An Information-Theoretic Analysis of Deduplication

Deduplication finds and removes long-range data duplicates. It is commonly used in cloud and enterprise server settings and has been successfully applied to primary, backup, and archival storage. Despite its practical importance as a source-coding technique, its analysis from the point of view of information theory is missing. This paper provides such an information-theoretic analysis of data deduplication. It introduces a new source model adapted to the deduplication setting. It formalizes the two standard fixed-length and variable-length deduplication schemes, and it introduces a novel multi-chunk deduplication scheme. It then provides an analysis of these three deduplication variants, emphasizing the importance of boundary synchronization between source blocks and deduplication chunks. In particular, under fairly mild assumptions, the proposed multi-chunk deduplication scheme is shown to be order optimal.Comment: 27 page

A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice

Using infinite projected entangled pair states, we study the ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice directly in the thermodynamic limit. We find an unexpected partially nematic partially magnetic phase in between the antiferroquadrupolar and ferromagnetic regions. Furthermore, we describe all observed phases and discuss the nature of the phase transitions involved.Comment: 27 pages, 15 figures; v3: adjusted sections 1 and 3, and added a paragraph to section 5.2.

Emergent Haldane phase in the S=1S=1 bilinear-biquadratic Heisenberg model on the square lattice

Infinite projected entangled pair states simulations of the $S=1$ bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and 3-sublattice 120$^\circ$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled $S=1$ chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.Comment: 5 pages, 4 figures, plus supplemental materia

A ground state study of the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice using tensor networks

Making use of infinite projected entangled pair states, we investigate the ground state phase diagram of the nearest-neighbor spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. In agreement with previous studies, we find the ferromagnetic, 120 degree magnetically ordered, ferroquadrupolar and antiferroquadrupolar phases, and confirm that all corresponding phase transitions are first order. Moreover, we provide an accurate estimate of the location of the ferroquadrupolar to 120 degree magnetically ordered phase transition, thereby fully establishing the phase diagram. Also, we do not encounter any signs of the existence of a quantum paramagnetic phase. In particular, contrary to the equivalent square lattice model, we demonstrate that on the triangular lattice the one-dimensional Haldane phase does not reach all the way up to the two-dimensional limit. Finally, we investigate the possibility of an intermediate partially-magnetic partially-quadrupolar phase close to $\theta = \pi/2$, and we show that, also contrary to the square lattice case, this phase is not present on the triangular lattice.Comment: 14 pages, 15 figures; v2: shortened section II.B and added a paragraph to section IV.

The Approximate Capacity of the Gaussian N-Relay Diamond Network

We consider the Gaussian "diamond" or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N^2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log^4(N)) of capacity with pre-constant in the order notation independent of the channel gains.Comment: 23 pages, to appear in IEEE Transactions on Information Theor

Evaluating the Evans function: Order reduction in numerical methods

We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss--Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schr\"odinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss--Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.Comment: 21 pages, 3 figures; removed superfluous material (+/- 1 page), added paragraph to conclusion and two reference