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On multi-partition communication complexity

Pavol Ďuriš, Juraj Hromkovič, Stasys Jukna, Martin Sauerhoff, Georg Schnitger

2004
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Information and Computation
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We study k-partition communication protocols, an extension of the standard two-party best-partition model to k input partitions. The main results are as follows. 1. A strong explicit hierarchy on the degree of non-obliviousness is established by proving that, using k + 1 partitions instead of k may decrease the communication complexity from Θ(n) to Θ(log k). 2. Certain linear codes are hard for k-partition protocols even when k may be exponentially large (in the input size). On the other hand,
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... ne can show that all characteristic functions of linear codes are easy for randomized OBDDs. 3. It is proved that there are subfunctions of the triangle-freeness function and the function ⊕ Clique 3,n that are hard for multi-partition protocols. As an application, strongly exponential lower bounds on the size of nondeterministic read-once branching programs for these functions are obtained, solving an open problem of Razborov [22] . once branching programs. tion f if there is an assignment to the nondeterministic bits such that the protocol outputs 1 if and only if f (x, y) = 1. The complexity of a nondeterministic protocol P is the maximum of the number of exchanged bits taken over all inputs, including the nondeterministic bits. The nondeterministic communication complexity of f with respect to Π, ncc Π (f ), and the (best-partition) nondeterministic communication complexity of f , ncc(f ), are defined analogously to the deterministic case. For the following, it is important to mention an alternative, combinatorial characterization of nondeterministic communication complexity. For a partition Π = (X 1 , X 2 ) of the input variables, a (combinatorial) rectangle (with respect to Π) is a function r : {0, 1} n → {0, 1} that can be written as r = r (1) ∧r (2) , where the functions r (1) , r (2) : {0, 1} n → {0, 1} only depend on the variables in X 1 and X 2 , resp. A collection of such rectangles r 1 , . . . , r t with respect to Π is said to form a rectangle cover with respect to Π of a boolean function f defined on X if f = r 1 ∨ · · · ∨ r t . It is a well-known fact [9,16] that each nondeterministic communication protocol P for f with respect to a partition Π using m bits of communication yields a rectangle cover of f with respect to Π with 2 m rectangles and vice versa. In particular, ncc Π (f ) is equal to the logarithm (rounded up) of the minimum number of rectangles in a rectangle cover of f with respect to Π. We may regard two-party communication protocols as an oblivious model because they work with a fixed partition of the set of input variables for all inputs. Thus it is not surprising that a straightforward application of communication complexity for proving lower bounds only works for oblivious models of computation. As an example, we mention the situation for branching programs, where the first exponential lower bounds on the size using communication complexity have been for the oblivious variant of the model (Alon and Maass [3], see [12] for a generalized variant of their approach and [27] for a more detailed history of results). As an important step on the way to lower bounds for more general variants of branching programs, Okolnishnikova [20] and Borodin, Razborov, and Smolensky [6] succeeded in deriving exponential lower bounds on the size of the non-oblivious models of deterministic and nondeterministic syntactic read-k branching programs, resp. ¿From the perspective of communication complexity

doi:10.1016/j.ic.2004.05.002
fatcat:omgqvptmdfcctf5nmbs32xbnhi