Fraction Of Insertions Research Articles

Deterministic Document Exchange Protocols and Almost Optimal Binary Codes for Edit Errors

We study two basic problems regarding edit errors, document exchange and error correcting codes. Here, two parties try to exchange two strings with length roughly n and edit distance at most k , or one party tries to send a string of length n to another party through a channel that can introduce at most k edit errors. The goal is to use the least amount of communication or redundancy possible. Both problems have been extensively studied for decades, and in this article, we focus on deterministic document exchange protocols and binary codes for insertions and deletions (insdel codes). It is known that for small k (e.g., k ≤ n/4 ), in both problems the optimal communication or redundancy size is Θ ( k log n/k). In particular, this implies the existence of binary codes that can correct ε fraction of edit errors with rate 1-Θ (ε log 1/ε )). However, known constructions are far from achieving these bounds. In this article, we significantly improve previous results on both problems. For document exchange, we give an efficient deterministic protocol with communication complexity O ( k log 2 n/k . This significantly improves the previous best-known deterministic protocol, which has communication complexity O ( k 2 + k log 2 n ) [ 4 ]. For binary insdel codes, we obtain the following results: (1) An explicit binary insdel code with redundancy O ( k log 2 n/k). In particular this implies an explicit family of binary insdel codes that can correct ε fraction of insertions and deletions with rate 1-O(ε log 2 (1/ε))=1-Õ(ε). This significantly improves the previous best-known result, which only achieves rate 1-Õ(√ ε) [ 14 ], [ 15 ], and is optimal up to a log (1/ε factor. (2) An explicit binary insdel code with redundancy O ( k log n ). This significantly improves the previous best-known result of Reference [ 6 ], which only works for constant k and has redundancy O ( k 2 log k log n ); and that of Reference [ 4 ], which has redundancy O ( k 2 + k log 2 n ). Our code has optimal redundancy for k ≤ n 1-α , any constant 0< α < 1. This is the first explicit construction of binary insdel codes that has optimal redundancy for a wide range of error parameters k . In obtaining our results, we introduce several new techniques. Most notably, we introduce the notion of ε-self-matching hash functions and ε-synchronization hash functions . We believe our techniques can have further applications in the literature.

On the List Decodability of Insertions and Deletions

In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $\Omega(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $\Omega(1)$ that are list-decodable from a $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions, where $q$ depends only on $\tau_\mathsf{I}$ and $\tau_\mathsf{D}$. We also provide efficient encoding and decoding algorithms for list-decoding from $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions for any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric.

Polarity of Tn5 insertion mutations in Escherichia coli.

We assessed the effect of insertions of the kanamycin resistance transposon Tn5 in the lac operon of Escherichia coli on the expression of distal genes lacY and lacA (melibiose fermentation at 41 degrees C and thiogalactoside transacetylase synthesis, respectively). Every insertion mutation tested (41 in lacZ and 23 in lacY) was strongly polar. However, approximately one-third of the insertion mutants expressed distal genes at low levels due to a promoter associated with Tn5. To localize this promoter, we (i) reversed the orientation of Tn5 at several sites and (ii) replaced wild-type Tn5 with several substitution derivatives which lack Tn5's central region. Neither alteration changed the expression of distal genes. Thus, in contrast to transposons IS2 and TnA. Tn5's ability to turn on distal gene expression is not due to a promoter in its central region and therefore is not dependent on the overall orientation of Tn5 in the operon. Our results suggest that the promoter is within 186 base pairs of the ends of Tn5. It is possible that the promoter is detected in only a fraction of insertions because it overlaps Tn5-target sequence boundary.