Mikael Goldmann

Majority gates vs. general weighted threshold gates

In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following:1.A single threshold gate with weights cannot in general be replaced by a polynomial fan-in unweighted threshold gate of parity gates.2.On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depthd polynomial size threshold circuit with weights can be computed by a depthd+1 polynomial size unweighted threshold circuit, whered is an arbitrary fixed integer.3.A polynomial fan-in threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size.

The complexity of solving equations over finite groups

We study the computational complexity of solving systems of equations over a finite group. An equation over a group G is an expression of the form w1.w2....wk = 1G, where each wi is either a variable, an inverted variable, or a group constant and 1G is the identity element of G . A solution to such an equation is an assignment of the variables (to values in G) which realizes the equality. A system of equations is a collection of such equations; a solution is then an assigmnent which simultaneously realizes each equation. We show that the problem of determining if a (single) equation has a solution is NP-complete for all nonsolvable groups G. For nilpotent groups, this same problem is shown to be in P. The analogous problem for systems of such equations is shown to be NP-complete if G is non-Abelian, and in P otherwise. Finally, we observe some connections between these problems and the theory of nonuniform automata.

On the power of small-depth threshold circuits

The power of threshold circuits of small depth is investigated. In particular, functions that require exponential-size unweighted threshold circuits of depth 3 when the bottom fan-in is restricted are given. It is proved that there are monotone functions f/sub k/ that can be computed on depth k and linear size AND, OR circuits but require exponential-size to be computed by a depth-(k-1) monotone weighted threshold circuit.<<ETX>>

Simulating Threshold Circuits by Majority Circuits

We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size, depth-2 majority circuit. In general we show that a polynomial-size, depth-d threshold circuit can be simulated uniformly by a polynomial-size majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [Comput. Complexity, 2 (1992), pp. 277--300] that a nonuniform simulation exists. Our construction answers two open questions posed by them: we give an explicit construction, whereas they use a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (their simulation gives polynomial-size circuits only when d is constant).

Simulating threshold circuits by majority circuits

We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size, depth-2 majority circuit. In general we show that a polynomial-size, depth-d threshold circuit can be simulated uniformly by a polynomial-size majority circuit of depth d + 1. Goldmann, H astad, and Razborov showed in (Comput. Complexity, 2 (1992), pp. 277{300) that a nonuniform simulation exists. Our construction answers two open questions posed by them: we give an explicit construction, whereas they use a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (their simulation gives polynomial-size circuits only when d is constant).

A simple lower bound for monotone clique using a communication game

Abstract We give a simple proof that a monotone circuit for the k-clique problem in an n-vertex graph requires depth Ω( k ) , when k ⩽( n 2 3 ) 2 . The proof is based on an equivalence between the depth of a Boolean circuit for a function and the number of rounds required to solve a related communication problem. This equivalence was shown by Karchmer and Wigderson.

Measurements on the spotify peer-assisted music-on-demand streaming system

Spotify is a streaming service offering low-latency access to a large library of music. Streaming is performed by a combination of client-server access and a peer-to-peer protocol. The service currently has a user base of over 10 million and is available in seven European countries. We provide a background on the Spotify protocol with emphasis on the formation of the peer-to-peer overlay. Using measurement data collected over a week by instrumenting Spotify clients, we analyze general network properties such as the correspondence between individual user accounts and the number of IP addresses they connect from and the prevalence of Network Address Translation devices (NATs). We also discuss the performance of one of the two peer discovery mechanisms used by Spotify.

On the power of a threshold gate at the top

The discriminator lemma is normally used to prove lower bounds for circuits with small-weight threshold gates. In this note we adapt the lemma to circuits with a general (large-weight) threshold gate at the top. The new lemma is used to give a new proof of a previously known lower bound for the size of a threshold of parity gates that computes inner product mod 2, and to prove that a small-depth AND-OR circuit for parity must have exponential size even if we allow a threshold gate at the top. The latter result is a generalization to large weights of a result by Green, and depends heavily on Hastads switching-lemma.

Monotone Circuits for Connectivity Have Depth (log n ) 2- o (1)

We prove that a monotone circuit of size nd recognizing connectivity must have depth

On average time hierarchies

\Omega((\log n)^2/\log d)