Efficient Indexing of Uncertain Strings

Strings in the real world often have some level of uncertainty. This can happen for various reasons, such as unreliable data measurements, flexible sequence modeling, or noise introduced for privacy protection.

In the character-level uncertainty model, an uncertain string is a sequence of probability distributions over certain letters. For example, given an uncertain string and a weight threshold, we say that a certain pattern occurs in the string at a certain position if the combined probabilities of the letters of that pattern meet or exceed the threshold.

Indexing standard strings for searches can be done quickly and efficiently. However, indexing uncertain strings poses more of a challenge due to their complexity. Currently, the best method for indexing uncertain strings has a large size, requires significant time and space to be created, and can answer Queries in an optimal time. For large strings, this method may not be feasible because of the resources required, which can overshadow the benefits of using it.

Thus, there is a need to create a more space-efficient index, even if this means that the search queries may take a bit longer. Our approach reveals that if we have a lower limit on the length of the patterns we want to search for, we can significantly reduce the index size and the space required to create it.

We propose a new index that can be built with expected space, while supporting fast pattern search in expectation for patterns of a certain length. We have tested several versions of our index, and the best performing version is much smaller than the current leading method, while often providing faster or competitive response times to queries.

In many real-world database systems, a large amount of data is stored in text form, which consists of sequences of letters over an alphabet. This includes data types such as DNA sequences, natural language text from users, IDs created by software, or sensor measurements. Given the growing size of string data, it is important to represent this information in a compact way, while still allowing for efficient searches.

The classic text indexing problem involves preparing a string into a structure that supports fast pattern matching queries, meaning it will report where in the string occurrences of a specific pattern begin. In text indexing, we focus on four aspects of efficiency:

1. The size of the index for a string of a given length.
2. How fast we can find the answer to a query of a certain length.
3. The amount of memory needed to build the index.
4. How long it takes to construct the index.

The typical structure for indexing, called a suffix tree, has a size that is directly related to the length of the string and allows for optimal search times as well. However, when it comes to uncertain strings, the data may not be as clear-cut, leading to additional complications.

The uncertainty in these strings can arise from:

1. Inaccurate or incomplete data measurements, like those from sensors or trajectory tracking.
2. Intentional flexible sequence modeling, such as analyzing closely related genomes together.
3. The presence of confidential information, which may have been altered for privacy.

While many solutions exist for indexing standard texts or responding to queries for diverse types of uncertain data, effective indexing methods specifically for uncertain strings are still being developed. Our work aims to move toward creating practical and space-efficient indexes for these types of strings.

#Our Data Model and Motivation

We utilize a standard character-level uncertainty model. An uncertain string or weighted string is a sequence of sets. Each set contains ordered pairs, where a letter represents a possible character at a particular position and its accompanying probability signifies how likely that character is to appear in that position.

We define the problem of Weighted Indexing as follows: Given a weighted string and a weight threshold, we want to prepare it into a compact structure that allows for efficient pattern matching queries. This means reporting all positions in the string where the pattern appears with a probability that meets or exceeds the threshold.

The current best methods for indexing weighted strings have received much attention from researchers but can be impractical for larger datasets. For example, if we have a weighted string that is lengthy, the constant factors involved could require many terabytes of memory to store the index, which is clearly not feasible.

We aim to find a way to trade off space for query time, focusing on allowing for smaller index sizes while still enabling effective pattern queries. A useful input parameter is the minimum length of any queried pattern, as it is often realistic to expect knowledge of this beforehand.

In fields like bioinformatics, for instance, the length of sequencing patterns can range widely. Knowing this can help in constructing more space-efficient indexes.

#Our Techniques and Results

We start by taking a weighted string and using it to create a set of standard strings, each of a specified length. For each string, we examine all its suffixes and sample them using a technique called minimizers. These suffixes correspond to sets of suffixes and reversed suffixes.

Next, we index these suffixes in two Suffix Trees, connecting them using a grid structure. This enables us to handle pattern matching queries for patterns that meet our specified minimum length. When a query pattern is provided, we find its leftmost minimizer, which corresponds to suffixes that we can query using our index. The index then merges the outcomes efficiently and provides valid occurrences of the pattern.

We show that the resulting index can be compact, thanks to new methods of encoding edge labels in the suffix trees. A key part of our approach is that after construction, we can discard certain strings, leading to a significantly smaller index size.

However, our method still requires constructing standard strings initially, which results in a certain amount of space being required. Our further development introduces an efficient algorithm to build the index without explicitly creating these strings, allowing us to sample an implicit representation instead. This method consumes space equivalent to the final index size.

