Greg N. Frederickson

The complexity of selection and ranking in X + Y and matrices with sorted columns

The complexity of selection is analyzed for two sets, X + Y and matrices with sorted columns. Algorithms are presented that run in time which depends nontrivially on the rank k of the element to be selected and which is sublinear with respect to set cardinality. Identical bounds are also shown for the problem of ranking elements in these sets, and all bounds are shown to be optimal to within a constant multiplicative factor.

Finding kth paths and p-centers by generating and searching good data structures

A succinct and easily searchable representation of the set of intervertex distances of a tree is given. Algorithms are presented for generating this representation, for searching it to select a kth longest path, and for searching it to locate a p-center. The complete algorithm for path selection is asymptotically optimal in the worst case, and the algorithms for p-center location improve on previous methods. The p-center results are extended to networks with independent cycles.

Sequencing Tasks with Exponential Service Times to Minimize the Expected Flow Time or Makespan

The problems of minimizing the expected makespan and minimizing the expected tic for a finite set of independent tasks with exponential service-time distributions on m ~ 2 it processors are considered. It is shown that a scheduling policy minimizes the expected flow timq only if it is shortest expected processing time tint, and that a policy minimizes the expected make and only if it is longest expected processing time fast. rE,, WORDS ArCD PHRASES: scheduling, flow time, makespan, sequencing, policy C R C A T E G O R I E S : 5.39, 5.42

On the relationship between the biconnectivity augmentation and travelling salesman problems

Abstract A strategy for solving the traveling salesman problem is adapted to the problem of finding a biconnected subgraph of a weighted graph whose cost function satisfies the triangle inequality. An approximation algorithm similar to Christofides algorithm [5] for the traveling salesman problem is shown to possess the same worst-case bound of 3 2 when applied to the biconnectivity augmentation problem. A tight inequality is derived relating the cost of an optimal traveling salesman tour to the cost of an optimal biconnection.

Implicit Data Structures for the Dictionary Problem

Several new data structures for dictionaries are presented that use just one location in addition to those required for key values. The structures are generahzations of a rotated sorted list, with the best realizing a search tune of O(log n) and insemon and deletion tunes of O(n ~ ( l o g n) 3/2) Structures adapted to allow fast average search times and structures that allow pamal match retrieval on records wRh d keys, d > 1, are also considered.

Self-organizing heuristics for implicit data structures

Self-organizing heuristics are presented for data structures containing elements with different weights (access probabilities). The structures use just a constant number of locations in addition to those required for the values of the elements, and support average access times that are within a constant multiplicative factor of optimal. Data structures and corresponding heuristics with essentially optimal average unsuccessful search times are also given for the case in which there are probabilities of access associated with the intervals between consecutive element values.

A Note on Expected Makespans for Largest-First Sequences of Independent Tasks on Two Processors

We consider the scheduling of sets of n simultaneously available tasks on two identical processors. The task execution times are assumed to be independent samples from the uniform distribution on [0, 1]. We analyze the expected makespan schedule-length performance of two well-known largest-task-first, nonpreemptive approximation rules. With the largest-first LF rule, tasks are assigned in nonincreasing order of execution time to the processors as they become available. With the restricted largest first RLF rule, tasks are assigned in pairs, one to a processor. The larger task of a pair is the first to be scheduled. If n is odd, the last task is simply assigned to the earner finishing processor in the schedule for the first n-1 tasks. We prove that the expected makespan for LF is bounded by

Generalized selection and ranking (Preliminary Version)

frac{n}4 + frac{e}{2n+1},