Réduction (complexité) | EPFL Graph Search

En calculabilité et en théorie de la complexité, une réduction est un algorithme transformant une instance d'un problème algorithmique en une ou plusieurs instances d'un autre problème. S'il existe une telle réduction d'un problème A à un problème B, on dit que le problème A se réduit au problème B. Dans ce cas, le problème B est plus difficile que le problème A, puisque l'on peut résoudre le problème A en appliquant la réduction puis un algorithme pour le problème B. On écrit alors A ≤ B. Il y a deux utilisations des réductions.

Montrer qu'un problème est intrinsèquement difficile. Par exemple, on utilise une réduction pour montrer que des problèmes sont indécidables : on montre alors que le problème est tellement algorithmiquement difficile, qu'il n'y a pas d'algorithmique qui le décide. Les réductions polynomiales sont utilisées pour démontrer que des problèmes sont NP-difficiles. Plus généralement, on démontre avec des réductions qu'un problème est parmi les plus difficiles

Recompression Of Hadamard Products Of Tensors In Tucker Format

Daniel Kressner, Lana Perisa

The Hadamard product features prominently in tensor-based algorithms in scientific computing and data analysis. Due to its tendency to significantly increase ranks, the Hadamard product can represent a major computational obstacle in algorithms based on low-rank tensor representations. It is therefore of interest to develop recompression techniques that mitigate the effects of this rank increase. In this work, we investigate such techniques for the case of the Tucker format, which is well suited for tensors of low order and small to moderate multilinear ranks. Fast algorithms are attained by combining iterative methods, such as the Lanczos method and randomized algorithms, with fast matrix-vector products that exploit the structure of Hadamard products. The resulting complexity reduction is particularly relevant for tensors featuring large mode sizes I and small to moderate multilinear ranks R. To implement our algorithms, we have created a new Julia library for tensors in Tucker format.

Siam Publications2017

Filtering Random Graph Processes Over Random Time-Varying Graphs

Andreas Loukas

Graph filters play a key role in processing the graph spectra of signals supported on the vertices of a graph. However, despite their widespread use, graph filters have been analyzed only in the deterministic setting, ignoring the impact of stochasticity in both the graph topology and the signal itself. To bridge this gap, we examine the statistical behavior of the two key filter types, finite impulse response and autoregressive moving average graph filters, when operating on random time-varying graph signals (or random graph processes) over random time-varying graphs. Our analysis shows that 1) in expectation, the filters behave as the same deterministic filters operating on a deterministic graph, being the expected graph, having as input signal a deterministic signal, being the expected signal, and 2) there are meaningful upper bounds for the variance of the filter output. We conclude this paper by proposing two novel ways of exploiting randomness to improve (joint graph-time) noise cancellation, as well as to reduce the computational complexity of graph filtering. As demonstrated by numerical results, these methods outperform the disjoint average and denoise algorithm and yield a (up to) four times complexity reduction, with a very little difference from the optimal solution.

Institute of Electrical and Electronics Engineers2017

Reducing Time Complexity in RFID Systems

Gildas Avoine, Etienne Dysli, Philippe Oechslin

Radio frequency identification systems based on low-cost computing devices is the new plaything that every company would like to adopt. Its goal can be either to improve the productivity or to strengthen the security. Specific identification protocols based on symmetric challenge-response have been developed in order to assure the privacy of the device bearers. Although these protocols fit the devices' constraints, they always suffer from a large time complexity. Existing protocols require O(n) cryptographic operations to identify one device among n. Molnar and Wagner suggested a method to reduce this complexity to O(log n). We show that their technique could degrade the privacy if the attacker has the possibility to tamper with at least one device. Because low-cost devices are not tamper-resistant, such an attack could be feasible. We give a detailed analysis of their protocol and evaluate the threat. Next, we extend an approach based on time-memory trade-offs whose goal is to improve Ohkubo, Suzuki, and Kinoshita's protocol. We show that in practice this approach reaches the same performances as Molnar and Wagner's method, without degrading privacy. Radio frequency identification systems based on low-cost computing devices is the new plaything that every company would like to adopt. Its goal can be either to improve the productivity or to strengthen the security. Specific identification protocols based on symmetric challenge-response have been developed in order to assure the privacy of the device bearers. Although these protocols fit the devices' constraints, they always suffer from a large time complexity. Existing protocols require O(n) cryptographic operations to identify one device among n. Molnar and Wagner suggested a method to reduce this complexity to O(log n). We show that their technique could degrade the privacy if the attacker has the possibility to tamper with at least one device. Because low-cost devices are not tamper-resistant, such an attack could be feasible. We give a detailed analysis of their protocol and evaluate the threat. Next, we extend an approach based on time-memory trade-offs whose goal is to improve Ohkubo, Suzuki, and Kinoshita's protocol. We show that in practice this approach reaches the same performances as Molnar and Wagner's method, without degrading privacy.

2006

Reducing Time Complexity in RFID Systems

Gildas Avoine, Etienne Dysli, Philippe Oechslin

2006