Abstract. The parametric maximum flow problem is an extension of the classical maximum flow problem in which the capacities of certain arcs are not fixed but are functions of a single parameter. Gallo et al. [6] showed that certain versions of the push-relabel algorithm for ordinary maximum flow can be extended to the parametric problem while only increasing the worst-case time bound by a constant factor. Recently Zhang et al. [14,13] proposed a novel, simple balancing algorithm for the parametric problem on bipartite networks. They claimed good performance for their algorithm on networks arising from a real-world application. We describe the results of an experimental study comparing the performance of the balancing algorithm, the GGT algorithm, and a simplified version of the GGT algorithm, on networks related to those of the application of Zhang et al. as well as networks designed to be hard for the balancing algorithm. Our implementation of the balancing algorithm beats both versions of the GGT algorithm on networks related to the application, thus supporting the observations of Zhang et al. On the other hand, the GGT algorithm is more robust; it beats the balancing algorithm on some natural networks, and by asymptotically increasing amount on networks designed to be hard for the balancing algorithm.

The Dynamic Optimality Conjecture [ST85] states that splay trees are competitive (within a constant competitive factor) among the class of all binary search tree (BST) algorithms. Despite 20 years of research this conjecture is still unresolved. Recently, Demaine et al. [DHIP04] suggested searching for alternative algorithms which have small but non-constant competitive factors. They proposed Tango, a BST algorithm which is nearly dynamically optimal-its competitive ratio is O(log log n) instead of a constant. Unfortunately, for many access patterns, such as random and sequential, Tango is worse than other BST algorithms by a factor of log log n. In this paper, we introduce the multi-splay tree (MST) data structure, which is the first O(log log n)competitive BST to simultaneously achieve O(log n) amortized cost and O(log 2 n) worst-case cost per query. We also prove the sequential access lemma for MSTs, which states that sequentially accessing all keys takes linear time. Thus, MSTs are O(log log n)-competitive like Tango but, unlike Tango, require only O(log n) amortized time per access in an arbitrary sequence and only O(1) amortized time per access during a sequential access sequence. Furthermore, we generalize the standard framework for competitive analysis of BST algorithms to include updates (insertions and deletions) in addition to queries. In doing so, we extend the lower bound of Wilber [Wil89] and Demaine et al. [DHIP04] to handle these update operations. We show how MSTs can be modified to support these update operations and be O(log log n)-competitive in the new framework while maintaining the rest of the properties above.

Abstract. We present skip-splay, the first binary search tree algorithm known to have a running time that nearly achieves the unified bound. Skip-splay trees require only O(m lg lg n + UB(σ)) time to execute a query sequence σ = σ1 . . . σm. The skip-splay algorithm is simple and similar to the splay algorithm.