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Title: Recorder 2.0: Efficient Parallel I/O Tracing and Analysis
Recorder is a multi-level I/O tracing tool that captures HDF5, MPI-I/O, and POSIX I/O calls. In this paper, we present a new version of Recorder that adds support for most metadata POSIX calls such as stat, link, and rename. We also introduce a compressed tracing format to reduce trace file size and run time overhead incurred from collecting the trace data. Moreover, we add a set of post-mortem and visualization routines to our new version of Recorder that manage the compressed trace data for users. Our experiments with four HPC applications show a file size reduction of over 2× and reduced post-processing time by 20% when using our new compressed trace file format.  more » « less
Award ID(s):
1763540
NSF-PAR ID:
10155948
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
The IEEE International Workshop on High-Performance Storage
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  1. Modern applications use storage systems in complex and often surprising ways. Tracing system calls is a common approach to understanding applications' behavior, allowing offline analysis and enabling replay in other environments. But current system-call tracing tools have drawbacks: (1) they often omit some information---such as raw data buffers---needed for full analysis; (2) they have high overheads; (3) they often use non-portable trace formats; and (4) they may not offer useful and scalable analysis and replay tools. We have developed Re-Animator, a powerful system-call tracing tool that focuses on storage-related calls and collects maximal information, capturing complete data buffers and writing all traces in the standard DataSeries format. We also created a prototype replayer that focuses on calls related to file-system state. We evaluated our system on long-running server applications such as key-value stores and databases. Our tracer has an average overhead of only 1.8-2.3×, but the overhead can be as low as 5% for I/O-bound applications. Our replayer verifies that its actions are correct, and faithfully reproduces the logical file system state generated by the original application. 
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  2. Obeid, I. (Ed.)
    The Neural Engineering Data Consortium (NEDC) is developing the Temple University Digital Pathology Corpus (TUDP), an open source database of high-resolution images from scanned pathology samples [1], as part of its National Science Foundation-funded Major Research Instrumentation grant titled “MRI: High Performance Digital Pathology Using Big Data and Machine Learning” [2]. The long-term goal of this project is to release one million images. We have currently scanned over 100,000 images and are in the process of annotating breast tissue data for our first official corpus release, v1.0.0. This release contains 3,505 annotated images of breast tissue including 74 patients with cancerous diagnoses (out of a total of 296 patients). In this poster, we will present an analysis of this corpus and discuss the challenges we have faced in efficiently producing high quality annotations of breast tissue. It is well known that state of the art algorithms in machine learning require vast amounts of data. Fields such as speech recognition [3], image recognition [4] and text processing [5] are able to deliver impressive performance with complex deep learning models because they have developed large corpora to support training of extremely high-dimensional models (e.g., billions of parameters). Other fields that do not have access to such data resources must rely on techniques in which existing models can be adapted to new datasets [6]. A preliminary version of this breast corpus release was tested in a pilot study using a baseline machine learning system, ResNet18 [7], that leverages several open-source Python tools. The pilot corpus was divided into three sets: train, development, and evaluation. Portions of these slides were manually annotated [1] using the nine labels in Table 1 [8] to identify five to ten examples of pathological features on each slide. Not every pathological feature is annotated, meaning excluded areas can include focuses particular to these labels that are not used for training. A summary of the number of patches within each label is given in Table 2. To maintain a balanced training set, 1,000 patches of each label were used to train the machine learning model. Throughout all sets, only annotated patches were involved in model development. The performance of this model in identifying all the patches in the evaluation set can be seen in the confusion matrix of classification accuracy in Table 3. The highest performing labels were background, 97% correct identification, and artifact, 76% correct identification. A correlation exists between labels with more than 6,000 development patches and accurate performance on the evaluation set. Additionally, these results indicated a need to further refine the annotation of invasive ductal carcinoma (“indc”), inflammation (“infl”), nonneoplastic features (“nneo”), normal (“norm”) and suspicious (“susp”). This pilot experiment motivated changes to the corpus that will be discussed in detail in this poster presentation. To increase the accuracy of the machine