By combining the best aspects of existing tree and array-based index paradigms, our indexes perform significantly better than current methods, being much smaller in both index size and construction space. For instance, we show that when indexing bacterial samples, our indexes only need a small fraction of the memory compared to existing methods.

#Paper Organization

1. Background: Introduction to the basic concepts and terms used.
2. Our Index: A detailed description of the new indexing method.
3. Space-Efficient Algorithm: Explanation of how to construct the index efficiently.
4. Practical Querying: Discussion on how to query the index without excessive overhead.
5. Related Work: Overview of existing methods and research in the field.
6. Experimental Evaluation: Presentation of results from testing our methods.
7. Conclusion: Summary of findings and potential future directions.

#Preliminaries and Problem Definition

To understand our work, we first define what a string is within this framework. A string is essentially a sequence of characters from a specific set called an alphabet. We use different lengths and combinations of these characters in our operations.

We also look at Sampling, which involves selecting specific starting positions of fragments of strings. This allows us to create sets of indices based on selected points.

A weighted string consists of sets where each element denotes the probability of certain letters appearing at given positions. The occurrences of a string within a weighted string only count if the probability exceeds a defined threshold. We categorize factors of a weighted string based on their likelihood of appearing.

Understanding solid occurrences and their relationships is crucial. A factor is termed valid if its probability meets the required threshold.

#The New Index: Minimizer-Based

This section introduces our new index method for handling uncertain strings. We rely on random access to the string data, which allows us to discard unnecessary information during construction.

Our main idea involves creating an estimation of the string that gathers expected information in a manageable size. We then employ the minimizer mechanism to identify positions corresponding to the minimum length necessary for our queries.

By building two suffix trees that contain the solid factors starting at these minimizer positions, we ensure efficient retrieval of valid occurrences. The final index size is achievable through the specific encoding of information, meaning we can achieve a compact representation without excessive data.

#Exploiting 2D Range Reporting

This part explains how to use a geometric data structure to connect our suffix trees to pair corresponding nodes based on shared minimizer positions. We create this pairing using a grid structure that allows for efficient queries by organizing search points in a 2D format.

This method helps us effectively navigate through potential candidates for valid occurrences of a pattern, making the retrieval process swift and efficient.

#Main Result

We conclude that after construction, we can expect our new index to be smaller in size and to respond more quickly to queries than previous methods. Our techniques collectively help in creating a tool that can manage large sets of uncertain data while providing practical and quick access for users wanting to extract meaningful patterns.

#Space-Efficient Construction of the Index

In this section, we detail a more efficient way to build our index, which reduces the space required during the process. Our method involves simulating a larger structure called an extended solid factor tree. By only keeping necessary parts during construction, we manage to reduce the overall memory used.

This approach also focuses on ensuring we maintain acceptable construction times while trading off some speed for lower space usage.

#Practically Fast Querying Without a Grid

Here, we present a simplified query method that does not rely on the complex grid structure. While this simpler method might not provide the same guarantees as the grid-based approach, it can perform better in practice for many use cases.

We demonstrate how to locate potential occurrences of a pattern efficiently and verify their validity without needing the additional overhead from the grid.

#Related Work

Previously, others have explored various methods for indexing uncertain or probabilistic data. However, many of these methods have certain limitations that our approach seeks to address. By developing a more practical indexing scheme for uncertain strings, we provide a much-needed advancement in the field of data indexing.

#Experimental Evaluation

In our experiments, we assess the practical performance of our methods using real-world data sets. By testing different indexes, we confirm that our new approach can outperform existing methods in terms of size, speed, and construction time.

#Conclusion of Our Experimental Evaluation

Based on our tests, it becomes clear that our algorithms provide the necessary efficiency and practicality for working with uncertain strings. The results demonstrate that our work significantly contributes to the field, paving the way for future advancements.

Overall, our focus on minimizing space and time while maximizing efficiency gives us a strong advantage in handling uncertain string data.

#Discussion: Limitations and Future Work

While our work has shown promise, it also has limitations. We're aware that our indexes could benefit from better guarantees in cases where the distribution of minimizers can lead to size issues. Future work could explore alternative sampling mechanisms or leverage probabilities more effectively to refine our indexes further.

By considering domain knowledge and applying it to our methods, we might achieve even better results in real-world applications. Additionally, we can further investigate how different data models might improve our indexing strategy to better suit natural complexities inherent in uncertain strings.

Our research will continue to evolve as we seek to address these challenges and enhance the efficiency and applicability of our indexing methods.