Bandwidth Constraints on Problems Complete for Polynomial Time

A graph G = ( b’. El has bandwidth k under a layout L : V 4’ ’ { 1. . . . , 1 VJ} if, for all {s. y} E E. jL(x 1 -L(y)] s k. Bandwidth constraints on several problems that are complete for [Fp (under log space reductions) are considered. In particular, the solvable path system problem and the and/or graph accessibility problem under various bandwidth constraints are used to prove results about subclasses of IFP. In general. restricting the bandwidth of problems complete for IFP results in complete problems for subclasses of IFP defined by simultaneous time-space bounds or defined by space bounds on alternating Turing machines. For instance, these results are used to show that the class SC, of sets accepted in polynomial time and simultaneous polylog space, can be characterized as the class reducible by log space transformations to qets accepted by one-wa) log log ~1 space bounded alternating Turing machines. An upper bound on the space requirements for tlie solvable path system problem under various bandwidth constraints is given by SPS( I (n \I E DSPACE( {(II ) log II 1. This yields, as a corollary, the result ASF’ACE(f(tr 1~ c [Jr, .,, DSPACE.t2 ““I ’ I for functions f that are suitably constructible and 40 .lot grow more rapidly than some logarithm function. This extends the known result: ASPACEtfrn )I = Jr, ,,, DTIME( 2 ““’ ’ 1. which only zpplles to functions that grow at least as rapidly as a logarithm function.