Dynamic graph research is an essential subject in Computer Science. The shortest path problem is still the central in this field; moreover there is a variety of applications in practical projects. In this paper, we select the transportation activity as a paradigm to study. Of course there exists the most clear and appealing practical application over the intelligent navigation, also which is an important part in intelligent transportation system (ITS). Totally speaking, this activity is fairly simple and well-behavior under our consideration, for which, there is a lot of sophisticated theories and techniques to motivate our researches although this problem relates to interdisciplinary; so that we can be dedicated to computing aspect. Below, a host of practical problems will by the way be discussed in theory and empiricism including correctness, sampling, accuracy, compatibility, quick query and the application into other research; therein some have been scarcely argued intentionally or not in literatures before. Through these contents, we are able to have a smart system built which easy to incorporate other modules to realize a sophisticated industrial system.
In this aspect of transportation, a seminal work has been accomplished by Chabini and his students; they have done a lot of celebrated works especially to regulate the context of transportation problem. In @cite_0 , Chabini and Ganugapati intensively summarizes these premises for transportation problem: (1) versus . (2) versus . (3) ph One-to-all versus . (4) versus ph unallowable . (5) () versus .
@cite_0
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In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms WGAN for approximating probability measures with multiple clusters in low dimensional space. Abstract We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Świȩch (1998) [15] . We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results in Gangbo and Świȩch (1998) [15] . The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011) [1] . The classic optimal transportation problem consists in finding the most cost-effective way of moving masses from one set of locations to another, minimizing its transportation cost. The formulation of this problem and its solution have been useful to understand various mathematical, economical, and control theory phenomena, such as, e.g., Witsenhausen's counterexample in stochastic control theory, the principal-agent problem in microeconomic theory, location and planning problems, etc. In this work, we incorporate the effect of network congestion to the optimal transportation problem and we are able to find a closed form expression for its solution. As an application of our work, we focus on the mobile association problem in cellular networks (the determination of the cells corresponding to each base station). In the continuum setting, this problem corresponds to the determination of the locations at which mobile terminals prefer to connect (by also considering the congestion they generate) to a given base station rather than to other base stations. Two types of problems have been addressed: a global optimization problem for minimizing the total power needed by the mobile terminals over the whole network (global optimum), and a user optimization problem, in which each mobile terminal chooses to which base station to connect in order to minimize its own cost (user equilibrium). This work combines optimal transportation with strategic decision making to characterize both solutions. We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely the periodically-varying graphs (the PV-graphs, modeling public transportation systems, among others). These are defined by a set of carriers following infinitely their prescribed route along the stations of the network. Flocchini, Mans, and Santoro [FMS09] (ISAAC 2009) studied this problem in the case when the agent must always travel on the carriers and thus cannot wait on a station. They described the necessary and sufficient conditions for the problem to be solvable and proved that the optimal number of steps (and thus of moves) to explore a n-node PV-graph of k carriers and maximal period p is in Θ(k·p2) in the general case. In this paper, we study the impact of the ability to wait at the stations. We exhibit the necessary and sufficient conditions for the problem to be solvable in this context, and we prove that waiting at the stations allows the agent to reduce the worst-case optimal number of moves by a multiplicative factor of at least Θ(p), while the time complexity is reduced to Θ(n·p). (In any connected PV-graph, we have n≤k·p.) We also show some complementary optimal results in specific cases (same period for all carriers, highly connected PV-graphs). Finally this new ability allows the agent to completely map the PV-graph, in addition to just explore it.