The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the $\ell_1$-norm, showing that a makespan of at most $5r$ can always be achieved, where $r$ is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most $5r$ can be computed in optimal time $O(n)$. Both bounds, the time and the makespan are optimal. This implies a new upper bound of $5\sqrt{2}r \approx 7.07r$ on the makespan in the $\ell_2$-norm, improving the best known bound so far $(5+2\sqrt{2}+\sqrt{5})r \approx 10.06r$.
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Nanopore sequencing is a promising technology for DNA sequencing. In this paper, we investigate a specific model of the nanopore sequencer, which takes a $q$-ary sequence of length $n$ as input and outputs a vector of length $n+\ell-1$ referred to as an $\ell$-read vector where the $i$-th entry is a multi-set composed of the $\ell$ elements located between the $(i-\ell+1)$-th and $i$-th positions of the input sequence. Considering the presence of substitution errors in the output vector, we study $\ell$-read codes under the Hamming metric. An $\ell$-read $(n,d)_q$-code is a set of $q$-ary sequences of length $n$ in which the Hamming distance between $\ell$-read vectors of any two distinct sequences is at least $d$. We first improve the result of Banerjee \emph{et al.}, who studied $\ell$-read $(n,d)_q$-codes with the constraint $\ell\geq 3$ and $d=3$. Then, we investigate the bounds and constructions of $2$-read codes with a minimum distance of $3$, $4$, and $5$, respectively. Our results indicate that when $d \in \{3,4\}$, the optimal redundancy of $2$-read $(n,d)_q$-codes is $o(\log_q n)$, while for $d=5$ it is $\log_q n+o(\log_q n)$. Additionally, we establish an equivalence between $2$-read $(n,3)_q$-codes and classical $q$-ary single-insertion reconstruction codes using two noisy reads. We improve the lower bound on the redundancy of classical $q$-ary single-insertion reconstruction codes as well as the upper bound on the redundancy of classical $q$-ary single-deletion reconstruction codes when using two noisy reads. Finally, we study $\ell$-read codes under the reconstruction model.
GNN-based approaches for learning general policies across planning domains are limited by the expressive power of $C_2$, namely; first-order logic with two variables and counting. This limitation can be overcomed by transitioning to $k$-GNNs, for $k=3$, wherein object embeddings are substituted with triplet embeddings. Yet, while $3$-GNNs have the expressive power of $C_3$, unlike $1$- and $2$-GNNs that are confined to $C_2$, they require quartic time for message exchange and cubic space for embeddings, rendering them impractical. In this work, we introduce a parameterized version of relational GNNs. When $t$ is infinity, R-GNN[$t$] approximates $3$-GNNs using only quadratic space for embeddings. For lower values of $t$, such as $t=1$ and $t=2$, R-GNN[$t$] achieves a weaker approximation by exchanging fewer messages, yet interestingly, often yield the $C_3$ features required in several planning domains. Furthermore, the new R-GNN[$t$] architecture is the original R-GNN architecture with a suitable transformation applied to the input states only. Experimental results illustrate the clear performance gains of R-GNN[$1$] and R-GNN[$2$] over plain R-GNNs, and also over edge transformers that also approximate $3$-GNNs.
Conventional Task and Motion Planning (TAMP) approaches rely on manually crafted interfaces connecting symbolic task planning with continuous motion generation. These domain-specific and labor-intensive modules are limited in addressing emerging tasks in real-world settings. Here, we present LLM^3, a novel Large Language Model (LLM)-based TAMP framework featuring a domain-independent interface. Specifically, we leverage the powerful reasoning and planning capabilities of pre-trained LLMs to propose symbolic action sequences and select continuous action parameters for motion planning. Crucially, LLM^3 incorporates motion planning feed- back through prompting, allowing the LLM to iteratively refine its proposals by reasoning about motion failure. Consequently, LLM^3 interfaces between task planning and motion planning, alleviating the intricate design process of handling domain- specific messages between them. Through a series of simulations in a box-packing domain, we quantitatively demonstrate the effectiveness of LLM^3 in solving TAMP problems and the efficiency in selecting action parameters. Ablation studies un- derscore the significant contribution of motion failure reasoning to the success of LLM^3. Furthermore, we conduct qualitative experiments on a physical manipulator, demonstrating the practical applicability of our approach in real-world settings.
Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions $\mathbb{R}^3$, position and orientations $\mathbb{R}^3 {\times} S^2$, and the group $SE(3)$ itself. Among these, $\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to represent directional information, which $\mathbb{R}^3$ methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full $SE(3)$ group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.
Foundation models, such as Large language Models (LLMs), have attracted significant amount of interest due to their large number of applications. Existing works show that appropriate prompt design, such as Chain-of-Thoughts, can unlock LLM's powerful capacity in diverse areas. However, when handling tasks involving repetitive sub-tasks and/or deceptive contents, such as arithmetic calculation and article-level fake news detection, existing prompting strategies either suffers from insufficient expressive power or intermediate errors triggered by hallucination. To make LLM more discerning to such intermediate errors, we propose to guide LLM with a Divide-and-Conquer program that simultaneously ensures superior expressive power and disentangles task decomposition, sub-task resolution, and resolution assembly process. Theoretic analysis reveals that our strategy can guide LLM to extend the expressive power of fixed-depth Transformer. Experiments indicate that our proposed method can achieve better performance than typical prompting strategies in tasks bothered by intermediate errors and deceptive contents, such as large integer multiplication, hallucination detection and misinformation detection.
