/**************************************************************************
*
* The code below is solely for testing correctness of the data type.
*
**************************************************************************/
// Determines whether a digraph has an Eulerian path using necessary
// and sufficient conditions (without computing the path itself):
// - indegree(v) = outdegree(v) for every vertex,
// except one vertex v may have outdegree(v) = indegree(v) + 1
// (and one vertex v may have indegree(v) = outdegree(v) + 1)
// - the graph is connected, when viewed as an undirected graph
// (ignoring isolated vertices)
// This method is solely for unit testing.
private static bool hasEulerianPath(Digraph G)
{
if (G.E == 0)
{
return(true);
}
// Condition 1: indegree(v) == outdegree(v) for every vertex,
// except one vertex may have outdegree(v) = indegree(v) + 1
int deficit = 0;
for (int v = 0; v < G.V; v++)
{
if (G.Outdegree(v) > G.Indegree(v))
{
deficit += (G.Outdegree(v) - G.Indegree(v));
}
}
if (deficit > 1)
{
return(false);
}
// Condition 2: graph is connected, ignoring isolated vertices
Graph H = new Graph(G.V);
for (int v = 0; v < G.V; v++)
{
foreach (int w in G.Adj(v))
{
H.AddEdge(v, w);
}
}
// check that all non-isolated vertices are connected
int s = nonIsolatedVertex(G);
BreadthFirstPaths bfs = new BreadthFirstPaths(H, s);
for (int v = 0; v < G.V; v++)
{
if (H.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
private LinkedStack <int> cycle = null; // Eulerian cycle; null if no such cylce
/// <summary>
/// Computes an Eulerian cycle in the specified digraph, if one exists.
/// </summary>
/// <param name="G">the digraph</param>
///
public DirectedEulerianCycle(Digraph G)
{
// must have at least one edge
if (G.E == 0)
{
return;
}
// necessary condition: indegree(v) = outdegree(v) for each vertex v
// (without this check, DFS might return a path instead of a cycle)
for (int v = 0; v < G.V; v++)
{
if (G.Outdegree(v) != G.Indegree(v))
{
return;
}
}
// create local view of adjacency lists, to iterate one vertex at a time
IEnumerator <int>[] adj = new IEnumerator <int> [G.V];
for (int v = 0; v < G.V; v++)
{
adj[v] = G.Adj(v).GetEnumerator();
}
// initialize stack with any non-isolated vertex
int s = nonIsolatedVertex(G);
LinkedStack <int> stack = new LinkedStack <int>();
stack.Push(s);
// greedily add to putative cycle, depth-first search style
cycle = new LinkedStack <int>();
while (!stack.IsEmpty)
{
int v = stack.Pop();
while (adj[v].MoveNext())
{
stack.Push(v);
v = adj[v].Current;
}
// add vertex with no more leaving edges to cycle
cycle.Push(v);
}
// check if all edges have been used
// (in case there are two or more vertex-disjoint Eulerian cycles)
if (cycle.Count != G.E + 1)
{
cycle = null;
}
Debug.Assert(certifySolution(G));
}
private int[] rank; // rank[v] = order where vertex v appers in order
/// <summary>
/// Determines whether the digraph <c>G</c> has a topological order and, if so,
/// finds such a topological order.
/// </summary>
/// <param name="G">the digraph</param>
///
public TopologicalX(Digraph G)
{
// indegrees of remaining vertices
int[] indegree = new int[G.V];
for (int v = 0; v < G.V; v++)
{
indegree[v] = G.Indegree(v);
}
// initialize
rank = new int[G.V];
order = new LinkedQueue <int>();
int count = 0;
// initialize queue to contain all vertices with indegree = 0
LinkedQueue <int> queue = new LinkedQueue <int>();
for (int v = 0; v < G.V; v++)
{
if (indegree[v] == 0)
{
queue.Enqueue(v);
}
}
for (int j = 0; !queue.IsEmpty; j++)
{
int v = queue.Dequeue();
order.Enqueue(v);
rank[v] = count++;
foreach (int w in G.Adj(v))
{
indegree[w]--;
if (indegree[w] == 0)
{
queue.Enqueue(w);
}
}
}
// there is a directed cycle in subgraph of vertices with indegree >= 1.
if (count != G.V)
{
order = null;
}
Debug.Assert(check(G));
}
/**************************************************************************
*
* The code below is solely for testing correctness of the data type.
