/// <summary>
/// Determines whether a graph has an Eulerian cycle using necessary
/// and sufficient conditions (without computing the cycle itself):
/// - at least one EdgeW
/// - degree(v) is even for every vertex v
/// - the graph is connected (ignoring isolated vertices)
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
private static bool HasEulerianCycle(Graph g)
{
// Condition 0: at least 1 EdgeW
if (g.E == 0)
{
return(false);
}
// Condition 1: degree(v) is even for every vertex
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) % 2 != 0)
{
return(false);
}
}
// Condition 2: graph is connected, ignoring isolated vertices
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(g, s);
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
/// <summary>
/// returns any non-isolated vertex; -1 if no such vertex
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
private static int NonIsolatedVertex(Graph g)
{
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) > 0)
{
return(v);
}
}
return(-1);
}
/// <summary>
/// Determines whether a graph has an Eulerian path using necessary
/// and sufficient conditions (without computing the path itself):
/// - degree(v) is even for every vertex, except for possibly two
/// - the graph is connected (ignoring isolated vertices)
/// This method is solely for unit testing.
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
private static bool HasEulerianPath(Graph g)
{
if (g.E == 0)
{
return(true);
}
// Condition 1: degree(v) is even except for possibly two
var oddDegreeVertices = 0;
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) % 2 != 0)
{
oddDegreeVertices++;
}
}
if (oddDegreeVertices > 2)
{
return(false);
}
// Condition 2: graph is connected, ignoring isolated vertices
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(g, s);
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
/// <summary>
/// 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.
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
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
var deficit = 0;
for (var 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
var h = new Graph(g.V);
for (var 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
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(h, s);
for (var v = 0; v < g.V; v++)
{
if (h.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
/// <summary>
/// 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)
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
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 (var v = 0; v < g.V; v++)
{
if (g.Outdegree(v) != g.Indegree(v))
{
return(false);
}
}
// Condition 2: graph is connected, ignoring isolated vertices
var h = new Graph(g.V);
for (var 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
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(h, s);
for (var v = 0; v < g.V; v++)
{
if (h.Degree(v) > 0 && !bfs.HasPathTo(v))
{
return(false);
}
}
return(true);
}
/// <summary>
/// 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.
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
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
var deficit = 0;
for (var 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
var h = new Graph(g.V);
for (var 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
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(h, s);
for (var v = 0; v < g.V; v++)
if (h.Degree(v) > 0 && !bfs.HasPathTo(v))
return false;
return true;
}
private readonly Collections.Stack<Integer> _path; // Eulerian path; null if no suh path
#endregion Fields
#region Constructors
/// <summary>
/// Computes an Eulerian path in the specified graph, if one exists.
/// </summary>
/// <param name="g">g the graph</param>
public EulerianPath(Graph g)
{
// find vertex from which to start potential Eulerian path:
// a vertex v with odd degree(v) if it exits;
// otherwise a vertex with degree(v) > 0
var oddDegreeVertices = 0;
var s = NonIsolatedVertex(g);
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) % 2 != 0)
{
oddDegreeVertices++;
s = v;
}
}
// graph can't have an Eulerian path
// (this condition is needed for correctness)
if (oddDegreeVertices > 2) return;
// special case for graph 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
// the helper Edge data type is used to avoid exploring both copies of an edge v-w
var adj = new Collections.Queue<EdgeW>[g.V];
for (var v = 0; v < g.V; v++)
adj[v] = new Collections.Queue<EdgeW>();
for (var v = 0; v < g.V; v++)
{
var selfLoops = 0;
foreach (int w in g.Adj(v))
{
// careful with self loops
if (v == w)
{
if (selfLoops % 2 == 0)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
selfLoops++;
}
else if (v < w)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
}
}
// initialize stack with any non-isolated vertex
var stack = new Collections.Stack<Integer>();
stack.Push(s);
// greedily search through edges in iterative DFS style
_path = new Collections.Stack<Integer>();
while (!stack.IsEmpty())
{
int v = stack.Pop();
while (!adj[v].IsEmpty())
{
var edge = adj[v].Dequeue();
if (edge.IsUsed) continue;
edge.IsUsed = true;
stack.Push(v);
v = edge.Other(v);
}
// push vertex with no more leaving edges to path
_path.Push(v);
}
// check if all edges are used
if (_path.Size() != g.E + 1)
_path = null;
//assert certifySolution(G);
}
// returns any non-isolated vertex; -1 if no such vertex
private static int NonIsolatedVertex(Graph g)
{
for (var v = 0; v < g.V; v++)
if (g.Degree(v) > 0)
return v;
return -1;
}
/// <summary>
/// Determines whether a graph has an Eulerian path using necessary
/// and sufficient conditions (without computing the path itself):
/// - degree(v) is even for every vertex, except for possibly two
/// - the graph is connected (ignoring isolated vertices)
/// This method is solely for unit testing.
