The Intractability of Computing the Hamming Distance 1st ediiton by Bodo Manthey, Rudiger Reischuk ISBN 3540206958  9783540206958

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ISBN 10:       3540206958 
ISBN 13: 978-3540206958 
Author:         Bodo Manthey, Rüdiger Reischuk
 

Given a string x and a language L, the Hamming distance of x to L is the minimum Hamming distance of x to any string in L. The edit distance of a string to a language is analogously defined.

First, we prove that there is a language in AC ⁰ such that both Hamming and edit distance to this language are hard to approximate; they cannot be approximated with a factor <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="O(n13−ϵ)”>O(n13−ϵ), for any ε > 0, unless P = NP (n denotes the length of the input string).

Second, we show the parameterized intractability of computing the Hamming distance. We prove that for every t ∈ ℕ there exists a language in AC ⁰ for which computing the Hamming distance is W[t]-hard. Moreover, there is a language in P for which computing the Hamming distance is W[P]-hard.

Finally, we show that the problems of computing the Hamming distance and of computing the edit distance are in some sense equivalent by presenting reductions from the former to the latter and vice versa.

The Intractability of Computing the Hamming Distance 1st Table of contents:

1. Introduction

   • 1.1 Overview of Hamming Distance
   • 1.2 Importance of Hamming Distance in Computer Science and Information Theory
   • 1.3 Computational Complexity of Hamming Distance Calculation
   • 1.4 Motivation for Studying the Intractability of Hamming Distance
   • 1.5 Structure of the Paper

2. Background on Hamming Distance

   • 2.1 Definition and Basic Properties of Hamming Distance
   • 2.2 Applications of Hamming Distance in Error Detection and Correction
   • 2.3 Relationship Between Hamming Distance and Other Metrics (e.g., Edit Distance)
   • 2.4 Algorithms for Computing Hamming Distance in Classical Settings

3. Computational Complexity of Hamming Distance

   • 3.1 The Concept of Intractability in Computational Problems
   • 3.2 Formal Complexity Classifications of Hamming Distance Computation
   • 3.3 Connection to NP-Hardness and Other Hard Problems
   • 3.4 Hardness of Computing Hamming Distance for Large Data Sets
   • 3.5 Lower Bounds for Hamming Distance Computation

4. Special Cases of Hamming Distance Computation

   • 4.1 Hamming Distance for Fixed-Length Sequences
   • 4.2 Variations in Hamming Distance in Graphs and Network Models
   • 4.3 Hamming Distance in Quantum and Approximate Settings
   • 4.4 Hardness of Computing Hamming Distance in Specific Applications (e.g., Bioinformatics, Cryptography)
   • 4.5 On the Generalization of Hamming Distance to Higher Dimensions

5. Approximation and Heuristic Methods

   • 5.1 Approximation Algorithms for Hamming Distance Calculation
   • 5.2 Heuristic Approaches to Speed Up Hamming Distance Computation
   • 5.3 Trade-offs Between Exact and Approximate Solutions
   • 5.4 Greedy Algorithms for Large-Scale Problems
   • 5.5 Randomized and Probabilistic Approaches for Efficient Computation

6. Applications of Hamming Distance and Intractability

   • 6.1 Hamming Distance in Error Correction Codes (e.g., Hamming Codes, LDPC Codes)
   • 6.2 Bioinformatics Applications: Sequence Comparison and Alignment
   • 6.3 Hamming Distance in Cryptography and Security Protocols
   • 6.4 Use in Machine Learning and Data Mining (e.g., Nearest Neighbor Search)
   • 6.5 Applications in Communication Systems and Networking

7. Experimental Results and Benchmarking

   • 7.1 Experimental Setup and Case Studies
   • 7.2 Benchmarking Exact and Approximate Algorithms for Hamming Distance
   • 7.3 Performance Analysis of Hamming Distance Computation on Large Data Sets
   • 7.4 Impact of Sequence Length and Input Size on Computation Time
   • 7.5 Comparison with Alternative Metrics for Similar Problems

8. Challenges and Open Problems

   • 8.1 Limitations of Current Algorithms for Hamming Distance Computation
   • 8.2 Open Problems in Reducing the Intractability of Hamming Distance
   • 8.3 Connections to Other Computationally Hard Problems
   • 8.4 Potential Breakthroughs in Approximation Algorithms for Hamming Distance
   • 8.5 Future Directions in Complexity Theory Related to Hamming Distance

9. Conclusion

   • 9.1 Summary of Findings on the Intractability of Hamming Distance
   • 9.2 Implications for the Design of Algorithms and Applications
   • 9.3 Final Thoughts on Overcoming the Intractability of Hamming Distance
   • 9.4 Closing Remarks on Future Research in Hamming Distance Computation
10.  

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