*
**************************************************************************/
// Determines whether a digraph has an Eulerian cycle using necessary
// and sufficient conditions (without computing the cycle itself):
// - at least one edge
// - indegree(v) = outdegree(v) for every vertex
// - the graph is connected, when viewed as an undirected graph
// (ignoring isolated vertices)
private static bool hasEulerianCycle(Digraph G)
{
// Condition 0: at least 1 edge
if (G.E == 0)
{
return(false);
}
// Condition 1: indegree(v) == outdegree(v) for every vertex
for (int v = 0; v < G.V; v++)
{
if (G.Outdegree(v) != G.Indegree(v))
{
return(false);
}
}
// Condition 2: graph is connected, ignoring isolated vertices
Graph H = new Graph(G.V);
for (int v = 0; v < G.V; v++)
{
foreach (int w in G.Adj(v))
{
H.AddEdge(v, w);
}
}
// check that all non-isolated vertices are conneted
int s = nonIsolatedVertex(G);
BreadthFirstPaths bfs = new BreadthFirstPaths(H, s);
for (int v = 0; v < G.V; v++)
{
if (H.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
/// <summary>
/// Initializes a new digraph that is a deep copy of the specified digraph.</summary>
/// <param name="G">the digraph to copy</param>
///
public Digraph(Digraph G) : this(G.V)
{
numEdges = G.E;
for (int v = 0; v < V; v++)
{
this.indegree[v] = G.Indegree(v);
}
for (int v = 0; v < G.V; v++)
{
// reverse so that adjacency list is in same order as original
LinkedStack <int> reverse = new LinkedStack <int>();
foreach (int w in G.adj[v])
{
reverse.Push(w);
}
foreach (int w in reverse)
{
adj[v].Add(w);
}
}
}
private LinkedStack <int> cycle; // the directed cycle; null if digraph is acyclic
/// <summary>
/// Determines whether the digraph <c>G</c> has a directed cycle and, if so,
/// finds such a cycle.</summary>
/// <param name="G">the digraph</param>
///
public DirectedCycleX(Digraph G)
{
// indegrees of remaining vertices
int[] indegree = new int[G.V];
for (int v = 0; v < G.V; v++)
{
indegree[v] = G.Indegree(v);
}
// initialize queue to contain all vertices with indegree = 0
LinkedQueue <int> queue = new LinkedQueue <int>();
for (int v = 0; v < G.V; v++)
{
if (indegree[v] == 0)
{
queue.Enqueue(v);
}
}
for (int j = 0; !queue.IsEmpty; j++)
{
int v = queue.Dequeue();
foreach (int w in G.Adj(v))
{
indegree[w]--;
if (indegree[w] == 0)
{
queue.Enqueue(w);
}
}
}
// there is a directed cycle in subgraph of vertices with indegree >= 1.
int[] edgeTo = new int[G.V];
int root = -1; // any vertex with indegree >= -1
for (int v = 0; v < G.V; v++)
{
if (indegree[v] == 0)
{
continue;
}
else
{
root = v;
}
foreach (int w in G.Adj(v))
{
if (indegree[w] > 0)
{
edgeTo[w] = v;
}
}
}
if (root != -1)
{
// find any vertex on cycle
bool[] visited = new bool[G.V];
while (!visited[root])
{
visited[root] = true;
root = edgeTo[root];
}
// extract cycle
cycle = new LinkedStack <int>();
int v = root;
do
{
cycle.Push(v);
v = edgeTo[v];
} while (v != root);
cycle.Push(root);
}
Debug.Assert(check());
}
private LinkedStack <int> path = null; // Eulerian path; null if no suh path
/// <summary>
/// Computes an Eulerian path in the specified digraph, if one exists.</summary>
/// <param name="G">the digraph</param>
///
public DirectedEulerianPath(Digraph G)
{
// find vertex from which to start potential Eulerian path:
// a vertex v with outdegree(v) > indegree(v) if it exits;
// otherwise a vertex with outdegree(v) > 0
int deficit = 0;
int s = nonIsolatedVertex(G);
for (int v = 0; v < G.V; v++)
{
if (G.Outdegree(v) > G.Indegree(v))
{
deficit += (G.Outdegree(v) - G.Indegree(v));
s = v;
}
}
// digraph can't have an Eulerian path
// (this condition is needed)
if (deficit > 1)
{
return;
}
// special case for digraph with zero edges (has a degenerate Eulerian path)
if (s == -1)
{
s = 0;
}
// create local view of adjacency lists, to iterate one vertex at a time
IEnumerator <int>[] adj = new IEnumerator <int> [G.V];
for (int v = 0; v < G.V; v++)
{
adj[v] = G.Adj(v).GetEnumerator();
}
// greedily add to cycle, depth-first search style
LinkedStack <int> stack = new LinkedStack <int>();
stack.Push(s);
path = new LinkedStack <int>();
while (!stack.IsEmpty)
{
int v = stack.Pop();
while (adj[v].MoveNext())
{
stack.Push(v);
v = adj[v].Current;
}
// push vertex with no more available edges to path
path.Push(v);
}
// check if all edges have been used
if (path.Count != G.E + 1)
{
path = null;
}
Debug.Assert(check(G));
}
