Generalized Connectors

An $n$-connector is an acyclic directed graph having $n$ inputs and $n$ outputs and satisfying the following condition: given any one-to-one correspondence between inputs and distinct outputs, there exists a set of vertex-disjoint paths that join each input to the corresponding output. It is known that the minimum possible number of edges in an $n$-connector lies between lower and upper bounds that are asymptotic to $3n\log _3 n$ and $6n\log _3 n$ respectively. A generalized $n$-connector satisfies the following stronger condition: given any one-to-many correspondence between inputs and disjoint sets of outputs, there exists a set of vertex-disjoint trees that join each input to the corresponding set of outputs. It is shown that the minimum number of edges in a generalized $n$-connector is asymptotic to the minimum number in an $n$-connector

Practical Wide-Sense Nonblocking Generalized Connectors

In this note, we show that wide-sense nonblocking networks can be obtained by cascading a pair of Cantor networks or a pair of Clos networks. The only constraint placed on the routing algorithms is that branching be restricted to the second network in the cascade. This result yields practical network for multipoint communication with complexities O(N(logN)2 and O(N1+1/r)

Some Graph-Colouring Theorems with Applications to Generalized Connection Networks

With the aid of a new graph-colouring theorem, we give a simple explicit construction for generalized n-connectors with 2k - 1 stages and O( n1 + 1 / k (log n )( k - 1)/ 2 ) edges. This is asymptotically the best explicit construction known for generalized connectors

Wide-Sense Nonblocking Networks

A new method for constructing wide-sense nonblocking networks is presented. Application of this method yields (among other things) wide-sense nonblocking generalized connectors with n inputs and outputs and size O( n log n ), and with depth k and size O( n1 + 1/k ( log n )1 - 1/k )

Circuits with arbitrary gates for random operators

We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.Comment: 7 page

Mapping web personal learning environments

A recent trend in web development is to build platforms which are carefully designed to host a plurality of software components (sometimes called widgets or plugins) which can be organized or combined (mashed-up) at user's convenience to create personalized environments. The same holds true for the web development of educational applications. The degree of personalization can depend on the role of users such as in traditional virtual learning environment, where the components are chosen by a teacher in the context of a course. Or, it can be more opened as in a so-called personalized learning environment (PLE). It now exists a wide array of available web platforms exhibiting different functionalities but all built on the same concept of aggregating components together to support different tasks and scenarios. There is now an overlap between the development of PLE and the more generic developments in web 2.0 applications such as social network sites. This article shows that 6 more or less independent dimensions allow to map the functionalities of these platforms: the screen dimensionmaps the visual integration, the data dimension maps the portability of data, the temporal dimension maps the coupling between participants, the social dimension maps the grouping of users, the activity dimension maps the structuring of end users–interactions with the environment, and the runtime dimensionmaps the flexibility in accessing the system from different end points. Finally these dimensions are used to compare 6 familiar Web platforms which could potentially be used in the construction of a PLE