(MPQ-Tree Algorithm) Add a certifying algorithm for interval graph checking

[pdelvo]

Edit by PhoenixIra:
ToDo list:

• special case for Adj(u) is empty

To be done by Jiong:

• label tree
• implement doubly linked circular list required by MPQ tree node
• test path - if it succeeds, get path along with small N and big N
• test outer sections of q nodes

To be done by Dia:

• change path according to template/add vertex to leaf

To be done by Ira:

• the interval representation of the graph from the resulting MPQ Tree as a positive certificate
• the AT-Triple of the graph from the last MPQ Tree as a negative certificate

[pdelvo]

We want to use this algorithm. It refers to this algorithm.

More references: https://www.sciencedirect.com/science/article/pii/S0022000076800451

[PhoenixIra]

I have tried to comprehend the recognition algorithm to a more code-like algorithm
Some explenation:
T is an MPQ Tree
o is calculated by the ChordalityInspector Class of JgraphT
Adj(u) are the predecessors according to o
N/N./N* are nodes in the tree
V(N) are the nodes in the graph of the node N in the tree
A/B are sets of nodes the graph

The tree has a size of O(|V|) (according to the paper). The algorithm runs in O(|V|+|E|) somehow.

#init
-calculate perfect elimination order o
-Let T be the MPQ Tree
-for all vertex u in o
	-special case for Adj(u) is empty
	#labeling phase
	-label every vertex in T with 1 if it has a neighbour of u
	-label every vertex in T with inf if it has only neighbours of u
	#test label
	-all nodes with positive label are a path
	-all Q nodes N with positive label have an outer section containing Adj(u) intersection V(N)
	#else G is not an interval graph
	
	#update phase
	-Let Path be the path of all positive labels from root to a leaf
	-Let N. be the lowest positive node in Path
	-Let N* be the highest non inf P-Node/Section in Path if Path contains nonempty P-Node/section above N. without inf label, else N*=N.
	-case 1: Path only has nonempty inf labels. add u to the leaf of Path (fig 5, L1)
	-case 2: Path has nonempty not-inf label. traverse bottom-up from N. to N*, current Node N:
		-partition V(N) into A, adjacent to u, and B, not adjacent to u)
		-template match with a case (fig4-7)

Edit: Added some further explanation

[PhoenixIra]

We now can use the branch mpqIntervalRecognizer

[PhoenixIra]

I will work on computing the certificates:

• the interval representation of the graph from the resulting MPQ Tree as a positive certificate
• the AT-Triple of the graph from the last MPQ Tree as a negative certificate

[jiong-fu]

labelling and testing phases are completely implemented.

works left up to now

• integrate Q-node templates
• check the necessity of the HashSet usage on the bag, replace with it when necessary
• examine again the actual time complexity of the implementation
• write more test cases for the implementation