In this paper, a mechanism connecting all weighted migration routes with minimum cost with EGOSST is proposed. Weighted migration routes may be converted to weighted input edges considered as not only traces but also traffics or trip frequencies of moving object on communication lines, roads or railroads. Proposed mechanism can be used in more wide and practical area than mechanisms considering only moving object traces. In our experiments, edge number, maximum weight for input edges, and detail level for grid are used as input parameters. The mechanism made connection cost decrease average 1.07% and 0.43% comparing with the method using weight minimum spanning tree and weight steiner minimum tree respectively. When grid detail level is 0.1 and 0.001, while each execution time for a connecting solution increases average 97.02% and 2843.87% comparing with the method using weight minimum spanning tree, connecting cost decreases 0.86% and 1.13% respectively. This shows that by adjusting grid detail level, proposed mechanism might be well applied to the applications where designer must grant priority to reducing connecting cost or shortening execution time as well as that it can provide good solutions of connecting migration routes with weights.