Searching for a Compressed Polyline with a Minimum Number of Vertices (Discrete Solution)

Abstract

There are many practical applications that require the simplification of polylines. Some of the goals are to reduce the amount of information, improve processing time, or simplify editing. Simplification is usually done by removing some of the vertices, making the resultant polyline go through a subset of the source polyline vertices. If the resultant polyline is required to pass through original vertices, it often results in extra segments, and all segments are likely to be shifted due to fixed endpoints. Therefore, such an approach does not necessarily produce a new polyline with the minimum number of vertices. Using an algorithm that finds the compressed polyline with the minimum number of vertices reduces the amount of memory required and the postprocessing time. However, even more important, when the resultant polylines are edited by an operator, the polylines with the minimum number of vertices decrease the operator time, which reduces the cost of processing the data. A viable solution to finding a polyline within a specified tolerance with the minimum number of vertices is described in this paper.

Appendix: Lower Bounds for an Optimal Solution

Following the approach described in [11, Sect. 4]

$$\begin{aligned} \begin{aligned}&\min _ { \begin{aligned} k_1 \le k' \wedge k' \le k_2\\ 0 \le j' \wedge j'< N_{k'}\\ {{\mathrm{check}}}{ \left( \left( k', j' \right) , \left( k, j \right) \right) } \end{aligned} } { \left( \left\{ \begin{aligned}&T_{k', j'}^{\#} + 1\\&T_{k', j'}^{\epsilon } + \epsilon _ { \left( k', j' \right) , \left( k, j \right) } \end{aligned} \right\} \right) } \gtrapprox \\ \gtrapprox&\min _ { \begin{aligned} k_1 \le k' \wedge k' \le k_2\\ 0 \le j' \wedge j' < N_{k'} \end{aligned} } { \left( \left\{ \begin{aligned}&T_{k', j'}^{\#} + 1\\&T_{k', j'}^{\epsilon } + \epsilon _ { \left( k', j' \right) , \left( k, j \right) } ^ { \left( k_2 \right) } \end{aligned} \right\} \right) } , \end{aligned} \end{aligned}$$

(4)

where .

From (4), it follows that

$$\begin{aligned} \min _ { \begin{aligned} k_1 \le k' \wedge k' \le k_2\\ 0 \le j' \wedge j' < N_{k'}\\ {{\mathrm{check}}}{ \left( \left( k', j' \right) , \left( k, j \right) \right) } \end{aligned} } { \left( \left\{ \begin{aligned}&T_{k', j'}^{\#} + 1\\&T_{k', j'}^{\epsilon } + \epsilon _ { \left( k', j' \right) , \left( k, j \right) } \end{aligned} \right\} \right) } \gtrapprox \end{aligned}$$

$$\begin{aligned} \gtrapprox \min _ { \begin{aligned} 0 \le j_2 \wedge j_2 < N_{k_2}\\ {{\mathrm{check}}}{ \left( \left( k_2, j_2 \right) , \left( k, j \right) \right) } \end{aligned} } { \left( \left\{ \begin{aligned}&T_{k_2, j_2}^{\#}\\&T_{k_2, j_2}^{\epsilon } + \epsilon _ { \left( k_2, j_2 \right) , \left( k, j \right) } \end{aligned} \right\} \right) } \end{aligned}$$

(5)

and

$$\begin{aligned} \gtrapprox \min _ { 0 \le j_1 \wedge j_1< N_{k_1} } { \left( \left\{ \begin{aligned}&T_{k_1, j_1}^{\#}\\&T_{k_1, j_1}^{\epsilon } \end{aligned} \right\} \right) } + \left\{ \begin{aligned}&\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, 1\\&\min _ { \begin{aligned} 0 \le j_2 \wedge j_2 < N_{k_2}\\ {{\mathrm{check}}}{ \left( \left( k_2, j_2 \right) , \left( k, j \right) \right) } \end{aligned} } { \left( \epsilon _ { \left( k_2, j_2 \right) , \left( k, j \right) } \right) } \end{aligned} \right\} . \end{aligned}$$

(6)

The inequalities (5) and (6) are approximate due to the use of considered locations. However, this allows finding stricter limitations for the solution inside the interval and simultaneously finding the solution for breaking at vertex $k_2$.

It is possible to construct (5) and (6) with exact inequalities by constructing the optimal solution when the endpoint is not required to end in the considered location. Similarly, the part from vertex $k_2$ to $\left( k, j \right)$ should not be required to end in the considered locations for vertex $k_2$.