Abstract
We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any pair of sufficiently large sets is close to the expected number, we consider partitions and require this condition only for most of the pairs of blocks. As a result, the blocks can be substantially smaller. We show that for some range of parameters, to be a partition expander a random graph needs exponentially smaller degree than any expander would require in order to achieve similar expanding properties. We apply the concept of partition expanders in communication complexity. First, we construct an optimal pseudo-random generator (PRG) for the Simultaneous Message Passing (SMP) model: it needs n + log k random bits against protocols of cost Ω(k). Second, we compare the SMP model to that of Simultaneous Two-Way Communication, and give a new separation that is stronger both qualitatively and quantitatively than the previously known ones.
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Notes
We use “N” to denote the number of graph vertices and “n” for the input length throughout the paper.
We get quadratic improvement in terms of the partition size, and show that it is, essentially, the best possible general bound in terms of the spectral gap alone.
All previously known PRGs in communication complexity were given against stronger models, thus requiring exponentially larger “overhead” over n in terms of seed length—for details, see Section 5.
In those cases when we explicitly allow multiple edges, the edges of a graph will be viewed as a collection with repetitions.
Note that if v 1, v 2 ∈ S 1 ∩ S 2, then the edge (v 1, v 2) appears in E(S 1, S 2) twice: as ordered pairs (v 1, v 2) and (v 2, v 1).
Note also that the distribution \(\mathcal {G}_{N,d}^{\prime }\) is not uniform over its support—e.g., \(\mathcal {G}_{2,2}^{\prime }\) produces the graph with two parallel edges with probability 2/3.
Note that if e = (v 2i−1, v 2i ) ∈ E then (v 2i , v 2i−1) ∈ E as well.
For example, the models \(\mathcal {R}^{1}\) and \(\mathcal {R}^{\leftrightarrow }\) (and more generally, any two-party model where a k-bit message from one player reaches the other player, who also receives his own n bits of input) require seed length at least n + k − O(1) even with a non-uniform PRG, as witnessed by the protocol where the sender sends the first k bits of his input and the computationally-unlimited recipient outputs “1” only if the message together with his own n bits of input have Kolmogorov complexity n + k−O(1).
The same applies to many other fields of complexity, where also most of the following discussion remains valid—e.g., in the field of circuit complexity.
Note that we required both \(\mathcal {U}_{f^{-1}(0)}\) and \(\mathcal {U}_{f^{-1}(1)}\) to be k-PRGs for \(\mathcal {M}\) when f is k-pseudo-random in order not to require f to be balanced; if it is balanced, either condition implies the other.
The power of the model \(\mathcal {R}^{\leftrightarrow }\) is not affected by allowing public randomness.
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We thank Hartmut Klauck, Michael A. Forbes and anonymous reviewers for helpful comments.
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Partially funded by the grant P202/12/G061 of GA ČR and by RVO: 67985840.
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Gavinsky, D., Pudlák, P. Partition Expanders. Theory Comput Syst 60, 378–395 (2017). https://doi.org/10.1007/s00224-016-9738-5
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DOI: https://doi.org/10.1007/s00224-016-9738-5