We consider the problem of constructing binary codes to recover from k-bit deletions with efficient encoding/decoding, for a fixed k. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with ≈ 2n/n codewords of length n, i.e., at most log n bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than nΩ(1). For any fixed k, we construct a binary code with cklog n redundancy that can be decoded from k deletions in Ok(n log4n) time. The coefficient ckcan be taken to be O(k2log k), which is only quadratically worse than the optimal, non-constructive bound of O(k). We also indicate how to modify this code to allow for a combination of up to k insertions and deletions. We also note that among linear codes capable of correcting k deletions, the (k + 1)-fold repetition code is essentially the best possible.

An increasingly prevalent technique for improving response time in queueing systems is the use of redundancy. In a system with redundant requests, each job that arrives to the system is copied and dispatched to multiple servers. As soon as the first copy completes service, the job is considered complete, and all remaining copies are deleted. A great deal of empirical work has demonstrated that redundancy can significantly reduce response time in systems ranging from Google's BigTable service to kidney transplant waitlists. We propose a theoretical model of redundancy, the Redundancy-d system, in which each job sends redundant copies to d servers chosen uniformly at random. We derive the first exact expressions for mean response time in Redundancy-d systems with any finite number of servers. We also find asymptotically exact expressions for the distribution of response time as the number of servers approaches infinity.

We consider the problem of constructing binary codes to recover from k–bit deletions with efficient encoding/decoding, for a fixed k. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with ≈ 2n/n codewords of length n, i.e., at most log n bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than nΩ(1). For any fixed k, we construct a binary code with ck log n redundancy that can be decoded from k deletions in Ok(n log4 n) time. The coefficient ck can be taken to be O(k2 log k), which is only quadratically worse than the optimal, non-constructive bound of O(k). We also indicate how to modify this code to allow for a combination of up to k insertions and deletions. We also note that among linear codes capable of correcting k deletions, the (k + 1)-fold repetition code is essentially the best possible.

Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a request on multiple servers and wait for the first completion (discarding all remaining copies of the request). However there is no exact analysis of systems with redundancy.

Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the nonzero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erdös in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.