Tight Pounds on Oblivious Chaining

The chaining problem is deened as follows. = maxfj j a j = 1; j < ig. (Deene maxfg = 0) The chaining problem appears as a subproblem in many contexts. There are algorithms known that solve the chaining problem on CRCW PRAMs in O((n)) time, where (n) is the inverse of Ackerman's function, and is a very slowly growing function. We study a class of algorithms (called oblivious algorithms) for this problem. We present a simple oblivious chaining algorithm running in O((n)) time. More importantly, we demonstrate the optimality of the algorithm by showing a matching lower bound for oblivious algorithms using n processors. We also provide the rst steps towards a lower bound for all chaining algorithms by showing that any chaining algorithm that runs in two steps must use a superlinear number of processors. Our proofs use preex graphs and weak superconcentrators. We demonstrate an interesting connection between the two and use this idea to obtain improved bounds on the size of preex graphs.