We show how to support the finger search operation on degree-balanced search trees in a space-efficient manner that retains a worst-case time bound of O(log d), where d is the difference in rank between successive search targets. While most existing tree-based designs allocate linear extra storage in the nodes (e.g., for side links and parent pointers), our design maintains a compact auxiliary data structure called the “hand ” during the lifetime of the tree and imposes no other storage requirement within the tree. The hand requires O(log n) space for an n-node tree and has a relatively simple structure. It can be updated synchronously during insertions and deletions with time proportional to the number of structural changes in the tree. The auxiliary nature of the hand also makes it possible to introduce finger searches into any existing implementation without modifying the underlying data representation (e.g., any implementation of Red-Black trees can be used). Together these factors make finger searches more appealing in practice. Our design also yields a simple yet optimal inorder walk algorithm with worst-case O(1) work per increment (again without any extra storage requirement in the nodes), and we believe our algorithm can be used in database applications when the overall performance is very sensitive to retrieval latency. 1

We consider the problem of finding the minimum cost of a feasible flow in directed series-parallel networks. We allow real-valued lower and upper bounds for the flows on edges. While strongly polynomial-time algorithms are known for this problem on arbitrary networks, it is known to be "hard" for parallelization. We develop, for the first time, an efficient NC algorithm to solve the min-cost flow problem on directed seriesparallel networks partially solving a problem posed by Booth and Tarjan [6, 5]. Our algorithm takes O(log 2 m) time using O(m= log m) processors on an EREW PRAM and it is optimal with respect to Booth and Tarjan's algorithm with running time O(m log m). The algorithm owes it's efficiency to the tree contraction technique and using simple data structures for flow list manipulations as opposed to finger search trees. 1 Introduction Let G = (V; E) be a directed network with two distinguished vertices s and t called the source and the sink respectively. For each e = ...

Introduction. The one-dimensional channel compaction problem with automatic jog insertion may be described informally as follows. We are given n horizontal wire segments organized into t tracks. Segments on the same track have the same y-coordinate and the ranges of their x-coordinates do not overlap. Each segment has a via at each end, connecting it to vertical wires on another layer. The goal is to minimize channel height subject to design rule constraints on the distances between wires and between wires and vias. The relative vertical position of wires is not allowed to change during compaction. Figure 1(a) shows an initial layout with 7 tracks. The horizontal wires to be compacted are shown as thin solid lines while vertical wires on another layer are thick dashed lines. Dots represent vias. Relative vertical position must be maintained due to what are called vertical constraints (see e.g. [PL88]). The left part of net