15,362 research outputs found
Geometry Helps to Compare Persistence Diagrams
Exploiting geometric structure to improve the asymptotic complexity of
discrete assignment problems is a well-studied subject. In contrast, the
practical advantages of using geometry for such problems have not been
explored. We implement geometric variants of the Hopcroft--Karp algorithm for
bottleneck matching (based on previous work by Efrat el al.) and of the auction
algorithm by Bertsekas for Wasserstein distance computation. Both
implementations use k-d trees to replace a linear scan with a geometric
proximity query. Our interest in this problem stems from the desire to compute
distances between persistence diagrams, a problem that comes up frequently in
topological data analysis. We show that our geometric matching algorithms lead
to a substantial performance gain, both in running time and in memory
consumption, over their purely combinatorial counterparts. Moreover, our
implementation significantly outperforms the only other implementation
available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX
201
Computing the interleaving distance is NP-hard
We show that computing the interleaving distance between two multi-graded
persistence modules is NP-hard. More precisely, we show that deciding whether
two modules are -interleaved is NP-complete, already for bigraded, interval
decomposable modules. Our proof is based on previous work showing that a
constrained matrix invertibility problem can be reduced to the interleaving
distance computation of a special type of persistence modules. We show that
this matrix invertibility problem is NP-complete. We also give a slight
improvement of the above reduction, showing that also the approximation of the
interleaving distance is NP-hard for any approximation factor smaller than .
Additionally, we obtain corresponding hardness results for the case that the
modules are indecomposable, and in the setting of one-sided stability.
Furthermore, we show that checking for injections (resp. surjections) between
persistence modules is NP-hard. In conjunction with earlier results from
computational algebra this gives a complete characterization of the
computational complexity of one-sided stability. Lastly, we show that it is in
general NP-hard to approximate distances induced by noise systems within a
factor of 2.Comment: 25 pages. Several expository improvements and minor corrections. Also
added a section on noise system
Multi-agent quality of experience control
In the framework of the Future Internet, the aim of the Quality of Experience (QoE) Control functionalities is to track the personalized desired QoE level of the applications. The paper proposes to perform such a task by dynamically selecting the most appropriate Classes of Service (among the ones supported by the network), this selection being driven by a novel heuristic Multi-Agent Reinforcement Learning (MARL) algorithm. The paper shows that such an approach offers the opportunity to cope with some practical implementation problems: in particular, it allows to face the so-called “curse of dimensionality” of MARL algorithms, thus achieving satisfactory performance results even in the presence of several hundreds of Agents
A multistage linear array assignment problem
The implementation of certain algorithms on parallel processing computing architectures can involve partitioning contiguous elements into a fixed number of groups, each of which is to be handled by a single processor. It is desired to find an assignment of elements to processors that minimizes the sum of the maximum workloads experienced at each stage. This problem can be viewed as a multi-objective network optimization problem. Polynomially-bounded algorithms are developed for the case of two stages, whereas the associated decision problem (for an arbitrary number of stages) is shown to be NP-complete. Heuristic procedures are therefore proposed and analyzed for the general problem. Computational experience with one of the exact problems, incorporating certain pruning rules, is presented with one of the exact problems. Empirical results also demonstrate that one of the heuristic procedures is especially effective in practice
Performance tradeoffs in static and dynamic load balancing strategies
The problem of uniformly distributing the load of a parallel program over a multiprocessor system was considered. A program was analyzed whose structure permits the computation of the optimal static solution. Then four strategies for load balancing were described and their performance compared. The strategies are: (1) the optimal static assignment algorithm which is guaranteed to yield the best static solution, (2) the static binary dissection method which is very fast but sub-optimal, (3) the greedy algorithm, a static fully polynomial time approximation scheme, which estimates the optimal solution to arbitrary accuracy, and (4) the predictive dynamic load balancing heuristic which uses information on the precedence relationships within the program and outperforms any of the static methods. It is also shown that the overhead incurred by the dynamic heuristic is reduced considerably if it is started off with a static assignment provided by either of the other three strategies
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