learning model, we modified how we addressed underperforming labels. One common source of error arose with how non-background labels were converted into patches. Large areas of background within other labels were isolated within a patch resulting in connective tissue misrepresenting a non-background label. In response, the annotation overlay margins were revised to exclude benign connective tissue in non-background labels. Corresponding patient reports and supporting immunohistochemical stains further guided annotation reviews. The microscopic diagnoses given by the primary pathologist in these reports detail the pathological findings within each tissue site, but not within each specific slide. The microscopic diagnoses informed revisions specifically targeting annotated regions classified as cancerous, ensuring that the labels “indc” and “dcis” were used only in situations where a micropathologist diagnosed it as such. Further differentiation of cancerous and precancerous labels, as well as the location of their focus on a slide, could be accomplished with supplemental immunohistochemically (IHC) stained slides. When distinguishing whether a focus is a nonneoplastic feature versus a cancerous growth, pathologists employ antigen targeting stains to the tissue in question to confirm the diagnosis. For example, a nonneoplastic feature of usual ductal hyperplasia will display diffuse staining for cytokeratin 5 (CK5) and no diffuse staining for estrogen receptor (ER), while a cancerous growth of ductal carcinoma in situ will have negative or focally positive staining for CK5 and diffuse staining for ER [9]. Many tissue samples contain cancerous and non-cancerous features with morphological overlaps that cause variability between annotators. The informative fields IHC slides provide could play an integral role in machine model pathology diagnostics. Following the revisions made on all the annotations, a second experiment was run using ResNet18. Compared to the pilot study, an increase of model prediction accuracy was seen for the labels indc, infl, nneo, norm, and null. This increase is correlated with an increase in annotated area and annotation accuracy. Model performance in identifying the suspicious label decreased by 25% due to the decrease of 57% in the total annotated area described by this label. A summary of the model performance is given in Table 4, which shows the new prediction accuracy and the absolute change in error rate compared to Table 3. The breast tissue subset we are developing includes 3,505 annotated breast pathology slides from 296 patients. The average size of a scanned SVS file is 363 MB. The annotations are stored in an XML format. A CSV version of the annotation file is also available which provides a flat, or simple, annotation that is easy for machine learning researchers to access and interface to their systems. Each patient is identified by an anonymized medical reference number. Within each patient’s directory, one or more sessions are identified, also anonymized to the first of the month in which the sample was taken. These sessions are broken into groupings of tissue taken on that date (in this case, breast tissue). A deidentified patient report stored as a flat text file is also available. Within these slides there are a total of 16,971 total annotated regions with an average of 4.84 annotations per slide. Among those annotations, 8,035 are non-cancerous (normal, background, null, and artifact,) 6,222 are carcinogenic signs (inflammation, nonneoplastic and suspicious,) and 2,714 are cancerous labels (ductal carcinoma in situ and invasive ductal carcinoma in situ.) The individual patients are split up into three sets: train, development, and evaluation. Of the 74 cancerous patients, 20 were allotted for both the development and evaluation sets, while the remain 34 were allotted for train. The remaining 222 patients were split up to preserve the overall distribution of labels within the corpus. This was done in hope of creating control sets for comparable studies. Overall, the development and evaluation sets each have 80 patients, while the training set has 136 patients. In a related component of this project, slides from the Fox Chase Cancer Center (FCCC) Biosample Repository (https://www.foxchase.org/research/facilities/genetic-research-facilities/biosample-repository -facility) are being digitized in addition to slides provided by Temple University Hospital. This data includes 18 different types of tissue including approximately 38.5% urinary tissue and 16.5% gynecological tissue. These slides and the metadata provided with them are already anonymized and include diagnoses in a spreadsheet with sample and patient ID. We plan to release over 13,000 unannotated slides from the FCCC Corpus simultaneously with v1.0.0 of TUDP. Details of this release will also be discussed in this poster. Few digitally annotated databases of pathology samples like TUDP exist due to the extensive data collection and processing required. The breast corpus subset should be released by November 2021. By December 2021 we should also release the unannotated FCCC data. We are currently annotating urinary tract data as well. We expect to release about 5,600 processed TUH slides in this subset. We have an additional 53,000 unprocessed TUH slides digitized. Corpora of this size will stimulate the development of a new generation of deep learning technology. In clinical settings where resources are limited, an assistive diagnoses model could support pathologists’ workload and even help prioritize suspected cancerous cases. ACKNOWLEDGMENTS This material is supported by the National Science Foundation under grants nos. CNS-1726188 and 1925494. