ACM Transactions on Algorithms 

About: ACM Transactions on Algorithms is an academic journal published by Association for Computing Machinery. The journal publishes majorly in the area(s): Approximation algorithm & Time complexity. It has an ISSN identifier of 1549-6325. Over the lifetime, 751 publications have been published receiving 26378 citations. The journal is also known as: ACM Trans. Algorithms.

Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm

Abstract: The problem of maximizing a concave function f(x) in the unit simplex Δ can be solved approximately by a simple greedy algorithm. For given k, the algorithm can find a point x(k) on a k-dimensional face of Δ, such that f(x(k) ≥ f(xa) − O(1/k). Here f(xa) is the maximum value of f in Δ, and the constant factor depends on f. This algorithm and analysis were known before, and related to problems of statistics and machine learning, such as boosting, regression, and density mixture estimation. In other work, coming from computational geometry, the existence of ϵ-coresets was shown for the minimum enclosing ball problem by means of a simple greedy algorithm. Similar greedy algorithms, which are special cases of the Frank-Wolfe algorithm, were described for other enclosure problems. Here these results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.

Multiplierless multiple constant multiplication

Abstract: A variable can be multiplied by a given set of fixed-point constants using a multiplier block that consists exclusively of additions, subtractions, and shifts. The generation of a multiplier block from the set of constants is known as the multiple constant multiplication (MCM) problem. Finding the optimal solution, namely, the one with the fewest number of additions and subtractions, is known to be NP-complete. We propose a new algorithm for the MCM problem, which produces solutions that require up to 20p less additions and subtractions than the best previously known algorithm. At the same time our algorithm, in contrast to the closest competing algorithm, is not limited by the constant bitwidths. We present our algorithm using a unifying formal framework for the best, graph-based MCM algorithms and provide a detailed runtime analysis and experimental evaluation. We show that our algorithm can handle problem sizes as large as 100 32-bit constants in a time acceptable for most applications. The implementation of the new algorithm is available at www.spiral.net.

Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets

Abstract: We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,…,m − 1} for some integer m while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i), which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n, m) p o(n) p O(lg lg m) bits to store a set of size n, where B(n, m) e ⌊lg (m/n)⌋ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001]. We present extensions and applications of our indexable dictionary data structure, including: —an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time; —a representation of a multiset of size n from {0,…,m − 1} in B(n, m p n) p o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time; and p O(lg lg m) —a representation of a sequence of n nonnegative integers summing up to m in B(n, m p n) p o(n) bits that supports prefix sum queries in constant time.

Compressed representations of sequences and full-text indexes

Abstract: Given a sequence S = s1s2…sn of integers smaller than r = O(polylog(n)), we show how S can be represented using nH0(S) p o(n) bits, so that we can know any sq, as well as answer rank and select queries on S, in constant time. H0(S) is the zero-order empirical entropy of S and nH0(S) provides an information-theoretic lower bound to the bit storage of any sequence S via a fixed encoding of its symbols. This extends previous results on binary sequences, and improves previous results on general sequences where those queries are answered in O(log r) time. For larger r, we can still represent S in nH0(S) p o(n log r) bits and answer queries in O(log r/log log n) time.Another contribution of this article is to show how to combine our compressed representation of integer sequences with a compression boosting technique to design compressed full-text indexes that scale well with the size of the input alphabet Σ. Specifically, we design a variant of the FM-index that indexes a string T[1, n] within nHk(T) p o(n) bits of storage, where Hk(T) is the kth-order empirical entropy of T. This space bound holds simultaneously for all k ≤ α logvΣvn, constant 0

Algorithmic construction of sets for k-restrictions

Abstract: This work addresses k-restriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σm that satisfies a given set of k-wise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science.The standard approach for deterministically solving such problems is via almost k-wise independence or k-wise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor et al. [1995]. Among other results, we enhance the combinatorial objects in the heart of their method, called splitters, and construct multi-way splitters, using a new discrete version of the topological Necklace Splitting Theorem [Alon 1987].We utilize our methods to show improved constructions for group testing [Ngo and Du 2000] and generalized hashing [Alon et al. 2003], and an improved inapproximability result for SET-COVER under the assumption P ≠ NP.