Succinct and I/O efficient data structures for traversal in trees

We present two results for path traversal in trees, where the traversal is performed in an asymptotically optimal number of I/Os and the tree structure is represented succinctly. Our first result is for bottom-up traversal that starts with a node in a tree on N nodes and traverses a path to the root. We show how a tree T on N nodes with q-bit keys, where q = O(lgN), can be blocked in a succinct fashion such that a bottom-up traversal requires O(K/B + 1) I/Os using only (2 + q)N + q . [2τN(q+2 lgB) w + o(N)] + 8tN lgB w bits to store T for any constant 0 < τ <1, where K is the path length and w is the word size. This data structure is succinct because the above space cost is at most (2+q)N +q . (?N +o(N)) bits for any arbitrarily selected constant, ?, such that 0<?<1. When storing keys with tree nodes is not required, we can represent T in 2N + εN lgB w + o(N) bits, where ε is an arbitrarily selected constant such that 0 < ε <1, while providing the same support for queries. Our second result is for top-down traversal in binary trees. We store the tree in (3 + q)N + o(N) bits, while top-down traversal can still be performed in an asymptotically optimal number of I/Os

I/O-Efficient Path Traversal in Succinct Planar Graphs

We present a technique for representing bounded-degree planar graphs in a succinct fashion while permitting I/O-efficient traversal of paths. Using our representation, a graph with N vertices, (In this paper (Formula presented.) denotes (Formula presented.)) each with an associated key of (Formula presented.) bits, can be stored in (Formula presented.) bits and traversing a path of length K takes (Formula presented.) I/Os, where B denotes the disk block size. By applying our construction to the dual of a terrain represented as a triangular irregular network, we can represent the terrain in the above space bounds and support path traversals on the terrain using (Formula presented.) I/Os, where K is the number of triangles visited by the path. This is useful for answering a number of queries on the terrain, such as reporting terrain profiles, trickle paths, and connected components

Point location in well-shaped meshes using jump-and-walk

We present results on executing point location queries in well-shaped meshes in R2 and R3 using the Jump- And-Walk paradigm. If the jump step is performed on a nearest-neighbour search structure built on the vertices of the mesh, we demonstrate that the walk step can be performed in guaranteed constant time. Constant time for the walk step holds even if the jump step starts with an approximate nearest neighbour