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. REFERENCES [1] N. Shawki et al., “The Temple University Digital Pathology Corpus,” in Signal Processing in Medicine and Biology: Emerging Trends in Research and Applications, 1st ed., I. Obeid, I. Selesnick, and J. Picone, Eds. New York City, New York, USA: Springer, 2020, pp. 67 104. https://www.springer.com/gp/book/9783030368432. [2] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, “MRI: High Performance Digital Pathology Using Big Data and Machine Learning.” Major Research Instrumentation (MRI), Division of Computer and Network Systems, Award No. 1726188, January 1, 2018 – December 31, 2021. https://www. isip.piconepress.com/projects/nsf_dpath/. [3] A. Gulati et al., “Conformer: Convolution-augmented Transformer for Speech Recognition,” in Proceedings of the Annual Conference of the International Speech Communication Association (INTERSPEECH), 2020, pp. 5036-5040. https://doi.org/10.21437/interspeech.2020-3015. [4] C.-J. Wu et al., “Machine Learning at Facebook: Understanding Inference at the Edge,” in Proceedings of the IEEE International Symposium on High Performance Computer Architecture (HPCA), 2019, pp. 331–344. https://ieeexplore.ieee.org/document/8675201. [5] I. Caswell and B. Liang, “Recent Advances in Google Translate,” Google AI Blog: The latest from Google Research, 2020. [Online]. Available: https://ai.googleblog.com/2020/06/recent-advances-in-google-translate.html. [Accessed: 01-Aug-2021]. [6] V. Khalkhali, N. Shawki, V. Shah, M. Golmohammadi, I. Obeid, and J. Picone, “Low Latency Real-Time Seizure Detection Using Transfer Deep Learning,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2021, pp. 1 7. https://www.isip. piconepress.com/publications/conference_proceedings/2021/ieee_spmb/eeg_transfer_learning/. [7] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, “MRI: High Performance Digital Pathology Using Big Data and Machine Learning,” Philadelphia, Pennsylvania, USA, 2020. https://www.isip.piconepress.com/publications/reports/2020/nsf/mri_dpath/. [8] I. Hunt, S. Husain, J. Simons, I. Obeid, and J. Picone, “Recent Advances in the Temple University Digital Pathology Corpus,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2019, pp. 1–4. https://ieeexplore.ieee.org/document/9037859. [9] A. P. Martinez, C. Cohen, K. Z. Hanley, and X. (Bill) Li, “Estrogen Receptor and Cytokeratin 5 Are Reliable Markers to Separate Usual Ductal Hyperplasia From Atypical Ductal Hyperplasia and Low-Grade Ductal Carcinoma In Situ,” Arch. Pathol. Lab. Med., vol. 140, no. 7, pp. 686–689, Apr. 2016. https://doi.org/10.5858/arpa.2015-0238-OA. 
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  3. Obeid, Iyad ; Picone, Joseph ; Selesnick, Ivan (Ed.)
    The Neural Engineering Data Consortium (NEDC) is developing a large open source database of high-resolution digital pathology images known as the Temple University Digital Pathology Corpus (TUDP) [1]. Our long-term goal is to release one million images. We expect to release the first 100,000 image corpus by December 2020. The data is being acquired at the Department of Pathology at Temple University Hospital (TUH) using a Leica Biosystems Aperio AT2 scanner [2] and consists entirely of clinical pathology images. More information about the data and the project can be found in Shawki et al. [3]. We currently have a National Science Foundation (NSF) planning grant [4] to explore how best the community can leverage this resource. One goal of this poster presentation is to stimulate community-wide discussions about this project and determine how this valuable resource can best meet the needs of the public. The computing infrastructure required to support this database is extensive [5] and includes two HIPAA-secure computer networks, dual petabyte file servers, and Aperio’s eSlide Manager (eSM) software [6]. We currently have digitized over 50,000 slides from 2,846 patients and 2,942 clinical cases. There is an average of 12.4 slides per patient and 10.5 slides per case with one report per case. The data is organized by tissue type as shown below: Filenames: tudp/v1.0.0/svs/gastro/000001/00123456/2015_03_05/0s15_12345/0s15_12345_0a001_00123456_lvl0001_s000.svs tudp/v1.0.0/svs/gastro/000001/00123456/2015_03_05/0s15_12345/0s15_12345_00123456.docx Explanation: tudp: root directory of the corpus v1.0.0: version number of the release svs: the image data type gastro: the type of tissue 000001: six-digit sequence number used to control directory complexity 00123456: 8-digit patient MRN 2015_03_05: the date the specimen was captured 0s15_12345: the clinical case name 0s15_12345_0a001_00123456_lvl0001_s000.svs: the actual image filename consisting of a repeat of the case name, a site code (e.g., 0a001), the type and depth of the cut (e.g., lvl0001) and a token number (e.g., s000) 0s15_12345_00123456.docx: the filename for the corresponding case report We currently recognize fifteen tissue types