Separating the communication complexities of MOD m and MOD p circuits

We prove in this paper that it is much harder to evaluate depth--2, size--$N$ circuits with MOD $m$ gates than with MOD $p$ gates by $k$--party communication protocols: we show a $k$--party protocol which communicates $O(1)$ bits to evaluate circuits with MOD $p$ gates, while evaluating circuits with MOD $m$ gates needs $\Omega(N)$ bits, where $p$ denotes a prime, and $m$ a composite, non-prime power number. Let us note that using $k$--party protocols with $k\geq p$ is crucial here, since there are depth--2, size--$N$ circuits with MOD $p$ gates with $p>k$, whose $k$--party evaluation needs $\Omega(N)$ bits. As a corollary, for all $m$, we show a function, computable with a depth--2 circuit with MOD $m$ gates, but not with any depth--2 circuit with MOD $p$ gates. It is easy to see that the $k$--party protocols are not weaker than the $k'$--party protocols, for $k'>k$. Our results imply that if there is a prime $p$ between $k$ and $k'$: $k<p\leq k'$, then there exists a function which can be computed by a $k'$--party protocol with a constant number of communicated bits, while any $k$--party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi--party protocols

Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity

In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between quantum branching programs and nonuniform quantum Turing machines are presented which allow to transfer lower and upper bound results between the two models. In the second part of the paper, different variants of quantum OBDDs are compared with their deterministic and randomized counterparts. In the third part, quantum branching programs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte

Communication Memento: Memoryless Communication Complexity

We study the communication complexity of computing functions $F:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}$ in the memoryless communication model. Here, Alice is given $x\in \{0,1\}^n$, Bob is given $y\in \{0,1\}^n$ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least $c \log n$? Note that $c\geq 2+\varepsilon$ would imply a $\Omega(n^{2+\varepsilon})$ lower bound for general formula size, improving upon the best lower bound by Ne\v{c}iporuk in 1966.Comment: 30 Pages; several improvements to the presentation