6455009
The Randomized Complexity of Maintaining the Minimum
The complexity of maintaining a set under the operations Insert, Delete and FindMin is considered. In the comparison model it is shown that any randomized algorithm with expected amortized cost t comparisons per Insert and Delete has expected cost at least n/(e22t)-1 comparisons for FindMin. If FindMin is replaced by a weaker operation. FindAny, then it is shown that a randomized algorithm with constant expected cost per operation exists; in contrast, it is shown that no deterministic algorithm can have constant cost per operation. Finally, a deterministic algorithm with constant amortized cost per operation for an offline version of the problem is given.
7826204
Probabilistic Recurrence Relations Revisited
The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(x)=a(x)+T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size x. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high probability guarantees on the performance of such randomised algorithms, it suffices to obtain bounds on the tail of the distribution of T(x). Karp derived tight bounds on this tail distribution, when the distribution of H(x) satisfies certain restrictions. However, his proof is quite difficult to understand. In this paper, we derive bounds similar to Karp's using standard tools from elementary probability theory, such as Markov's inequality, stochastic dominance and a variant of Chernoff bounds applicable to unbounded variables. Further, we extend the results, showing that similar bounds hold under weaker restrictions on H(x). As an application, we derive performance bounds for an interesting class of algorithms that was outside the scope of the previous results.
2155084
One-write algorithms for multivalued regular and atomic registers
Abstract. This paper presents an algorithm for implementing a k-valued regular register (the logical register) using 
$k(k-1)/2$ binary regular registers (the physical registers) that requires only one physical write per logical write. The same algorithm using binary atomic registers implements a k-valued atomic register. The algorithm is simple to describe and depends on properties of paths in a related graph. Two lower bounds are given on the number of registers required by one-write implementations in the regular case. The first lower bound, 
$2k - 1 - \lfloor \log k \rfloor$, holds for a fairly general class of algorithms. The second lower bound holds for a restricted class of implementations and implies that our algorithm is optimal for this class. Both lower bounds improve on the best previously known lower bound, which was k. The two lower bounds also hold for the atomic case under further restrictions.
17598554
Prefix Graphs and Their Applications
The range product problem is, for a given set S equipped with an associative operator o, to preprocess a sequence a1,⋯, an of elements from S so as to enable efficient subsequent processing of queries of the form: Given a pair (s, t) of integers with 1≤s≤t≤n, return a s oas+1 o⋯o a t . The generic range product problem and special cases thereof, usually with o computing the maximum of its arguments according to some linear order on S, have been extensively studied. We show that a large number of previous sequential and parallel algorithms for these problems can be unified and simplified by means of prefix graphs.
206818151
Graph bisection algorithms with good average case behavior
In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixedd≧3, all sufficiently largen and allb=o(n1−1/[(d+1)/2]), the algorithm finds the minimum bisection for almost alld-regular labelled simple graphs with 2n nodes and bisection widthb. For example, the algorithm succeeds for almost all 5-regular graphs with 2n nodes and bisection widtho(n2/3). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably:(i)the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection,(ii)whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection),(iii)the algorithm works well both theoretically and experimentally,(iv)the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and(v)the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).