in the first installment of the corpus. The raw image data is stored in Aperio’s “.svs” format, which is a multi-layered compressed JPEG format [3,7]. Pathology reports containing a summary of how a pathologist interpreted the slide are also provided in a flat text file format. A more complete summary of the demographics of this pilot corpus will be presented at the conference. Another goal of this poster presentation is to share our experiences with the larger community since many of these details have not been adequately documented in scientific publications. There are quite a few obstacles in collecting this data that have slowed down the process and need to be discussed publicly. Our backlog of slides dates back to 1997, meaning there are a lot that need to be sifted through and discarded for peeling or cracking. Additionally, during scanning a slide can get stuck, stalling a scan session for hours, resulting in a significant loss of productivity. Over the past two years, we have accumulated significant experience with how to scan a diverse inventory of slides using the Aperio AT2 high-volume scanner. We have been working closely with the vendor to resolve many problems associated with the use of this scanner for research purposes. This scanning project began in January of 2018 when the scanner was first installed. The scanning process was slow at first since there was a learning curve with how the scanner worked and how to obtain samples from the hospital. From its start date until May of 2019 ~20,000 slides we scanned. In the past 6 months from May to November we have tripled that number and how hold ~60,000 slides in our database. This dramatic increase in productivity was due to additional undergraduate staff members and an emphasis on efficient workflow. The Aperio AT2 scans 400 slides a day, requiring at least eight hours of scan time. The efficiency of these scans can vary greatly. When our team first started, approximately 5% of slides failed the scanning process due to focal point errors. We have been able to reduce that to 1% through a variety of means: (1) best practices regarding daily and monthly recalibrations, (2) tweaking the software such as the tissue finder parameter settings, and (3) experience with how to clean and prep slides so they scan properly. Nevertheless, this is not a completely automated process, making it very difficult to reach our production targets. With a staff of three undergraduate workers spending a total of 30 hours per week, we find it difficult to scan more than 2,000 slides per week using a single scanner (400 slides per night x 5 nights per week). The main limitation in achieving this level of production is the lack of a completely automated scanning process, it takes a couple of hours to sort, clean and load slides. We have streamlined all other aspects of the workflow required to database the scanned slides so that there are no additional bottlenecks. To bridge the gap between hospital operations and research, we are using Aperio’s eSM software. Our goal is to provide pathologists access to high quality digital images of their patients’ slides. eSM is a secure website that holds the images with their metadata labels, patient report, and path to where the image is located on our file server. Although eSM includes significant infrastructure to import slides into the database using barcodes, TUH does not currently support barcode use. Therefore, we manage the data using a mixture of Python scripts and manual import functions available in eSM. The database and associated tools are based on proprietary formats developed by Aperio, making this another important point of community-wide discussion on how best to disseminate such information. Our near-term goal for the TUDP Corpus is to release 100,000 slides by December 2020. We hope to continue data collection over the next decade until we reach one million slides. We are creating two pilot corpora using the first 50,000 slides we have collected. The first corpus consists of 500 slides with a marker stain and another 500 without it. This set was designed to let people debug their basic deep learning processing flow on these high-resolution images. We discuss our preliminary experiments on this corpus and the challenges in processing these high-resolution images using deep learning in [3]. We are able to achieve a mean sensitivity of 99.0% for slides with pen marks, and 98.9% for slides without marks, using a multistage deep learning algorithm. While this dataset was very useful in initial debugging, we are in the midst of creating a new, more challenging pilot corpus using actual tissue samples annotated by experts. The task will be to detect ductal carcinoma (DCIS) or invasive breast cancer tissue. There will be approximately 1,000 images per class in this corpus. Based on the number of features annotated, we can train on a two class problem of DCIS or benign, or increase the difficulty by increasing the classes to include DCIS, benign, stroma, pink tissue, non-neoplastic etc. Those interested in the corpus or in participating in community-wide discussions should join our listserv, nedc_tuh_dpath@googlegroups.com, to be kept informed of the latest developments in this project. You can learn more from our project website: https://www.isip.piconepress.com/projects/nsf_dpath. 
