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 firstinternal 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 .
We present the first succinct distance oracles for (unweighted) interval graphs and related classes of graphs, using a novel succinct data structure for ordinal trees that supports the mapping between preorder (i.e., depth-first) ranks and level-order (breadth-first) ranks of nodes in constant time. Our distance oracles for interval graphs also support navigation queries -- testing adjacency, computing node degrees, neighborhoods, and shortest paths -- all in optimal time. Our technique also yields optimal distance oracles for proper interval graphs (unit-interval graphs) and circular-arc graphs. Our tree data structure supports all operations provided by different approaches in previous work, as well as mapping to and from level-order ranks and retrieving the last (first) internal node before (after) a given node in a level-order traversal , all in constant time .
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distance oracles for (unweighted) interval graphs and related classes of graphs, using a novel succinct
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the
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It also provides constant-time support for all the
Our distance oracles for interval graphs also support navigation queries -- testing adjacency, computing node degrees, neighborhoods, and shortest paths -- all in optimal time. Our technique also yields optimal distance oracles for proper interval graphs (unit-interval graphs) and circular-arc graphs. Our tree data structure supports all
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new operations that allow us to retrieve the last internal node before or the firstinternal node after
mapping to and from level-order ranks and retrieving the last (first) internal node before (after)
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. 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
, all in constant time