We propose overcoming the memory capacity limitation of GPUs with high-capacity Storage-Class Memory (SCM) and DRAM cache. By significantly increasing the memory capacity with SCM, the GPU can capture a larger fraction of the memory footprint than HBM for workloads that oversubscribe memory, achieving high speedups. However, the DRAM cache needs to be carefully designed to address the latency and BW limitations of the SCM while minimizing cost overhead and considering GPU's characteristics. Because the massive number of GPU threads can thrash the DRAM cache, we first propose an SCM-aware DRAM cache bypass policy for GPUs that considers the multi-dimensional characteristics of memory accesses by GPUs with SCM to bypass DRAM for data with low performance utility. In addition, to reduce DRAM cache probes and increase effective DRAM BW with minimal cost, we propose a Configurable Tag Cache (CTC) that repurposes part of the L2 cache to cache DRAM cacheline tags. The L2 capacity used for the CTC can be adjusted by users for adaptability. Furthermore, to minimize DRAM cache probe traffic from CTC misses, our Aggregated Metadata-In-Last-column (AMIL) DRAM cache organization co-locates all DRAM cacheline tags in a single column within a row. The AMIL also retains the full ECC protection, unlike prior DRAM cache's Tag-And-Data (TAD) organization. Additionally, we propose SCM throttling to curtail power and exploiting SCM's SLC/MLC modes to adapt to workload's memory footprint. While our techniques can be used for different DRAM and SCM devices, we focus on a Heterogeneous Memory Stack (HMS) organization that stacks SCM dies on top of DRAM dies for high performance. Compared to HBM, HMS improves performance by up to 12.5x (2.9x overall) and reduces energy by up to 89.3% (48.1% overall). Compared to prior works, we reduce DRAM cache probe and SCM write traffic by 91-93% and 57-75%, respectively.
The rise of cloud computing has spurred a trend of transferring data storage and computational tasks to the cloud. To protect confidential information such as customer data and business details, it is essential to encrypt this sensitive data before cloud storage. Implementing encryption can prevent unauthorized access, data breaches, and the resultant financial loss, reputation damage, and legal issues. Moreover, to facilitate the execution of data mining algorithms on the cloud-stored data, the encryption needs to be compatible with domain computation. The $k$-nearest neighbor ($k$-NN) computation for a specific query vector is widely used in fields like location-based services. Sanyashi et al. (ICISS 2023) proposed an encryption scheme to facilitate privacy-preserving $k$-NN computation on the cloud by utilizing Asymmetric Scalar-Product-Preserving Encryption (ASPE). In this work, we identify a significant vulnerability in the aforementioned encryption scheme of Sanyashi et al. Specifically, we give an efficient algorithm and also empirically demonstrate that their encryption scheme is vulnerable to the ciphertext-only attack (COA).
We consider a distributed setup for reinforcement learning, where each agent has a copy of the same Markov Decision Process but transitions are sampled from the corresponding Markov chain independently by each agent. We show that in this setting, we can achieve a linear speedup for TD($\lambda$), a family of popular methods for policy evaluation, in the sense that $N$ agents can evaluate a policy $N$ times faster provided the target accuracy is small enough. Notably, this speedup is achieved by ``one shot averaging,'' a procedure where the agents run TD($\lambda$) with Markov sampling independently and only average their results after the final step. This significantly reduces the amount of communication required to achieve a linear speedup relative to previous work.
Conformer-based attention models have become the de facto backbone model for Automatic Speech Recognition tasks. A blank symbol is usually introduced to align the input and output sequences for CTC or RNN-T models. Unfortunately, the long input length overloads computational budget and memory consumption quadratically by attention mechanism. In this work, we propose a "Skip-and-Recover" Conformer architecture, named Skipformer, to squeeze sequence input length dynamically and inhomogeneously. Skipformer uses an intermediate CTC output as criteria to split frames into three groups: crucial, skipping and ignoring. The crucial group feeds into next conformer blocks and its output joint with skipping group by original temporal order as the final encoder output. Experiments show that our model reduces the input sequence length by 31 times on Aishell-1 and 22 times on Librispeech corpus. Meanwhile, the model can achieve better recognition accuracy and faster inference speed than recent baseline models. Our code is open-sourced and available online.
This paper introduces AL$\ell_0$CORE, a new form of probabilistic non-negative tensor decomposition. AL$\ell_0$CORE is a Tucker decomposition where the number of non-zero elements (i.e., the $\ell_0$-norm) of the core tensor is constrained to a preset value $Q$ much smaller than the size of the core. While the user dictates the total budget $Q$, the locations and values of the non-zero elements are latent variables and allocated across the core tensor during inference. AL$\ell_0$CORE -- i.e., $allo$cated $\ell_0$-$co$nstrained $core$-- thus enjoys both the computational tractability of CP decomposition and the qualitatively appealing latent structure of Tucker. In a suite of real-data experiments, we demonstrate that AL$\ell_0$CORE typically requires only tiny fractions (e.g.,~1%) of the full core to achieve the same results as full Tucker decomposition at only a correspondingly tiny fraction of the cost.
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