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  4. This dataset contains monthly average output files from the iCAM6 simulations used in the manuscript "Enhancing understanding of the hydrological cycle via pairing of process-oriented and isotope ratio tracers," in review at the Journal of Advances in Modeling Earth Systems. A file corresponding to each of the tagged and isotopic variables used in this manuscript is included. Files are at 0.9° latitude x 1.25° longitude, and are in NetCDF format. Data from two simulations are included: 1) a simulation where the atmospheric model was "nudged" to ERA5 wind and surface pressure fields, by adding an additional tendency (see section 3.1 of associated manuscript), and 2) a simulation where the atmospheric state was allowed to freely evolve, using only boundary conditions imposed at the surface and top of atmosphere. Specific information about each of the variables provided is located in the "usage notes" section below. Associated article abstract: The hydrologic cycle couples the Earth's energy and carbon budgets through evaporation, moisture transport, and precipitation. Despite a wealth of observations and models, fundamental limitations remain in our capacity to deduce even the most basic properties of the hydrological cycle, including the spatial pattern of the residence time (RT) of water in the atmosphere and the mean distance traveled from evaporation sources to precipitation sinks. Meanwhile, geochemical tracers such as stable water isotope ratios provide a tool to probe hydrological processes, yet their interpretation remains equivocal despite several decades of use. As a result, there is a need for new mechanistic tools that link variations in water isotope ratios to underlying hydrological processes. Here we present a new suite of “process-oriented tags,” which we use to explicitly trace hydrological processes within the isotopically enabled Community Atmosphere Model, version 6 (iCAM6). Using these tags, we test the hypotheses that precipitation isotope ratios respond to parcel rainout, variations in atmospheric RT, and preserve information regarding meteorological conditions during evaporation. We present results for a historical simulation from 1980 to 2004, forced with winds from the ERA5 reanalysis. We find strong evidence that precipitation isotope ratios record information about atmospheric rainout and meteorological conditions during evaporation, but little evidence that precipitation isotope ratios vary with water vapor RT. These new tracer methods will enable more robust linkages between observations of isotope ratios in the modern hydrologic cycle or proxies of past terrestrial environments and the environmental processes underlying these observations.   Details about the simulation setup can be found in section 3 of the associated open-source manuscript, "Enhancing understanding of the hydrological cycle via pairing of process‐oriented and isotope ratio tracers." In brief, we conducted two simulations of the atmosphere from 1980-2004 using the isotope-enabled version of the Community Atmosphere Model 6 (iCAM6) at 0.9x1.25° horizontal resolution, and with 30 vertical hybrid layers spanning from the surface to ~3 hPa. In the first simulation, wind and surface pressure fields were "nudged" toward the ERA5 reanalysis dataset by adding a nudging tendency, preventing the model from diverging from observed/reanalysis wind fields. In the second simulation, no additional nudging tendency was included, and the model was allowed to evolve 'freely' with only boundary conditions provided at the top (e.g., incoming solar radiation) and bottom (e.g., observed sea surface temperatures) of the model. In addition to the isotopic variables, our simulation included a suite of 'process-oriented tracers,' which we describe in section 2 of the manuscript. These variables are meant to track a property of water associated with evaporation, condensation, or atmospheric transport. Metadata are provided about each of the files below; moreover, since the attached files are NetCDF data - this information is also provided with the data files. NetCDF metadata can be accessed using standard tools (e.g., ncdump). Each file has 4 variables: the tagged quantity, and the associated coordinate variables (time, latitude, longitude). The latter three are identical across all files, only the tagged quantity changes. Twelve files are provided for the nudged simulation, and an additional three are provided for the free simulations: Nudged simulation files iCAM6_nudged_1980-2004_mon_RHevap: Mass-weighted mean evaporation source property: RH (%) with respect to surface temperature. iCAM6_nudged_1980-2004_mon_Tevap: Mass-weighted mean evaporation source property: surface temperature in Kelvin iCAM6_nudged_1980-2004_mon_Tcond: Mass-weighted mean condensation property: temperature (K) iCAM6_nudged_1980-2004_mon_columnQ: Total (vertically integrated) precipitable water (kg/m2).  