Tree Compression Research Articles

Entropy Bounds for Grammar-Based Tree Compressors

The definition of $\text {k}^{th}$ -order empirical entropy of strings is extended to node-labelled binary trees: A notion of $\text {k}^{th}$ -order empirical entropy for node-labelled binary trees is proposed that is able to capture regularities in both labels and structure of a tree. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the $\text {k}^{th}$ -order empirical entropy plus some lower order terms. This result is then extended from node-labelled binary trees to node-labelled unranked trees. This generalizes recent results for grammar-based string compression to grammar-based tree compression. Additionally, experimental results with real XML document trees are presented, in which the proposed notion of $\text {k}^{th}$ -order empirical tree entropy is computed and compared to the performance of grammar-based tree compressors for those XML document trees.

Top Tree Compression of Tries

We present a compressed representation of tries based on top tree compression [ICALP 2013] that works on a standard, comparison-based, pointer machine model of computation and supports efficient prefix search queries. Namely, we show how to preprocess a set of strings of total length n over an alphabet of size \(\sigma\) into a compressed data structure of worst-case optimal size \(O(n/\log _\sigma n)\) that given a pattern string P of length m determines if P is a prefix of one of the strings in time \(O(\min (m\log \sigma ,m + \log n))\). We show that this query time is in fact optimal regardless of the size of the data structure. Existing solutions either use \(\Omega (n)\) space or rely on word RAM techniques, such as tabulation, hashing, address arithmetic, or word-level parallelism, and hence do not work on a pointer machine. Our result is the first solution on a pointer machine that achieves worst-case o(n) space. Along the way, we develop several interesting data structures that work on a pointer machine and are of independent interest. These include an optimal data structures for random access to a grammar-compressed string and an optimal data structure for a variant of the level ancestor problem.

Ascending cholangitis in patients with malignant obstructive jaundice associated with biliary tree compression

Ascending cholangitis in patients with malignant obstructive jaundice associated with biliary tree compression

Balancing Straight-line Programs

We show that a context-free grammar of size that produces a single string of length (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for of size , whose unique derivation tree has depth . This solves an open problem in the area of grammar-based compression, improves many results in this area, and greatly simplifies many existing constructions. Similar results are shown for two formalisms for grammar-based tree compression: top dags and forest straight-line programs. These balancing results can be all deduced from a single meta-theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to with the cost of a constant multiplicative size increase. Here, refers to the size of the unfolding (or unravelling) of the circuit. In particular, this results applies to standard arithmetic circuits over (noncommutative) semirings.

The Effects of Compression on the Detection of Atrial Fibrillation in ECG Signals

Although Atrial Fibrillation (AF) is the most frequent cause of cardioembolic stroke, the arrhythmia remains underdiagnosed, as it is often asymptomatic or intermittent. Automated detection of AF in ECG signals is important for patients with implantable cardiac devices, pacemakers or Holter systems. Such resource-constrained systems often operate by transmitting signals to a central server where diagnostic decisions are made. In this context, ECG signal compression is being increasingly investigated and employed to increase battery life, and hence the storage and transmission efficiency of these devices. At the same time, the diagnostic accuracy of AF detection must be preserved. This paper investigates the effects of ECG signal compression on an entropy-based AF detection algorithm that monitors R-R interval regularity. The compression and AF detection algorithms were applied to signals from the MIT-BIH AF database. The accuracy of AF detection on reconstructed signals is evaluated under varying degrees of compression using the state-of-the-art Set Partitioning In Hierarchical Trees (SPIHT) compression algorithm. Results demonstrate that compression ratios (CR) of up to 90 can be obtained while maintaining a detection accuracy, expressed in terms of the area under the receiver operating characteristic curve, of at least 0.9. This highlights the potential for significant energy savings on devices that transmit/store ECG signals for AF detection applications, while preserving the diagnostic integrity of the signals, and hence the detection performance.

Regular matching and inclusion on compressed tree patterns with constrained context variables

We study the complexity of regular matching and inclusion for compressed tree patterns with context variables subject to regular constraints. Context variables with regular constraints permit to properly generalize on unranked tree patterns with hedge variables. Regular inclusion on unranked tree patterns is relevant to certain query answering on Xml streams with references. We show that regular matching and inclusion with regular constraints can be reduced in polynomial time to the corresponding problem without regular constraints.

