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 in O (( m + n ) 6/5 + ε ) expected time, for any ε > 0. When shortcuts are allowed in both curves, we give an O (( m 2/3 n 2/3 + m + n )log 3 ( m + n ))-time deterministic algorithm. We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points 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 ) 2/3 m 2/3 n 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 contain 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 approximating 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 be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.

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 in O((m+n)6/5+ϵ) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an O((m2/3n2/3 + m + n) log3(m + n))-time deterministic algorithm.

The Fréchet distance measures similarity between two curves $f$ and $g$ that takes into account the ordering of the points along the two curves: Informally, it is the minimum length of a leash required to connect a dog, walking along $f$, and its owner, walking along $g$, as they walk without backtracking along their respective curves from one endpoint to the other. The discrete Fréchet distance replaces the dog and its owner by a pair of frogs that can only reside on $m$ and $n$ specific stones, respectively. The stones are in fact sequences of points, typically sampled from the respective curves $f$ and $g$. These frogs hop from one stone to the next without backtracking, and the discrete Fréchet distance is the minimum length of a “leash” that connects the frogs and allows them to execute such a sequence of hops from the starting points to the terminal points of their sequences. The discrete Fréchet distance can be computed in $O(mn)$ time by a straightforward dynamic programming algorithm. We present the first subquadratic algorithm for computing the discrete Fréchet distance between two sequences of points in the plane. Assuming $m\le n$, the algorithm runs in $O(\frac{mn\log\log n}{\log n})$ time, in the word RAM model, using $O(n)$ storage. Our approach uses the geometry of the problem in a subtle way to encode legal positions of the frogs as states of a finite automaton.

The Fréchet distance is a similarity measure between two curves A and B that takes into account the location and ordering of the points along the two curves: Informally, it is the minimum length of a leash required to connect a dog, walking along A, and its owner, walking along B, as they walk without backtracking along their respective curves from one endpoint to the other. The discrete Fréchet distance replaces the dog and its owner by a pair of frogs that can only reside on n and m specific stones on the curves A and B, respectively. These frogs hop from one stone to the next without backtracking, and the discrete Fréchet distance is the minimum length of a “leash” that connects the frogs and allows them to execute such a sequence of hops. It can be computed in quadratic time by a straightforward dynamic programming algorithm. We present the first subquadratic algorithm for computing the discrete Fréchet distance between two sequences of points in the plane. Assuming m ≤ n, the algorithm runs in time, in the standard RAM model, using O(n) storage. Our approach uses the geometry of the problem in a subtle way to encode legal positions of the frogs as states of a finite automaton.

Let P be a set of n points in ℜ3. The 2-center problem for P is to find two congruent balls of the minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n3 log8 n) expected time, and the second algorithm runs in O(n2 log8 n/(1--r*/r0)3) expected time, where r* is the radius of the 2-center of P and r0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r* is not very close to r0, which is equivalent to the condition of the centers of the two balls in the 2-center of P not being very close to each other.