Write Channel Model for Bit-Patterned Media Recording

We propose a new write channel model for bit-patterned media recording that reflects the data dependence of write synchronization errors. It is shown that this model accommodates both substitution-like errors and insertion-deletion errors whose statistics are determined by an underlying channel state process. We study information theoretic properties of the write channel model, including the capacity, symmetric information rate, Markov-1 rate and the zero-error capacity.Comment: 11 pages, 12 figures, journa

Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits

A complete orthonormal basis of N-qutrit unitary operators drawn from the Pauli Group consists of the identity and 9^N-1 traceless operators. The traceless ones partition into 3^N+1 maximally commuting subsets (MCS's) of 3^N-1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3^N are isomorphic to all MCS's, and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure allowing for the coexistence of separable, partially-entangled, and totally entangled (GHZ-like) bases. However, a minimum of 16 GHZ-like bases must occur. Every basis state is described by an N-digit trinary number consisting of the eigenvalues of N observables constructed from the corresponding MCS.Comment: LaTeX, 10 pages, 2 references adde

Hardness of Games and Graph Sampling

The work presented in this document is divided into two parts. The �rst part presents the hardness of games and the second part presents Graph sampling. Non-deterministic constraint logic[1] is used to prove the hardness of games. The games which are considered in this work is Reversi (2 player bounded game), Peg Solitaire (single player bounded game), Badland (single player bounded game). It also contains a theoretical study of peg solitaire on special graph classes. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire is proved to be NP-Complete using Bounded NCL. Badland is proved to be NP-Complete by a reduction from 3-SAT. The objective of study of peg solitaire of special graph classes is to �nd the maximum number of marbles we can remove from a fully �lled board, if the player is given the privilege to remove a marble from any cell initially, then following the rules after the initial move. The second part of the work is dedicated to graph sampling. Given a graph G, we try to sample a represen- tative subgraph Gs which is similar to the original graph G. The properties that are being studied are Degree Distribution, Clustering Coefficient, Average Shortest Path Length, Largest Connected Component Size. To measure the similarity between the original graph and sample we use the metrics Kolmogorov - Smirnov test and Kullback - Leibler divergence test. Tightly Induced Edge Sampling performs well on general graphs but it's performance decreases when the graph is a tree. Overall TIBFS and KARGER produces a sample which closely matches the distribution of original graphs.

An FPT algorithm for Matching Cut and d-cut

Given a positive integer $d$, the $d$-CUT problem is to decide if an undirected graph $G=(V,E)$ has a non trivial bipartition $(A,B)$ of $V$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). When $d=1$, this is the MATCHING CUT problem. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for $d$-CUT, when parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for the MATCHING CUT (and more generally for $d$-CUT) for general graphs with running time $2^{O(k\log k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut. This is the first FPT algorithm for the MATCHING CUT (and $d$-CUT) with an explicit dependence on this parameter. We also observe a lower bound of $2^{\Omega(k)}n^{O(1)}$ with same parameter for MATCHING CUT assuming ETH