## 2 Related Works  
The vector map watermarking algorithm was first proposed and designed by [PERSON] [14], who encoded watermark information directly on the coordinate values of each vertex. The method is also the first spatial domain watermarking for vector maps, but it is vulnerable to various simple attacks and is poorly concealed. In order to improve concealment and security, most vector map watermarking in the existing literature on a spatial domain would use the geometric statistical characteristics as the watermark embedding domain, including topological relations [15], angle values between feature vertices [16], element perimeters and areas [17], the spatial distance between different elements [18], etc. This collection of spatial statistical eigenvalues possesses a wide range of geometric invariants that can serve as a solid foundation for the applicable watermarking methods. However, all of these watermarking techniques accomplish watermark embedding by directly altering the coordinate values, which leaves them vulnerable to precision bit erasure and low robustness [19]. In contrast to the spatial domain watermarking method of directly manipulating coordinate values, the frequency domain watermarking of vector maps converts the coordinate domain to the frequency domain and inserts watermark information by adjusting the frequency coefficient [19]. Typical frequency transform methods include discrete Fourier transform (DFT), discrete wavelet transform (DWT), discrete cosine transform (DCT), and singular value decomposition (SVD) [19; 20; 21; 22]. The transformation coefficients of these methods all provide a certain degree of geometric invariance, hence ensuring the robustness of related watermarking. Both spatial domain watermarking and frequency domain watermarking embed watermarks by modifying vertex coordinates. Many research used deliberate interference reduction strategies, but the quality and availability of watermarked vector maps are still affected to varying degrees. In this case, the zero-watermarking algorithm with lossless characteristics has attracted more and more attention [20; 23; 24; 25]. Rather than altering the protected electronic data in any way of embedded watermarking, zero-watermarking extracts feature information from it to build the watermark and then utilizes the watermark pictures themselves as the secret key for copyright verification, with the help of a trusted third party [26]. The non-embedded way subverts the functional mechanism of conventional watermarking and achieves no interference with data in a true sense, which is ideal for the precise usage requirements of vector maps [27]. Depending on the type of feature information, vector map zero watermarking may be broken down into two categories: spatial domain statistical features-based zero watermarking and improved frequency domain-based zero watermarking. In spatial statistical features-based zero watermarking, [PERSON] et al. [28] built zero-watermarking by taking the distance between feature points and cluster centers in cluster blocks as feature information. The method is resistant to data reduction or compression and geometric modification. [PERSON] et al. [29] divided the vector map into multiple rings with concentric circles, calculated the number of vertices in each ring as feature information, and constructed zero-watermarking images after the XOR operation with copyright information. This algorithm can effectively resist common geometric attacks and coordinate point attacks. [PERSON] et al. [30] employed the distance ratio between each feature point and the geometric center of all feature points as the feature information to construct zero watermarking, and this method can better resist geometric attacks, interpolation, and simplification attacks. Regarding the research of improved frequency domain zero watermarking, [PERSON] et al.