; // necessary because the do-while structure
do
{
pruned++;
at_node = queue[queue.Count - 1];
queue.RemoveAt(queue.Count - 1); // Removing at end of array saves time.
from = at_node.number;
} while (at_node.lower_bound > current_count && queue.Count > 0);
if (timer.Elapsed.TotalSeconds > ProblemAndSolver.TIME_LIMIT)
{
break;
}
} while (queue.Count > 0);
timer.Stop();
results[COST] = costOfBssf().ToString();
results[TIME] = timer.Elapsed.ToString();
results[COUNT] = count.ToString();
return(results);
}