Shortest path problem in rectangular complexes of global nonpositive curvature

Abstract

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x, y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ≥ 4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n 2 ) so that, given any two points x, y ∈ K, the shortest path γ(x, y) between x and y can be computed in O (d(p, q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x, y) ⊂ K(I(p, q)) and d(p, q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p, q) log ∆) time, where ∆ is the maximal degree of a vertex of G(K).Finally, if K is a squaregraph, then one can construct a data structure D of size O(n log n) and answer two-point shortest path queries in O(d(p, q)) time.

Geodesics in CAT(0) cubical complexes

“…Theorem 2.5 gives us a simple proof. (Brodzki, Campbell, Guentner, Niblo, and Wright [4] and Chepoi and Maftuleac [6] independently found proofs similar to ours.) Theorem 3.5.…”

“…For a detailed survey, see [15]. Recently, Chepoi and Maftuleac [6] gave a polynomial algorithm for computing the shortest path through a CAT(0) rectangular complex, in which each cell is 2-dimensional.…”

Shortest paths and convex hulls in 2D complexes with non-positive curvature

“…We show how to construct the last step shortest path map in O(n 2 ) preprocessing time and space for special cases. This generalizes and unifies two previous results: an algorithm by Chepoi and Maftuleac [19] for the case of 2D CAT(0) rectangular complexes; and an algorithm by Chen and Han [16] specialized to the case of a 2D CAT(0) complex that is a topological 2-manifold with boundary (i.e. every edge is incident to at most two faces).…”

“…We first show that the shortest path map may have exponential size for a general 2D CAT(0) complex. This contrasts with the fact that the shortest path map has size O(n 2 ) in the two special cases where the single-source shortest path problem is known to be efficiently solvable: when the complex is a topological 2-manifold with boundary, which we will call a 2-manifold for short [41]; and when the complex is rectangular [19].…”

“…Maftuleac [41] explicitly discussed this as a problem in a CAT(0) space and gave a Dijkstra-like algorithm with the same space and query time, but preprocessing time of O(n 2 log n). Chepoi and Maftuleac [19] used different methods to give a polynomial time algorithm for all-pair shortest path queries in any 2D CAT(0) rectangular complex, with preprocessing time O(n 2 ), space O(n 2 ), and query time proportional to the size of the output path.…”