arXiv subjects

Shrisha Rao

Publications and source records attributed to Shrisha Rao.

41 records · Page 3Linked to original sources

arXiv2009

An Algebra of Fault Tolerance

Every system of any significant size is created by composition from smaller sub-systems or components. It is thus fruitful to analyze the fault-tolerance of a system as a function of its composition. In this paper, two basic types of system composition are described, and an algebra to describe fault tolerance of composed systems is derived. The set of systems forms monoids under the two composition operators, and a semiring when both are concerned. A partial ordering relation between systems is used to compare their fault-tolerance behaviors.

math.GM↗

arXiv2009

The Semiring Properties of Boolean Propositional Algebras

This paper illustrates the relationship between boolean propositional algebra and semirings, presenting some results of partial ordering on boolean propositional algebras, and the necessary conditions to represent a boolean propositional subalgebra as equivalent to a corresponding boolean propositional algebra. It is also shown that the images of a homomorphic function on a boolean propositional algebra have the relationship of boolean propositional algebra and its subalgebra. The necessary and sufficient conditions for that homomorphic function to be onto-order preserving, and also an extension of boolean propositional algebra, are explored.

math.RA↗

arXiv2008

Fairness in Combinatorial Auctioning Systems

One of the Multi-Agent Systems that is widely used by various government agencies, buyers and sellers in a market economy, in such a manner so as to attain optimized resource allocation, is the Combinatorial Auctioning System (CAS). We study another important aspect of resource allocations in CAS, namely fairness. We present two important notions of fairness in CAS, extended fairness and basic fairness. We give an algorithm that works by incorporating a metric to ensure fairness in a CAS that uses the Vickrey-Clark-Groves (VCG) mechanism, and uses an algorithm of Sandholm to achieve optimality. Mathematical formulations are given to represent measures of extended fairness and basic fairness.

cs.GT↗

arXiv2008

Central Forests in Trees

A new 2-parameter family of central structures in trees, called central forests, is introduced. Minieka's $m$-center problem and McMorris's and Reid's central-$k$-tree can be seen as special cases of central forests in trees. A central forest is defined as a forest $F$ of $m$ subtrees of a tree $T$, where each subtree has $k$ nodes, which minimizes the maximum distance between nodes not in $F$ and those in $F$. An $O(n(m+k))$ algorithm to construct such a central forest in trees is presented, where $n$ is the number of nodes in the tree. The algorithm either returns with a central forest, or with the largest $k$ for which a central forest of $m$ subtrees is possible. Some of the elementary properties of central forests are also studied.

math.CO↗

arXiv2007

The $n$-Queens Problem in Higher Dimensions

A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is known to have been studied by Gauss, and can be generalized to an (n \times n) board, where (n \geq 4). We consider this problem in $d$-dimensional chess spaces, where (d \geq 3), and obtain the result that in higher dimensions, $n$ queens do not always suffice (in any arrangement) to attack all board positions. Our methods allow us to obtain the first lower bound on the number of queens that are necessary to attack all positions in a $d$-dimensional chess space of size $n$, and further to show that for any $k$, there are higher-dimensional chess spaces in which not all positions can be attacked by (n^k) queens.

math.CO↗