We study the sensitivity of the Lempel-Ziv 77 compression algorithm to edits, showing how modifying a string w can deteriorate or improve its compression. Our first result is a tight upper bound for k edits: ∀ w' ∈ B(w,k), we have C_LZ77(w') ≤ 3 · C_LZ77(w) + 4k. This result contrasts with Lempel-Ziv 78, where a single edit can significantly deteriorate compressibility, a phenomenon known as a *one-bit catastrophe*. We further refine this bound, focusing on the coefficient 3 in front of C_LZ77(w), and establish a surprising trichotomy based on the compressibility of w. More precisely we prove the following bounds: - if C_LZ77(w) ≲ k^3/2√(n), the compression may increase by up to a factor of ≈ 3, - if k^3/2√(n)≲ C_LZ77(w) ≲ k^1/3n^2/3, this factor is at most ≈ 2, - if C_LZ77(w) ≳ k^1/3n^2/3, the factor is at most ≈ 1. Finally, we present an ε-approximation algorithm to pre-edit a word w with a budget of k modifications to improve its compression. In favorable scenarios, this approach yields a total compressed size reduction by up to a factor of 3, accounting for both the LZ77 compression of the modified word and the cost of storing the edits, C_LZ77(w') + k log |w|.
What is the least integer $\text{sd}(n)$ such that every graph on $n$ vertices has fractional chromatic number $p /q$, where $p$ and $q$ are positive integers and $q \le \text{sd}(n)$? An upper bound on the determinants of Hadamard matrices implies that $\text{sd}(n)\le 2^{-n}(n+1)^{(n+1) /2}$. The only known lower bound on $\text{sd}(n)$ that is exponential in $n$ (asymptotically, roughly $1.346^n/\sqrt{\log n}$) was obtained using an iterated Mycielski construction [D. C. Fisher, J. Graph Theory 20 (1995), 403-409]. We improve on this bound by constructing a family of graphs which shows that $\text{sd}(n) \geq 2^{n/2}$.
BACKGROUND:Improved Physics models for the elastic and inelastic interactions of electrons in liquid water have recently been developed and implemented in the Geant4-DNA toolkit, allowing detailed electron transport simulations up to MeV energies. PURPOSE:To present a preliminary performance evaluation of a new set of Geant4-DNA Physics models, which will become the default Physics list for liquid water in the upcoming Geant4 public releases. METHODS:A new set of Physics models (constructor), denoted as « Option4 X », is implemented in Geant4-DNA that combines a newly developed inelastic model with two alternative models for the simulation of elastic scattering, one based on the ELSEPA software and another one based on the analytic Uehara Screened Rutherford model. The inelastic model includes all the low-energy developments of the existing « Option4 » Physics constructor, upgrades its internal consistency (via sum rules), and extends its application from 10 keV to 10 MeV using the Relativistic Plane Wave Born Approximation (RPWBA). RESULTS:Preliminary results for electron dose point kernels (DPKs), frequency-mean (yF) and dose-mean (yD) lineal energies in nanometric spheres, S-values in spherical cells, and the W-value of liquid water are simulated, and a comparison is presented between the new Physics constructor, « Option4 X », the current default Physics constructor, « Option2 », and the low-energy Physics constructor, « Option4 ». Significant improvement is offered by « Option4 X » in the simulation of DPK and the W-value, which have been two long-lasting performance limitations of the current default Physics constructor, while the choice of elastic model has a relatively small influence. CONCLUSIONS:«Option4 X» improves and extends the current Geant4-DNA transport capabilities up to 10 MeV and will become the default Geant4-DNA Physics constructor for liquid water in the upcoming public releases of Geant4. This development will allow Geant4-DNA users, along with users of Monte Carlo software that are built on Geant4 (like GATE and TOPAS-nBio), to perform more accurate electron track structure simulations over a wider energy range (10 eV ‒ 10 MeV) and cover radiotherapeutic applications of track structure simulations not reachable until now.
The SLEEPING LOCAL model introduces a new complexity parameter, the awake complexity, to make distributed algorithms energy-efficient. In the synchronous LOCAL model, nodes can now decide to be awake or asleep in each round. In a round, only awake nodes can communicate to share information, which consumes energy. The awake complexity is the maximum number of times a node is activated to produce an output. In particular, it often comes at the cost of the total number of rounds required to solve a problem, compared with algorithms in which every node is awake in every round. In this article, we adapt the notion of awaken rounds to the context of self-stabilization, introducing the Self-Stabilizing SLEEPING model. Nodes are no longer required to remain awake at all times. However, in self-stabilization, nodes must be activated infinitely often to detect any issue in the system's current state. In this model, the complexities are: * How many synchronous rounds are needed to reach a legitimate configuration? * How many times does a node need to be awake to reach this configuration? * How often does a node need to be awake once this configuration is reached? The goal is to minimize those three metrics, and we can expect different trade-offs. We present energy-efficient algorithms to solve the problems of finding a (Δ+1)-coloring, a Maximal Independent Set, and a Maximal Matching, thanks to new ad hoc sleeping techniques that reduce the awake complexity (i.e., energy consumption) during the convergence phase. We also propose two transformers that adapt silent self-stabilizing algorithms to the SLEEPING setup. The first transformer is pretty simple and deals with low-complexity algorithms. The second is more elaborate and is more energy-efficient when it transforms slow self-stabilizing algorithms.
