Computing the Discrete Fréchet Distance in Subquadratic Time (1204.5333v1)

Abstract: The Fr\'echet distance is a similarity measure between two curves $A$ and $B$: Informally, it is the minimum length of a leash required to connect a dog, constrained to be on $A$, and its owner, constrained to be on $B$, as they walk without backtracking along their respective curves from one endpoint to the other. The advantage of this measure on other measures such as the Hausdorff distance is that it takes into account the ordering of the points along the curves. The discrete Fr\'echet distance replaces the dog and its owner by a pair of frogs that can only reside on $n$ and $m$ specific pebbles on the curves $A$ and $B$, respectively. These frogs hop from a pebble to the next without backtracking. The discrete Fr\'echet distance can be computed by a rather straightforward quadratic dynamic programming algorithm. However, despite a considerable amount of work on this problem and its variations, there is no subquadratic algorithm known, even for approximation versions of the problem. In this paper we present a subquadratic algorithm for computing the discrete Fr\'echet distance between two sequences of points in the plane, of respective lengths $m\le n$. The algorithm runs in $O(\dfrac{mn\log\log n}{\log n})$ time and uses $O(n+m)$ 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 automata.

Computing the Discrete Fréchet Distance in Subquadratic Time

The paper under discussion presents a significant advancement in computational geometry by introducing a subquadratic time algorithm for computing the discrete Fréchet distance between two sequences of points in the plane. Development of this algorithm addresses a longstanding open question in the field, offering a more efficient solution to a problem that previously relied on a quadratic dynamic programming approach.

Core Contributions

The main contribution is an algorithm that computes the discrete Fréchet distance between two point sequences $A$ and $B$, with respective lengths $m \leq n$, in $O\left(\frac{mn \log \log n}{\log n}\right)$ time and uses $O(n + m)$ storage. This improvement leverages properties of geometric problems to encode legal states of a finite automaton, which optimizes the decision process relating to the Fréchet distance computation.

Problem and Methodology

The discrete Fréchet distance is an adaptation of the continuous Fréchet distance, emphasizing the ordered traversal of sequence points. Traditionally, this problem has been approached with a quadratic-time algorithm. However, by utilizing a fine-tuned combination of geometric properties and symbolic computation, the authors reduced computational complexity significantly.

1. Layer and Block Partitioning: The sequences are systematically divided into layers and blocks, facilitating efficient processing. Each layer contains multiple blocks, allowing for concurrent computation through a novel adaptive method that processes these blocks in parallel, maintaining state information via automata theory.
2. Automaton Construction: The algorithm forms a deterministic finite automaton ($^*$) to process compacted input strings. Each string represents possible paths through subsets of points, allowing efficient analysis of feasible paths under given constraints.
3. Point Location and Arrangement: A critical aspect of the methodology involves geometric arrangements of unit disks centered on the sequence points. Efficient point location algorithms are employed to determine the most relevant points dynamically, crucial for maintaining subquadratic runtime.

Implications and Future Directions

The proposed method resolves a long-standing computational geometry issue, offering significant implications for shape similarity evaluation in various applications, such as pattern recognition, computer graphics, and geographic information systems. From a theoretical perspective, this contributes to a deeper understanding of the relationship between geometric properties and algorithmic efficiency.

However, the extent to which this subquadratic solution can be generalized to the continuous Fréchet distance remains unresolved, providing an intriguing avenue for future exploration. Moreover, it raises questions about further reducing the complexity or extending these techniques to higher-dimensional datasets.

Conclusion

In sum, this paper marks a substantial step forward in the efficient computation of the discrete Fréchet distance. By integrating geometric insights with sophisticated algorithmic frameworks, the authors have set a new benchmark in the field. This work not only advances the theoretical landscape but also promises to enhance computational practices in analyzing and interpreting spatial data.

This concise and technically rich treatment allows expert readers to appreciate the innovative solutions offered while anticipating subsequent research directions engendered by this paper. The advancement sets a precedent for further exploration into efficient distance metrics for complex data structures.