Li, ZongpengFang, Wei2016-09-292016-09-2920162016Fang, W. (2016). A Study of the Multiple-Unicast Network Coding Conjecture Using Riemannian Manifolds (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/25044http://hdl.handle.net/11023/3351Network coding encourages information mixing at the intermediate nodes within a network. The multiple-unicast conjecture proposed by Li and Li [18] in 2004 is one of the most well-known unsolved problems in network coding field. The conjecture asserts that, for multiple independent unicast transmissions in an undirected network, network coding has no advantage over traditional routing. In this thesis, we study the conjecture by embedding graphs into Riemannian manifolds using a geometric framework developed by Xiahou el al. [32]. We prove that isometric embedding of graphs into a Riemannian manifold is impossible. Then, interestingly, we construct an embedding that achieves an infinitesimally small distortion. We show that if the multiple-unicast network coding conjecture is true on Riemannian manifolds, it is also true for undirected networks. Our hope is to develop a Riemannian geometry approach for making new progresses against the long-time open conjecture.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Computer Sciencenetwork codingmultiple-unicast conjectureRiemannian manifoldsmaster thesishttp://dx.doi.org/10.11575/PRISM/25044