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
private static bool HasEulerianPath(Graph g)
{
if (g.E == 0) return true;
// Condition 1: degree(v) is even except for possibly two
var oddDegreeVertices = 0;
for (var v = 0; v < g.V; v++)
if (g.Degree(v) % 2 != 0)
oddDegreeVertices++;
if (oddDegreeVertices > 2) return false;
// Condition 2: graph is connected, ignoring isolated vertices
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(g, s);
for (var v = 0; v < g.V; v++)
if (g.Degree(v) > 0 && !bfs.HasPathTo(v))
return false;
return true;
}
private readonly Collections.Stack <Integer> _cycle = new Collections.Stack <Integer>(); // Eulerian cycle; null if no such cycle
/// <summary>
/// Computes an Eulerian cycle in the specified graph, if one exists.
/// </summary>
/// <param name="g">g the graph</param>
public EulerianCycle(Graph g)
{
// must have at least one EdgeW
if (g.E == 0)
{
return;
}
// necessary condition: all vertices have even degree
// (this test is needed or it might find an Eulerian path instead of cycle)
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) % 2 != 0)
{
return;
}
}
// create local view of adjacency lists, to iterate one vertex at a time
// the helper EdgeW data type is used to avoid exploring both copies of an EdgeW v-w
var adj = new Collections.Queue <EdgeW> [g.V];
for (var v = 0; v < g.V; v++)
{
adj[v] = new Collections.Queue <EdgeW>();
}
for (var v = 0; v < g.V; v++)
{
var selfLoops = 0;
foreach (int w in g.Adj(v))
{
// careful with self loops
if (v == w)
{
if (selfLoops % 2 == 0)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
selfLoops++;
}
else if (v < w)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
}
}
// initialize Collections.Stack with any non-isolated vertex
var s = NonIsolatedVertex(g);
var stack = new Collections.Stack <Integer>();
stack.Push(s);
// greedily search through EdgeWs in iterative DFS style
_cycle = new Collections.Stack <Integer>();
while (!stack.IsEmpty())
{
int v = stack.Pop();
while (!adj[v].IsEmpty())
{
var edgeW = adj[v].Dequeue();
if (edgeW.IsUsed)
{
continue;
}
edgeW.IsUsed = true;
stack.Push(v);
v = edgeW.Other(v);
}
// push vertex with no more leaving EdgeWs to cycle
_cycle.Push(v);
}
// check if all EdgeWs are used
if (_cycle.Size() != g.E + 1)
{
_cycle = null;
}
//assert certifySolution(G);
}
private readonly Collections.Stack <Integer> _path; // Eulerian path; null if no suh path
/// <summary>
/// Computes an Eulerian path in the specified graph, if one exists.
