## 1 Introduction  
Two-dimensional (2D) vector maps have a broad range of applications, such as engineering construction, urban planning, military affairs, and navigation [1; 2; 3]. The production of vector maps is often expensive, and illegal copying, altering, and misuse of the data could result in tremendous losses to data owners. With increasing demands for copyright protection, digital watermarking techniques have been developed rapidly in recent years [4; 5; 6; 7]. Efforts have been made to design various watermark embedding algorithms for digital images to improve the robustness and imperceptibility of the watermark [8; 9; 10; 11; 12]. The abundant studies on image watermarking provide valuable information and references for other digital products as well [13; 14; 15; 16], and thus enhance the development of watermarking for vector maps. Compared with watermarking of digital images, watermarking of vector maps requires not only external graphic analysis but also internal numerical calculations and thus poses challenges to the watermarking schemes. Classic watermarking algorithms of vector data mainly include the coordinate-domain and the frequency-domain watermarking algorithms. [PERSON] and [PERSON] [PERSON] and [PERSON] (1977) first proposed the coordinate-domain watermarking algorithm of vector maps and advocated encoding watermark information directly at each vertex coordinate. To improve the robustness of watermarking against various attacks, later studies developed methods to embed watermarks by controlling the position, tolerance, and accuracy of the map elements [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977); [PERSON] (1977). However, there is a lack of discussion on the deformation of map elements. For example, the Least Significant Bit method, a common watermarking algorithm in vector maps [PERSON] and [PERSON] (1977); [PERSON] (1977); [PERSON] (1977), embeds a watermark in the coordinate precision bit, but it is vulnerable to the bit-erase attack. In recent years, the developments in coordinate-domain watermarking algorithms have accounted for geographical features, geometric shapes and topological relations of map elements. [PERSON] et al. [PERSON] et al. (2018) proposed a watermarking scheme for the vector map that modifies the statistics of non-feature vertices in data blocks while keeping the geometric shapes of geographical elements. [PERSON] et al. [PERSON] et al. (2018) applied a general rule of the actual geometric error (0.75 m) in the 1: 2500 scale maps to limit the vertex movements after watermarking. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) put forward a watermarking scheme for polylines and polygons using metrics of both relative and absolute position accuracies to ensure the vertex movements within a maximum tolerance range. [PERSON] et al. [PERSON] et al. (2018) used the Voronoi diagram and constraint Delaunay triangulation to generate a maximum perturbed region for each individual vertex, thereby limiting topological element changes during the watermarking process. The frequency-domain watermarking algorithm uses invariant geometric information in key frequency-domain coefficients to embed watermarks. Common frequency-domain algorithms include the discrete Fourier transform, discrete wavelet transform and discrete cosine transform. The discrete Fourier transform based watermarking algorithm for vector maps is to embed watermarks in the Fourier coefficient sequence, which take advantage of the Fourier description factor's geometric invariance on the scaling and rotation to promote the watermarking scheme's robustness [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018); [PERSON] et al. [PERSON] et al. (2018) divided vector maps into grid sets as analogue to the pixels in raster images and introduced discrete wavelet transform to embed watermark in vector maps.