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Succinct representations of weighted trees supporting path queries. (English) Zbl 1268.68069
Summary: We consider the problem of succinctly representing a given vertex-weighted tree of n vertices, whose vertices are labeled by integer weights from $$\{1,2,\ldots ,\sigma\}$$ and supporting the following path queries efficiently: (1) path median query: given two vertices $$i$$, $$j$$, return the median weight on the path from $$i$$ to $$j$$, (2) path selection query: given two vertices $$i$$, $$j$$ and a positive integer $$k$$, return the $$k$$th smallest weight on the path from $$i$$ to $$j$$, (3) path counting/reporting query: given two vertices $$i$$, $$j$$ and a range $$[a,b]$$, count/report the vertices on the path from $$i$$ to $$j$$ whose weights are in this range.
The previous best data structure supporting these queries takes $$O(n\log n)$$ bits space and can perform path median/selection/counting in $$O(\log\sigma )$$ time and path reporting in $$O(\log \sigma +occ\log \sigma )$$ time, where $$occ$$ represents the number of outputs [M. He et al., Lect. Notes Comput. Sci. 7074, 140–149 (2011; Zbl 1350.68073)].
We present a succinct data structure taking $$n\log \sigma +6n+o(n\log \sigma )$$ bits space that can perform the above mentioned queries in $$O(\log \sigma \log n)$$ and $$O(\log \sigma \log n+occ\log \sigma )$$ time respectively.

##### MSC:
 68P05 Data structures 68R10 Graph theory (including graph drawing) in computer science 05C05 Trees
##### Keywords:
succinct data structures
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##### References:
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