Framework-based Arithmetic Datapath Generation to Explore Parallel Binary Multipliers

Arithmetic modules usually have a significant impact on performance, circuit area, energy, and power in digital circuits of DSP (Digital Signal Processing). Exploring implementation trade-offs in these circuits is of utmost importance in low-power and low-cost devices such as sensors in IoT devices which often have stringent requirements. Multipliers are of particular concern due to their ubiquitous use in DSP algorithms and their inherent implementation complexity. This work proposes a framework to efficiently generalize and explore different compositions of arithmetic operators with an emphasis on parallel binary multipliers, guiding the designer through the micro-architecture development. Several partial product encoders were combined with multiple compression trees to generate multipliers that were synthesized in a commercial 65 nm to obtain area, power, and timing results.

LayerChain: A Hierarchical Edge-Cloud Blockchain for Large-Scale Low-Delay Industrial Internet of Things Applications

The combination of pervasive edge computing and blockchain technologies opens up significant possibilities for industrial Internet of Things (IIoT) applications, but there are several critical limitations regarding efficient storage and rapid response for large-scale low-delay IIoT scenarios. To address these limitations, in this article we propose a hierarchical edge-cloud blockchain called LayerChain . Specifically, to promote scalability, we design a layered structure to hierarchically store the blockchain data in multiple distributed clouds and edge nodes. Next, we propose a node classification method to accommodate differences between the edge nodes when deploying the blockchain. Moreover, to mitigate lengthy delays during block propagation, we propose a tree-based clustering algorithm where blocks are propagated through different clusters with a compressed tree depth. Simulation results show that our LayerChain efficiently reduces the system's resource requirements and block propagation time, making it well-suited for large-scale low-delay IIoT applications.

Smart Digital Signatures (SDS): A post-quantum digital signature scheme for distributed ledgers

The upcoming quantum era is believed to be an end for the elliptic curve digital signature algorithm (ECDSA) and other number-theoretic digital signature schemes. Hence, the technologies which incorporate ECDSA would be at risk once quantum computers are available at large scale. Distributed ledger technology is one of the potential victims of powerful quantum computers. Fortunately, post-quantum digital signature schemes are already available. Hash-based digital signatures (HBS) schemes due to their simplicity and efficiency have gained tremendous attention from the research community. However, large size of key and signature are the major drawbacks of HBS schemes. This paper proposes a compact and efficient HBS scheme “Smart Digital Signatures” (SDS), which is closer to an existing popular HBS scheme, XMSS. SDS incorporates a novel one-time signature (OTS) scheme in XMSS, namely SDS-OTS. Furthermore, SDS uses a slightly modified version of the key compression tree as compared to XMSS. We have compared SDS with XMSS-WOTS and XMSS-WOTS+. The results reveal a significant reduction in hash tree construction time compared to XMSS, and key and signature sizes compared to WOTS and WOTS+. Finally, we have also proposed a model for incorporating SDS into a distributed ledger, with the help of High-Level Petri-nets.

Grammar-Based Compression of Unranked Trees

We introduce forest straight-line programs (FSLPs) as a compressed representation of unranked ordered node-labelled trees. FSLPs are based on the operations of forest algebra and generalize tree straight-line programs. We compare the succinctness of FSLPs with two other compression schemes for unranked trees: top dags and tree straight-line programs of first-child/next sibling encodings. Efficient translations between these formalisms are provided. Finally, we show that equality of unranked trees in the setting where certain symbols are associative or commutative can be tested in polynomial time. This generalizes previous results for testing isomorphism of compressed unordered ranked trees.

