Succinct Data Structure for Path Graphs
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We consider the problem of designing a succinct data structure for path graphs (which are a proper subclass of chordal graphs and a proper superclass of interval graphs) with n vertices while supporting degree, adjacency, and neighborhood queries efficiently. We provide two solutions for this problem. Our first data structure is succinct and occupies n log n+o(n log n) bits while answering adjacency query in O(log n) time, and neighborhood and degree queries in O(d log^2 n) time where d is the degree of the queried vertex. Our second data structure answers adjacency queries faster at the expense of slightly more space. More specifically, we provide an O(n log^2 n) bit data structure that supports adjacency query in O(1) time, and the neighborhood query in O(d log n) time where d is the degree of the queried vertex. Central to our data structures is the usage of the classical heavy path decomposition, followed by a careful bookkeeping using an orthogonal range search data structure among others, which maybe of independent interest for designing succinct data structures for other graphs. It is the use of the results of Acan et al. in the second data structure that permits a simple and efficient implementation at the expense of more space.
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