We proposed a new criterion \textit{noise-stability}, which revised the
classical rigidity theory, for evaluation of MDS algorithms which can
truthfully represent the fidelity of global structure reconstruction; then we
proved the noise-stability of the cMDS algorithm in generic conditions, which
provides a rigorous theoretical guarantee for the precision and theoretical
bounds for Euclidean embedding and its application in fields including wireless
sensor network localization and satellite positioning.
Furthermore, we looked into previous work about minimum-cost globally rigid
spanning subgraph, and proposed an algorithm to construct a minimum-cost
noise-stable spanning graph in the Euclidean space, which enabled reliable
localization on sparse graphs of noisy distance constraints with linear numbers
of edges and sublinear costs in total edge lengths. Additionally, this
algorithm also suggests a scheme to reconstruct point clouds from pairwise
distances at a minimum of O(n) time complexity, down from O(n3) for cMDS