As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packa...As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packages such as LINDO, and furthermore this model is extendedinto the stochastic version, that is, the minimal spanning tree problem for the digraph with theweights is not constant but random variables. Several algorithms are also developed to solve themodels. Finally, a numerical demonstration is given.展开更多
Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted l...Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted length of the minimum spanning tree containing all the n nodes,where the edge length between nodes X_(i)and X_(j)is weighted by a factor that depends on the individual locations of X_(i)and X_(j).We use approximation methods to obtain variance estimates for MSTC_(n)and prove that if the cities are well connected in a certain sense,then MSTC_(n)appropriately centred and scaled converges to zero in probability.Using the above proof techniques we also study MST_(n),the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights.In this case,the variance of MST_(n)grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of MST_(n),appropriately centred and scaled.展开更多
文摘As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packages such as LINDO, and furthermore this model is extendedinto the stochastic version, that is, the minimal spanning tree problem for the digraph with theweights is not constant but random variables. Several algorithms are also developed to solve themodels. Finally, a numerical demonstration is given.
基金I thank Professors Rahul Roy,Jacob van den Berg,Anish Sarkar,Federico Camia and the referees for crucial comments that led to an improvement of the paper.I also thank Professors Rahul Roy,Federico Camia and IMSc for my fellowships。
文摘Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted length of the minimum spanning tree containing all the n nodes,where the edge length between nodes X_(i)and X_(j)is weighted by a factor that depends on the individual locations of X_(i)and X_(j).We use approximation methods to obtain variance estimates for MSTC_(n)and prove that if the cities are well connected in a certain sense,then MSTC_(n)appropriately centred and scaled converges to zero in probability.Using the above proof techniques we also study MST_(n),the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights.In this case,the variance of MST_(n)grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of MST_(n),appropriately centred and scaled.
