The discrete Fréchet distance with shortcuts via approximate distance counting and selection

Rinat Ben Avraham, Omrit Filtser, Haim Kaplan, Matthew J. Katz, Micha Sharir

פרסום מחקרי: פרק בספר / בדוח / בכנספרסום בספר כנסביקורת עמיתים

תקציר

The Fréchet distance is a well studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs inO((m+n)6/5+ε) expected time, for any ε > 0. When shortcuts are allowed in both curves, we give an O((m2/3n 2/3 + m + n) log3(m + n))-time deterministic algorithm. We also consider the semi-continuous Fréchet distance with onesided shortcuts, where we have a sequence ofmpoints and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O((m + n)2/3m2/3n 1/3 log(m + n)). Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I, we develop an algorithm that decides whether the number of pairs (x, y) ∈ A × B whose distance dist(x, y) is in I, is less than some given threshold L. The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I, we can get a small sample of pairs that contains a pair at approximate median distance (i.e., we can approximately "bisect" I). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximate distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can apply to optimization problems for which the decision procedure is very fast but standard techniques like parametric search make the optimization algorithm substantially slower.

שפה מקוריתאנגלית
כותר פרסום המארחProceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014
מוציא לאורAssociation for Computing Machinery
עמודים377-386
מספר עמודים10
מסת"ב (מודפס)9781450325943
מזהי עצם דיגיטלי (DOIs)
סטטוס פרסוםפורסם - 2014
פורסם באופן חיצוניכן
אירוע30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, יפן
משך הזמן: 8 יוני 201411 יוני 2014

סדרות פרסומים

שםProceedings of the Annual Symposium on Computational Geometry

כנס

כנס30th Annual Symposium on Computational Geometry, SoCG 2014
מדינה/אזוריפן
עירKyoto
תקופה8/06/1411/06/14

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