**Reference**:- Mihai Bădoiu, Erik D. Demaine, MohammadTaghi Hajiaghayi, and Piotr Indyk, “Low-Dimensional Embedding with Extra Information”, in
*Proceedings of the 20th Annual ACM Symposium on Computational Geometry (SoCG 2004)*, Brooklyn, New York, June 9–11, 2004, pages 320–329. **Abstract**:- A frequently arising problem in computational geometry is when a physical structure, such as an ad-hoc wireless sensor network or a protein backbone, can measure local information about its geometry (e.g., distances, angles, and/or orientations), and the goal is to reconstruct the global geometry from this partial information. More precisely, we are given a graph, the approximate lengths of the edges, and possibly extra information, and our goal is to assign coordinates to the vertices that satisfy the given constraints up to a constant factor away from the best possible. We obtain the first subexponential-time (quasipolynomial-time) algorithm for this problem given a complete graph of Euclidean distances with additive error and no extra information. For general graphs, the analogous problem is NP-hard even with exact distances. Thus, for general graphs, we consider natural types of extra information that make the problem more tractable, including approximate angles between edges, the order type of vertices, a model of coordinate noise, or knowledge about the range of distance measurements. Our quasipolynomial-time algorithm for no extra information can also be viewed as a polynomial-time algorithm given an “extremum oracle” as extra information. We give several approximation algorithms and contrasting hardness results for these scenarios.
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**Related papers**:- Embedding_DCG (Low-Dimensional Embedding with Extra Information)

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Last updated June 1, 2023 by Erik Demaine.