arXiv subjects

Neal E. Young

Publications and source records attributed to Neal E. Young.

46 records · Page 3Linked to original sources

arXiv2002

Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.

cs.DS↗

arXiv2002

On-Line File Caching

In the on-line file-caching problem problem, the input is a sequence of requests for files, given on-line (one at a time). Each file has a non-negative size and a non-negative retrieval cost. The problem is to decide which files to keep in a fixed-size cache so as to minimize the sum of the retrieval costs for files that are not in the cache when requested. The problem arises in web caching by browsers and by proxies. This paper describes a natural generalization of LRU called Landlord and gives an analysis showing that it has an optimal performance guarantee (among deterministic on-line algorithms). The paper also gives an analysis of the algorithm in a so-called ``loosely'' competitive model, showing that on a ``typical'' cache size, either the performance guarantee is O(1) or the total retrieval cost is insignificant.

cs.DS↗

arXiv2002

Competitive Paging Algorithms

The paging problem is that of deciding which pages to keep in a memory of k pages in order to minimize the number of page faults. This paper introduces the marking algorithm, a simple randomized on-line algorithm for the paging problem, and gives a proof that its performance guarantee (competitive ratio) is O(log k). In contrast, no deterministic on-line algorithm can have a performance guarantee better than k.

cs.DS↗

arXiv2002

Orienting Graphs to Optimize Reachability

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard.

cs.DS↗

arXiv2002

Sequential and Parallel Algorithms for Mixed Packing and Covering

Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does comparable work but finishes in polylogarithmic time. The results generalize previous work on pure packing and covering (the special case when the constraints are all "less-than" or all "greater-than") by Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998).

cs.DS↗

arXiv2002

Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.

cs.DS↗

arXiv2002

Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.

cs.DS↗

arXiv2002

Prefix Codes: Equiprobable Words, Unequal Letter Costs

Describes a near-linear-time algorithm for a variant of Huffman coding, in which the letters may have non-uniform lengths (as in Morse code), but with the restriction that each word to be encoded has equal probability. [See also ``Huffman Coding with Unequal Letter Costs'' (2002).]

cs.DS↗

arXiv2002

Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory

Von Neumann's Min-Max Theorem guarantees that each player of a zero-sum matrix game has an optimal mixed strategy. This paper gives an elementary proof that each player has a near-optimal mixed strategy that chooses uniformly at random from a multiset of pure strategies of size logarithmic in the number of pure strategies available to the opponent. For exponentially large games, for which even representing an optimal mixed strategy can require exponential space, it follows that there are near-optimal, linear-size strategies. These strategies are easy to play and serve as small witnesses to the approximate value of the game. As a corollary, it follows that every language has small ``hard'' multisets of inputs certifying that small circuits can't decide the language. For example, if SAT does not have polynomial-size circuits, then, for each n and c, there is a set of n^(O(c)) Boolean formulae of size n such that no circuit of size n^c (or algorithm running in time n^c) classifies more than two-thirds of the formulae succesfully.

cs.CC↗

arXiv2002

On-Line Paging against Adversarially Biased Random Inputs

In evaluating an algorithm, worst-case analysis can be overly pessimistic. Average-case analysis can be overly optimistic. An intermediate approach is to show that an algorithm does well on a broad class of input distributions. Koutsoupias and Papadimitriou recently analyzed the least-recently-used (LRU) paging strategy in this manner, analyzing its performance on an input sequence generated by a so-called diffuse adversary -- one that must choose each request probabilitistically so that no page is chosen with probability more than some fixed epsilon>0. They showed that LRU achieves the optimal competitive ratio (for deterministic on-line algorithms), but they didn't determine the actual ratio. In this paper we estimate the optimal ratios within roughly a factor of two for both deterministic strategies (e.g. least-recently-used and first-in-first-out) and randomized strategies. Around the threshold epsilon ~ 1/k (where k is the cache size), the optimal ratios are both Theta(ln k). Below the threshold the ratios tend rapidly to O(1). Above the threshold the ratio is unchanged for randomized strategies but tends rapidly to Theta(k) for deterministic ones. We also give an alternate proof of the optimality of LRU.

cs.DS↗