Hi Yasir,
Congratulations on being tough enough to work through all the details of our path algorithms. ;)
"In the partitioned plane, the scanned areas are made disjunct, i.e., scanning is canceled whenever the pre value of the currently scanned node equals the pre value of the next context node."
I agree the sentence does not contain all the details required to grasp what’s going on: “Scanning” means that the initial pre value (initially 1 in the example) will be incremented until a given condition is fulfilled. The condition in the example is: equality with the pre value of the next context node (6). Without partitioning, all remaining nodes of the document would be checked.
Hope this helps; have fun, Christian