A $2d - 1$ Lower Bound for Two-Layer Knock-Knee Channel Routing

Abstract

This paper describes a two-point net channel routing problem with density d that requires channel width 2d-1 in the two-layer knock-knee channel routing model. This means that the (2d-1)-track algorithms of Rivest, Baratz, and Miller [1981 CMU Conference on VLSI Systems and Computations, Oct. 198 1, pp. 153-159], Bolognesi and Brown [unpublished manuscript, Coordinated Science Laboratory, University of Illinois at Urbanna-Charnpaign, 1982], Frank [Combinatorica, 2 (1982), pp. 361-37], Mehlhorn, Preparata, and Sarrafzadeh [University of Saarbrucken Tech. Rep., Saarbrucken, Germany, Nov. 1984], and Berger et al. [J. Assoc. Comput. Mach., 1994, to appear], are, in some cases, optimal. Thus, any improvement of these algorithms must rely on problem features other than density (such as flux [Advances in Computing Research 2 (VLSI Theory), F. P. Preparata, ed., JAI Press, Greenwich, CT, 1984, pp. 205-229]) or must make fundamental changes in the wiring model (such as increasing the number of layers [IEEE Trans. Comput., C-33 (1984), pp. 427-437) or allowing wires to overlap [see Berger et al., above], [Algorithmica, 1 (1986), pp. 223-232]).