Not a tagged quantity, but necessary to calculate depletion times in section 4.3 (e.g., Fig. 11 and 12). iCAM6_nudged_1980-2004_mon_d18O: Precipitation d18O (‰ VSMOW) iCAM6_nudged_1980-2004_mon_d18Oevap_0: Mass-weighted mean evaporation source property - d18O of the evaporative flux (e.g., the 'initial' isotope ratio prior to condensation), (‰ VSMOW) iCAM6_nudged_1980-2004_mon_dxs: Precipitation deuterium excess (‰ VSMOW) - note that precipitation d2H can be calculated from this file and the precipitation d18O as d2H = d-excess - 8*d18O. iCAM6_nudged_1980-2004_mon_dexevap_0: Mass-weighted mean evaporation source property - deuterium excess of the evaporative flux iCAM6_nudged_1980-2004_mon_lnf: Integrated property - ln(f) calculated from the constant-fractionation d18O tracer (see section 3.2). iCAM6_nudged_1980-2004_mon_precip: Total precipitation rate in m/s. Note there is an error in the metadata in this file - it is total precipitation, not just convective precipitation. iCAM6_nudged_1980-2004_mon_residencetime: Mean atmospheric water residence time (in days). iCAM6_nudged_1980-2004_mon_transportdistance: Mean atmospheric water transport distance (in km). Free simulation files iCAM6_free_1980-2004_mon_d18O: Precipitation d18O (‰ VSMOW) iCAM6_free_1980-2004_mon_dxs: Precipitation deuterium excess (‰ VSMOW) - note that precipitation d2H can be calculated from this file and the precipitation d18O as d2H = d-excess - 8*d18O. iCAM6_free_1980-2004_mon_precip: Total precipitation rate in m/s. Note there is an error in the metadata in this file - it is total precipitation, not just convective precipitation. 
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  5. Mikolaj Bojanczyk ; Emanuela Merelli ; David P. Woodruff (Ed.)
    Two equal length strings are a parameterized match (p-match) iff there exists a one-to-one function that renames the symbols in one string to those in the other. The Parameterized Suffix Tree (PST) [Baker, STOC' 93] is a fundamental data structure that handles various string matching problems under this setting. The PST of a text T[1,n] over an alphabet Σ of size σ takes O(nlog n) bits of space. It can report any entry in (parameterized) (i) suffix array, (ii) inverse suffix array, and (iii) longest common prefix (LCP) array in O(1) time. Given any pattern P as a query, a position i in T is an occurrence iff T[i,i+|P|-1] and P are a p-match. The PST can count the number of occurrences of P in T in time O(|P|log σ) and then report each occurrence in time proportional to that of accessing a suffix array entry. An important question is, can we obtain a compressed version of PST that takes space close to the text’s size of nlogσ bits and still support all three functionalities mentioned earlier? In SODA' 17, Ganguly et al. answered this question partially by presenting an O(nlogσ) bit index that can support (parameterized) suffix array and inverse suffix array operations in O(log n) time. However, the compression of the (parameterized) LCP array and the possibility of faster suffix array and inverse suffix array queries in compact space were left open. In this work, we obtain a compact representation of the (parameterized) LCP array. With this result, in conjunction with three new (parameterized) suffix array representations, we obtain the first set of PST representations in o(nlog n) bits (when logσ = o(log n)) as follows. Here ε > 0 is an arbitrarily small constant. - Space O(n logσ) bits and query time O(log_σ^ε n); - Space O(n logσlog log_σ n) bits and query time O(log log_σ n); and - Space O(n logσ log^ε_σ n) bits and query time O(1). The first trade-off is an improvement over Ganguly et al.’s result, whereas our third trade-off matches the optimal time performance of Baker’s PST while squeezing the space by a factor roughly log_σ n. We highlight that our trade-offs match the space-and-time bounds of the best-known compressed text indexes for exact pattern matching and further improvement is highly unlikely. 
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