Hybridization preceded radiation in diploid wheats

Evolutionary relationships among the Aegilops-Triticum relatives of cultivated wheats have been difficult to resolve owing to incomplete lineage sorting and reticulate evolution. Recent studies have suggested that the wheat D-genome lineage (progenitor of Ae. tauschii) originated through homoploid hybridization between the A-genome lineage (progenitor of Triticum s.str.) and the B-genome lineage (progenitor of Ae. speltoides). This scenario of reticulation has been debated, calling for adequate phylogenetic analyses based on comprehensive sampling. To reconstruct the evolution of Aegilops-Triticum diploids, we here combined high-throughput sequencing of 38 nuclear low-copy loci of multiple accessions of all 13 species with inferences of the species phylogeny using the full-parameterized MCMC_SEQ method. Phylogenies recovered a monophyletic Aegilops-Triticum lineage that began diversifying ~6.6 Ma ago and gave rise to four sublineages, i.e. the A- (2 species), B- (1 species), D- (9 species) and T- (Ae. mutica) genome lineage. Full-parameterized phylogenies as well as patterns of tree dilation and tree compression supported a hybrid origin of the D-genome lineage from A and B ~3.0–4.0 Ma ago, and did not indicate additional hybridization events. Conflicting ABBA-BABA tests suggestive of further reticulation were shown here to result from ancestral population structure rather than hybridization. This comprehensive and dated phylogeny of wheat relatives indicates that the origin of the hybrid D-genome was followed by intense diversification into the majority of extant diploid as well as allopolyploid wild wheats.

Compressed range minimum queries

Given a string S of n integers in [0,σ), a range minimum query RMQ(i,j) asks for the index of the smallest integer in S[i…j]. It is well known that the problem can be solved with a succinct data structure of size 2n+o(n) and constant query-time. In this paper we show how to preprocess S into a compressed representation that allows fast range minimum queries. This allows for sublinear size data structures with logarithmic query time. The most natural approach is to use string compression and construct a data structure for answering range minimum queries directly on the compressed string. We investigate this approach in the context of grammar compression. We then consider an alternative approach. Instead of compressing S using string compression, we compress the Cartesian tree of S using tree compression. We show that this approach can be exponentially better than the former, is never worse by more than an O(σ) factor (i.e. for constant alphabets it is never asymptotically worse), and can in fact be worse by an Ω(σ) factor.

JViz.RNA 4.0-Visualizing pseudoknots and RNA editing employing compressed tree graphs.

Previously, we have introduced an improved version of jViz.RNA which enabled faster and more stable RNA visualization by employing compressed tree graphs. However, the new RNA representation and visualization method required a sophisticated mechanism of pseudoknot visualization. In this work, we present our novel pseudoknot classification and implementation of pseudoknot visualization in the context of the new RNA graph model. We then compare our approach with other RNA visualization software, and demonstrate jViz.RNA 4.0’s benefits compared to other software. Additionally, we introduce interactive editing functionality into jViz.RNA and demonstrate its benefits in exploring and building RNA structures. The results presented highlight the new high degree of utility jViz.RNA 4.0 now offers. Users are now able to visualize pseudoknotted RNA, manipulate the resulting automatic layouts to suit their individual needs, and change both positioning and connectivity of the RNA molecules examined. Care was taken to limit overlap between structural elements, particularly in the case of pseudoknots to ensure an intuitive and informative layout of the final RNA structure.Availability: The software is freely available at: https://jviz.cs.sfu.ca/.

Performance Analysis of Compression Techniques for Chronic Wound Image Transmission Under Smartphone-Enabled Tele-Wound Network

The healing status of chronic wounds is important for monitoring the condition of the wounds. This article designs and discusses the implementation of smartphone-enabled compression technique under a tele-wound network (TWN) system. Nowadays, there is a huge demand for memory and bandwidth savings for clinical data processing. Wound images are captured using a smartphone through a metadata application page. Then, they are compressed and sent to the telemedical hub with a set partitioning in hierarchical tree (SPIHT) compression algorithm. The transmitted image can then be reduced, followed by an improvement in the segmentation accuracy and sensitivity. Better wound healing treatment depends on segmentation and classification accuracy. The proposed framework is evaluated in terms of rates (bits per pixel), compression ratio, peak signal to noise ratio, transmission time, mean square error and diagnostic quality under telemedicine framework. A SPIHT compression technique assisted YDbDr-Fuzzy c-means clustering considerably reduces the execution time (105s), is simple to implement, saves memory (18 KB), improves segmentation accuracy (98.39%), and yields better results than the same without using SPIHT. The results favor the possibility of developing a practical smartphone-enabled telemedicine system and show the potential for being implemented in the field of clinical evaluation and the management of chronic wounds in the future.

