A large class of classical combinatorial problems, including most of the difficult problems in the literature of network flows and computational graph theory, are shown to be equivalent, in the sense that either all or none of them can be solved in polynomial time. Moreover, they are solvable in polynomial time if and only if every problem solvable by a polynomial-depth backtrack search is also solvable by a polynomial-time algorithm. The technique of deriving these results through problem transformations is explained, and some comments are made as to the probable effect of these results on research in the field of combinatorial algorithms.

On the Computational Complexity of Combinatorial Problems

Achieving effective run-time mapping on multiprocessor systems-on-chip (MPSoCs) is a challenging task, particularly since the arrival order of the target applications is not known a priori. This paper targets real-time applications which are dynamically mapped onto embedded MPSoCs, where communication happens via the Network-on-Chip (NoC) approach, and resources connected to the NoC have multiple voltage levels. We address precisely the energy- and performance-aware incremental mapping problem for NoCs with multiple voltage levels and propose an efficient technique (consisting of region selection and node allocation) to solve it. Moreover, the proposed technique allows for new applications to be added to the system with minimal in- terprocessor communication overhead. Experimental results show that the proposed technique is very fast, and as much as 50% communication energy savings can be achieved compared to using an arbitrary allocation scheme.

Energy- and Performance-Aware Incremental Mapping for Networks on Chip With Multiple Voltage Levels

The One-Dimensional Space Allocation Problem ODSAP arises when locating rooms along a corridor, when setting books on a shelf, when allocating information items among the "cylinders" of a magnetic disk, or when storing products in certain warehouses The rooms objects to be located have a known length or width and, for each pair of rooms, there is a given number of "trips" between them; the problem is to find a sequencing of the rooms which minimizes the total traveled distance This paper deals first with two particular cases In the rooted tree case, previous results obtained by Adolphson and T C Hu for the Linear Ordering Problem are extended and provide efficient solution methods In addition, it is shown that most of the assumptions made by Adolphson and Hu are necessary, since relaxing any of them leads to an NP-complete problem In the case of independent destinations solved by Bergmans and Pratt when all lengths are equal, a unimodality property is shown for an optimum solution, but this is not sufficient to lead to a "good" algorithm and the problem is shown to be NP-complete For the general problem, an exact algorithm using dynamic programming is described and computationally tested

On the One-Dimensional Space Allocation Problem

In this paper we study the problem of scheduling the read/write head movement to handle a batch of $nI/O$ requests in a 2-dimensional secondary storage device in minimum time. Two models of storage systems are assumed in which the access time of a record (being proportional to the “distance” between the position of the record and that of the read/write head) is measured in terms of $L_1 $ and $L_\infty $ metrics, respectively. The scheduling problem, referred to as the Open Path Problem (OPP), is equivalent to finding a shortest Hamiltonian path with a specified end point in a complete graph with n vertices. We first show in this paper that there exists a natural isometry between the $L_1 $ and $L_\infty $ metrics. Consequently, the existence of a polynomial time algorithm for the OPP in one metric implies the existence of a polynomial time algorithm for the same problem in the other metric. Based on a result by Garey, Graham and Johnson, it is easy to show that the OPP in $L_1 $ (hence in $L_\infty $) me...

Voronoi Diagrams in L1 (L∞) Metrics with 2-Dimensional Storage Applications

Given a set of records with known access probabilities, the problems of record placement, page allocation, and dynamic record placement to mimmlze read/write head movement for sequential as well as batched processing are discussed. The distance traveled by the head in the systems under consideration is modeled by the rectilinear (Ll), Euclidean (L2), and maximum (L=) metrics. Some open problems are mentioned at the end of the paper.

Minimizing Expected Head Movement in One-Dimensional and Two-Dimensional Mass Storage Systems

We consider the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized. A simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric. For the rectilinear and maximum metrics, we show that there is no such heuristic. For the special case in which all access probabilities are equal, however, heuristics within an additive constant of optimal do exist, and their implementation requires solution of differential equations for which we give numerical solutions.

A. C. McKellar

Papers

Near-Optimal Solutions to a 2-Dimensional Placement Problem