Breadth-First Rank/Select in Succinct Trees and Distance Oracles for Interval Graphs

We present the first succinct data structure for ordinal trees that supports the mapping between the preorder (i.e., depth-first) ranks and level-order (breadth-first) ranks of nodes in constant time. It also provides constant-time support for all the operations provided by different approaches in previous work, as well as new operations that allow us to retrieve the last internal node before or the first internal node after a given node in a level-order traversal. This new representation gives us the functionality needed to design the first succinct distance oracles for interval graphs, proper interval graphs and $k$-proper/$k$-improper interval graphs.