/// </summary>
/// <param name="g">g the graph</param>
public EulerianPath(Graph g)
{
// find vertex from which to start potential Eulerian path:
// a vertex v with odd degree(v) if it exits;
// otherwise a vertex with degree(v) > 0
var oddDegreeVertices = 0;
var s = NonIsolatedVertex(g);
for (var v = 0; v < g.V; v++)
{
if (g.Degree(v) % 2 != 0)
{
oddDegreeVertices++;
s = v;
}
}
// graph can't have an Eulerian path
// (this condition is needed for correctness)
if (oddDegreeVertices > 2)
{
return;
}
// special case for graph 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
// the helper Edge data type is used to avoid exploring both copies of an edge v-w
var adj = new Collections.Queue <EdgeW> [g.V];
for (var v = 0; v < g.V; v++)
{
adj[v] = new Collections.Queue <EdgeW>();
}
for (var v = 0; v < g.V; v++)
{
var selfLoops = 0;
foreach (int w in g.Adj(v))
{
// careful with self loops
if (v == w)
{
if (selfLoops % 2 == 0)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
selfLoops++;
}
else if (v < w)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
}
}
// initialize stack with any non-isolated vertex
var stack = new Collections.Stack <Integer>();
stack.Push(s);
// greedily search through edges in iterative DFS style
_path = new Collections.Stack <Integer>();
while (!stack.IsEmpty())
{
int v = stack.Pop();
while (!adj[v].IsEmpty())
{
var edge = adj[v].Dequeue();
if (edge.IsUsed)
{
continue;
}
edge.IsUsed = true;
stack.Push(v);
v = edge.Other(v);
}
// push vertex with no more leaving edges to path
_path.Push(v);
}
// check if all edges are used
if (_path.Size() != g.E + 1)
{
_path = null;
}
//assert certifySolution(G);
}
private readonly Collections.Stack<Integer> _cycle = new Collections.Stack<Integer>(); // Eulerian cycle; null if no such cycle
#endregion Fields
#region Constructors
/// <summary>
/// Computes an Eulerian cycle in the specified graph, if one exists.
/// </summary>
/// <param name="g">g the graph</param>
public EulerianCycle(Graph g)
{
// must have at least one EdgeW
if (g.E == 0) return;
// necessary condition: all vertices have even degree
// (this test is needed or it might find an Eulerian path instead of cycle)
for (var v = 0; v < g.V; v++)
if (g.Degree(v) % 2 != 0)
return;
// create local view of adjacency lists, to iterate one vertex at a time
// the helper EdgeW data type is used to avoid exploring both copies of an EdgeW v-w
var adj = new Collections.Queue<EdgeW>[g.V];
for (var v = 0; v < g.V; v++)
adj[v] = new Collections.Queue<EdgeW>();
for (var v = 0; v < g.V; v++)
{
var selfLoops = 0;
foreach (int w in g.Adj(v))
{
// careful with self loops
if (v == w)
{
if (selfLoops % 2 == 0)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
selfLoops++;
}
else if (v < w)
{
var e = new EdgeW(v, w, 0);
adj[v].Enqueue(e);
adj[w].Enqueue(e);
}
}
}
// initialize Collections.Stack with any non-isolated vertex
var s = NonIsolatedVertex(g);
var stack = new Collections.Stack<Integer>();
stack.Push(s);
// greedily search through EdgeWs in iterative DFS style
_cycle = new Collections.Stack<Integer>();
while (!stack.IsEmpty())
{
int v = stack.Pop();
while (!adj[v].IsEmpty())
{
var edgeW = adj[v].Dequeue();
if (edgeW.IsUsed) continue;
edgeW.IsUsed = true;
stack.Push(v);
v = edgeW.Other(v);
}
// push vertex with no more leaving EdgeWs to cycle
_cycle.Push(v);
}
// check if all EdgeWs are used
if (_cycle.Size() != g.E + 1)
_cycle = null;
//assert certifySolution(G);
}
/// <summary>
/// Determines whether a graph has an Eulerian cycle using necessary
/// and sufficient conditions (without computing the cycle itself):
/// - at least one EdgeW
/// - degree(v) is even for every vertex v
/// - the graph is connected (ignoring isolated vertices)
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
private static bool HasEulerianCycle(Graph g)
{
// Condition 0: at least 1 EdgeW
if (g.E == 0) return false;
// Condition 1: degree(v) is even for every vertex
for (var v = 0; v < g.V; v++)
if (g.Degree(v) % 2 != 0)
return false;
// Condition 2: graph is connected, ignoring isolated vertices
var s = NonIsolatedVertex(g);
var bfs = new BreadthFirstPaths(g, s);
for (var v = 0; v < g.V; v++)
if (g.Degree(v) > 0 && !bfs.HasPathTo(v))
return false;
return true;
}