A Subquadratic Algorithm for Computing the L1-distance between Two Terrains
Résumé
We study the problem of computing the L1-distance between two piecewise-linear bivariate functions f and g, defined over a common bounded domain M ⊂ R 2 , that is, computing the quantity ∥f -g∥1 = M |f (x, y)-g(x, y)| dx dy. If f and g are defined by linear interpolation over triangulations T f and Tg, respectively, of M with a total of n triangles, we show that ∥f -g∥1 can be computed in Õ(n (ω+1)/2 ) time, where Θ(n ω ) is the time required to multiply two n × n matrices and Õ notation hides polylogarithmic factors; this bound holds for the currently best known value of ω, which is approximately 2.37. The previously best known algorithm for computing ∥f -g∥1 takes Θ(n 2 ) time in the worst case. More generally, if the complexity of the overlay of T f and Tg is κ, then the runtime of our algorithm is Õ(κ (ω-1)/2 n (3-ω)/2 ).
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