Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.07615

On Algorithmic Cache Optimization

Neil Bhavikatti
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study matrix-matrix multiplication of two matrices, $A$ and $B$, each of size $n \times n$. This operation results in a matrix $C$ of size $n\times n$. Our goal is to produce $C$ as efficiently as possible given a cache: a 1-D limited set of data values that we can work with to perform elementary operations (additions, multiplications, etc.). That is, we attempt to reuse the maximum amount of data from $A$, $B$ and $C$ during our computation (or equivalently, utilize data in the fast-access cache as often as possible). Firstly, we introduce the matrix-matrix multiplication algorithm. Secondly, we present a standard two-memory model to simulate the architecture of a computer, and we explain the LRU (Least Recently Used) Cache policy (which is standard in most computers). Thirdly, we introduce a basic model Cache Simulator, which possesses an $\mathcal{O}(M)$ time complexity (meaning we are limited to small $M$ values). Then we discuss and model the LFU (Least Frequently Used) Cache policy and the explicit control cache policy. Finally, we introduce the main result of this paper, the $\mathcal{O}(1)$ Cache Simulator, and use it to compare, experimentally, the savings of time, energy, and communication incurred from the ideal cache-efficient algorithm for matrix-matrix multiplication. The Cache Simulator simulates the amount of data movement that occurs between the main memory and the cache of the computer. One of the findings of this project is that, in some cases, there is a significant discrepancy in communication values between an LRU cache algorithm and explicit cache control. We propose to alleviate this problem by ``tricking'' the LRU cache algorithm by updating the timestamp of the data we want to keep in cache (namely entries of matrix $C$). This enables us to have the benefits of an explicit cache policy while being constrained by the LRU paradigm (realistic policy on a CPU).

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Neil Bhavikatti. 2023-11-12. On Algorithmic Cache Optimization. https://arxiv.org/abs/2311.07615

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