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.

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 w/sub 1//spl middot/w/sub 2//spl middot//spl middot//spl middot//spl middot//spl middot/w/sub k/=id where each w/sub i/ is either a variable, an inverted variable, or group constant and id 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 assignment which simultaneously realizes each equation. We demonstrate 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 languages and the theory of nonuniform automata.

We prove that a monotone circuit of size n d recognizing connectivity must have depth $\Omega((\log n)^2/\log d)$. For formulas this implies depth $\Omega((\log n)^2/\log\log n)$. For polynomial-size circuits the bound becomes $\Omega((\log n)^2)$ which is optimal up to a constant.

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, Håstad, 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).

Article Monotone circuits for connectivity have depth (log n)2-o(1) (extended abstract) Share on Authors: Mikael Goldmann Royal Institute of Technology Royal Institute of TechnologyView Profile , Johan Håstad Royal Institute of Technology and Laboratory for Computer Science, MIT Royal Institute of Technology and Laboratory for Computer Science, MITView Profile Authors Info & Claims STOC '95: Proceedings of the twenty-seventh annual ACM symposium on Theory of computingMay 1995 Pages 569–574https://doi.org/10.1145/225058.225274Online:29 May 1995Publication History 3citation180DownloadsMetricsTotal Citations3Total Downloads180Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

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).

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.>