Lower bounds of network embedding dilations

Network embedding is widely used as a method for simulations between networks of different topological structures. Using an embedding of network G into network H, one can automatically transform any algorithm A_g developed for a multiprocessor system connected by G into an algorithm A_h for the multiprocessor system connected by H. One of the parameters used for determining the efficiency of simulation by network embedding is the dilation of the embedding. The dilation of an embedding must be as small as possible so that the communication delay in simulation can be minimized. In this paper, we present some lower bound results on the dilations of one-to-one embeddings from any arbitrary graph G into its optimum complete binary tree (the smallest complete binary tree with at least the same number of nodes in G). Furthermore, we show the equivalence between the problem of embedding a large tree into a small tree with balanced leaves and the problem of embedding a complete partial inorder tree into its optimum complete binary tree