Fix incorrect Euler path/cycle conditions and complexity statements

Graph/Connectivity/HeirHolzer_algo_for_Euler_Cycle_or_Path.md

@@ -1,38 +1,52 @@
 **Condition for Euler path or cycle in directed graph is** - 
-* Indegree[u] == outdegree[u], for atleast n-2 vertices
-* All vertices with non-zero indegree(or outdegree) should be part of same SCC (isolated vertices which doesnt contribute any edge maybe there)
+* Let `in[u]=indegree(u)` and `out[u]=outdegree(u)`. Isolated vertices with `in[u]+out[u]==0` can be ignored.
+* **Euler cycle (directed)** exists iff:
+  * `in[u]==out[u]` for all vertices `u`
+  * All vertices with `in[u]+out[u]>0` belong to the same SCC (strongly connected after ignoring isolated vertices)
+* **Euler path (directed)** exists iff:
+  * Exactly one vertex `s` has `out[s]=in[s]+1` (start vertex)
+  * Exactly one vertex `t` has `in[t]=out[t]+1` (end vertex)
+  * For all other vertices `u`, `in[u]==out[u]`
+  * All vertices with `in[u]+out[u]>0` belong to the same weakly connected component (connected if directions are ignored)
 
 **Condition for Euler path or cycle in undirected graph is** - 
-* All vertices should have even degree, atleast n-2 vertices.
-* All vertices with non-zero degree should be connected ( isolated vertices which doesnt contribute any edge maybe there)
+* Isolated vertices with `deg[u]==0` can be ignored.
+* All vertices with non-zero degree should be connected.
+* Following cases arise - 
+  * If number of odd degree vertices is greater than 2 then no euler path or cycle
+  * If no odd degree vertices then there exist a Euler cycle.
+  * If odd degree vertices are 2 then there exist a Euler path.
 
 **HeirHolzer algorithm for finding Euler path or cycle whichever exist** - 
-* Time Complexity = `O(m + nlogn)` for undirected graph, have to use vector of multiset to represent adjacency list
-* Time Complexity = `O(m)` for directed graph
-* Time Complexity = `O(m + n*n)` for adjacency matrix representation
-* Following cases arise - 
+* Time Complexity = `O(m)` if adjacency list is stored as `vector<vector<int>>` (or list) with pointer/iterator per node (no log deletions)
+* Time Complexity = `O(m log n)` if you use `vector<multiset<int>>` (because each erase/find is `log`)
+* Time Complexity = `O(n*n)` for adjacency matrix representation (matrix scan based implementation)
+* Undirected graph handling - 
   * If number of odd degree vertices is greater than 2 then no euler path or cycle
   * If no odd degree vertices then there exist a Euler cycle.
   * If odd degree vertices are 2 then there exist a Euler path. In this case **add an edge between odd vertices** and find euler cycle. See note below to further handle this case
 
 **NOTE** - In result array which we obtain from finding euler cycle, `result[0] == result.back()`
 
 **Code Description** - 
-* Push any node in stack
+* Push a valid start node in stack:
+  * If odd degree vertices are 2, start from an odd vertex
+  * Else start from any vertex with non-zero degree (if all are zero then answer is trivial)
 * While stack is not empty
   * u = st.top(); 
   * if `adj[u].size() == 0, push u to result array` and pop stack
   * else:
     * `v = adj[u].begin(); push v to stack`
     * remove v from adj[u] and remove u from adj[v] (in case of undirected graph)
-* Result array will contain the euler cycle, with `result[0] == result.back()`
+* Result array will contain the euler cycle (often in reverse order of traversal), with `result[0] == result.back()` for a cycle
 
 **NOTE** - 
 * If we added an edge to handle odd degree vertices then to print euler path do this - 
   * Let odd degree vertices be `v1` and `v2`
   * Find index result array such that `result[i] == v1 && result[i+1] == v2 or vice versa`, let call that index `id` 
   * `print result from (id + 1 to result.size() - 1)`
   * `print result from (1 to id)`
+* Also ensure you started Hierholzer from a non-zero degree vertex (or odd vertex in path case), otherwise the algorithm may not traverse the component containing edges.
 
 ```c++
 #include<bits/stdc++.h>
@@ -75,14 +89,32 @@ void solve(int test)
        cout<<"No euler path or cycle"<<endl;
        return;
    }
+
+   if(m==0)
+   {
+       cout<<"Euler cycle found"<<endl;
+       cout<<0<<endl;
+       return;
+   }
+
    if(odd == 2)
    {
        adj[v1].insert(v2);
        adj[v2].insert(v1);
    }
 
+   int start = -1;
+   if(odd==2) start = v1;
+   else
+   {
+       for(i=0;i<n;i++)
+       {
+           if(deg[i]>0){start=i;break;}
+       }
+   }
+
    stack<int>s;
-   s.push(0);
+   s.push(start);
 
    vector<int>res;
    while(!s.empty())
@@ -98,7 +130,12 @@ void solve(int test)
        {
            int v = *adj[u].begin();
            adj[u].erase(adj[u].begin());
-           auto it = adj[v].find(u);   // first find one iterator pointing to u. Because simply calling multiset.erase(value) removes all instances of that value.
+           auto it = adj[v].find(u);
+           if(it==adj[v].end())
+           {
+               cout<<"Invalid edge state"<<endl;
+               return;
+           }
            adj[v].erase(it);
            s.push(v);
        }
@@ -113,27 +150,34 @@ void solve(int test)
        }
    }
 
+   reverse(res.begin(),res.end());
 
    if(odd==2)
    {
        vector<int>path;
        int id = -1;
-       for(i=0;i<res.size()-1;i++)
+       for(i=0;i<(int)res.size()-1;i++)
        {
            if(res[i] == v1 && res[i+1] == v2){id=i;break;}
            if(res[i] == v2 && res[i+1] == v1){id=i; break;}
        }
 
-       for(i=id+1;i<res.size();i++)path.pb(res[i]);
-       for(i=1;i<=id;i++)path.pb(res[i]);    // start from i = 1 because res[res.size()-1] == res[0] because it was a euler cycle ******
+       if(id==-1)
+       {
+           cout<<"Logic error: added edge not found"<<endl;
+           return;
+       }
+
+       for(i=id+1;i<(int)res.size()-1;i++)path.pb(res[i]);
+       for(i=0;i<=id;i++)path.pb(res[i]);
 
        cout<<"Euler path found"<<endl;
        for(auto v:path)cout<<v<<" ";
        cout<<endl;
        return;
    }
-   cout<<"Euler cycle found"<<endl;
 
+   cout<<"Euler cycle found"<<endl;
    for(auto v:res)cout<<v<<" ";
    cout<<endl;