Design and Analysis of Approximate Multipliers with a Tree Compressor

Design and Analysis of Approximate Multipliers with a Tree Compressor

Area Optimized Synthesis of Compressor Trees on Xilinx FPGAs Using Generalized Parallel Counters

Early compressor trees based on carry-save adders and single-column parallel counters show good performance in ASIC design, but do not adapt well to modern field-programmable gate arrays (FPGAs). Recently, compressor trees built from generalized parallel counters (GPCs) were synthesized on FPGAs to address this issue. Despite the improved timing performance of GPC-based compressor trees, area reduction is not as significant as delay, and can be further optimized. In this paper, we propose improved GPC mappings as well as new approaches for GPC cascading and binding for Xilinx FPGAs. With these improvements, we develop an integer linear programming (ILP) method for FPGA synthesis of GPC-based compressor trees that supports cascading and binding between GPCs. Experimental results show that the single-cycle compressor trees produced by the proposed ILP can reduce the average area by 42.40% compared with those generated by existing heuristic method, but are 13.16% slower; the pipelined compressor trees produced by the proposed ILP can reduce the average area by 33.43% at the cost of an average 14.35% decrease in maximum clock frequency compared with those obtained by existing heuristic method.

Improved Synthesis of Compressor Trees in High-Level Synthesis for Modern FPGAs

In this paper, an approach to synthesize compressor trees in high-level synthesis is proposed. We target the modern field-programmable gate arrays, which integrate carry chains and support fast ternary adders. Two main improvements are achieved in our approach: 1) based on the proposed modified bitmask analysis, we perform bit-level numerical optimizations to shrink the scale of generated compressor trees and achieve a better area-delay performance; 2) by estimating the arrival time of each multi-input addition operand, we combine the use of generalized parallel counters and ternary adders in compressor trees to further reduce the area while maintaining a similar delay performance. A series of experiments shows that our approach reduces the area significantly while maintaining similar delay performance, as compared to the existing approaches.

Entropy and Optimal Compression of Some General Plane Trees

We continue developing the information theory of structured data. In this article, we study models generating d -ary trees ( d ≥ 2) and trees with unrestricted degree. We first compute the entropy which gives us the fundamental lower bound on compression of such trees. Then we present efficient compression algorithms based on arithmetic encoding that achieve the entropy within a constant number of bits. A naïve implementation of these algorithms has a prohibitive time complexity of O ( n d ) elementary arithmetic operations (each corresponding to a number f ( n , d ) of bit operations), but our efficient algorithms run in O ( n 2 ) of these operations, where n is the number of nodes. It turns out that extending source coding (i.e., compression) from sequences to advanced data structures such as degree-unconstrained trees is mathematically quite challenging and leads to recurrences that find ample applications in the information theory of general structures (e.g., to analyze the information content of degree-unconstrained non-plane trees).

Advanced Compressor Tree Synthesis for FPGAs

This work presents novel methods for the optimization of compressor trees for FPGAs as required in many arithmetic computations. As demonstrated in recent work, important key elements for the design of efficient but fast compressor trees are target-optimized 4:2 compressors as well as generalized parallel counters (GPCs). However, the optimization of a compressor tree for minimal resources using both compressors and GPCs has not been addressed so far. As this combined optimization is a non-trivial task, three methods are proposed to find best solutions for a given problem size: 1) a heuristic that obtains compressor trees with typically less resources and fewer stages than state-of-the-art heuristics, 2) an integer linear programming (ILP)-based methodology that finds optimal compressor trees using the fewest stages possible, 3) a combined approach that partially solves the problem heuristically to reduce the search space for the ILP-based method. In all methods, the cost for pipeline registers can be included. Synthesis experiments show that the proposed methods provide pipelined compressor trees with about 40 percent less LUTs compared to trees of 2-input adders at the cost of being about 12 ...20 percent slower.

An approach for fast compressed text matching and to avoid false matching using WBTC and wavelet tree

Text matching is a process of finding the frequency of occurrences of text pattern in a corpus. It's very costly to store, process, and retrieve a vast volume of text data. In this paper, we present a method to keep the massive text corpus in lesser memory space by using text compression and to retrieve the results by matching directly on this compressed corpus without decompression using compressed pattern matching (CPM). The proposed approach also helps to minimize the time taken to perform matching without compromising the false matching results. We used word-based tagged coding to perform text compression and Wavelet Trees for representing the compressed text in memory. The proposed Text Matching in Compressed text using Parallel Wavelet Tree (TMC_PWT) method is quite fast in comparison to other existing text matching algorithms that support CPM. In the context of CPM, the proposed method provides a good compression ratio and does not suffer from the problem of false matching.