Generalized dynamic storage allocation

In today's rapidly changing field of Management Information Systems (MIS) one problem faced by organizations is the consumption of storage capacity due to the growing base of software assets. Research has shown that very few firms effectively monitor program usage. Storage management issues arise due to the fact that many of the programs occupying valuable storage space are used infrequently. In this dissertation, we apply Simon's model of information usage in order to model the dynamic behavior of program usage. This methodology enables organizations to identify the changing usage frequencies of software assets. We propose a classification scheme which MIS personnel can use in order to effectively monitor their program usage tendencies. This classification scheme may then serve as a basis for storage allocation decision-making. Through a study of the dynamic behavior of programs, we have formulated a minimum cost model for hierarchical storage allocation. We will show the value of incorporating dynamic usage frequencies into algorithms which have traditionally considered only a static view.

Approximability of the Ground State Problem for Certain Ising Spin Glasses

The channel assignment problem in ad hoc wireless networks is investigated. The problem is to assign channels to hosts in such a way that interference among hosts is eliminated and the total number of channels is minimised. Interference is caused by direct collisions from hosts that can hear each other or indirect collisions from hosts that cannot hear each other, but simultaneously transmit to the same destination. A new class of disk graphs (FDD: interFerence Double Disk graphs) is proposed that include both kinds of interference edges. Channel assignment in wireless networks is a vertex colouring problem in FDD graphs. It is shown that vertex colouring in FDD graphs is NP-complete and the chromatic number of an FDD graph is bounded by its clique number times a constant. A polynomial time approximation algorithm is presented for channel assignment and an upper bound 14 on its performance ratio is obtained. Results from a simulation study reveal that the new graph model can provide a more accurate estimation of the number of channels required for collision avoidance than previous models.

A method is proposed for estimating the numerical rank of a sparse matrix. The method uses orthogonal factorization along with a one-norm incremental condition estimator that is an adaptation of the LINPACK estimator. This approach allows the use of static storage allocation as is used in SPARSPAK-B, whereas there is no known way to implement column pivoting without dynamic storage allocation. It is shown here that this approach is probably more accurate than the method presently used by SPARSPAK-B. The method is implemented with an overhead of $O(n_U \log n)$ operations, where $n_U $ is the number of nonzeros in the upper triangular factor of the matrix. In theory, it can be implemented in $O(\max \{ n_U ,n\log n\} )$ operations, but this requires the use of a complicated data structure.It is shown how a variant of this strategy may be implemented on a message-passing architecture. A prototype implementation is done and tests show that the method is accurate and efficient.Ways in which the condition esti...

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.

In this chapter we discuss the definitions, results and open questions that appear in Golumbic and Monma (1982) and Golumbic, Monma, and Trotter (1984), the papers which introduced the topic of tolerance graphs. We also present consequences of these results and related topics from more recent literature. Notation and observations Recall from Section 1.3 that a graph G is a tolerance graph if each vertex ν ∈ V ( G ) can be assigned a closed interval I ν and a tolerance t ν ∈ R + so that xy ∈ E ( G ) if and only if | I x ∩ I y | ≥ min{ t x , t y }. If graph G has a tolerance representation with t ν ≤ | I ν | for all ν ∈ V ( G ), then G is called a bounded tolerance graph . Many important graph properties are inherited by all induced subgraphs and thus called hereditary properties. Given a (bounded) tolerance representation 〈 I , t 〉 of a graph G , for any subset of vertices W ⊆ V ( G ) the intervals { I w | w ∈ W } and tolerances { t w | w ∈ W } give a representation of G W . Thus, induced subgraphs of tolerance graphs are also tolerance graphs and induced subgraphs of bounded tolerance graphs are also bounded tolerance graphs. We record this as a remark. Remark 2.1. The property of being a tolerance graph (resp. bounded tolerance graph) is hereditary. In a tolerance representation of a graph G , we may have intervals of the form I x = [ a x , a x ].

Given a graph G = ( V, E ) and positive integral vertex weights w : V → N , the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , …, C k , minimize ∑ i =1 k max v ∈ C i w ( v ). This problem, restricted to interval graphs, arises whenever there is a need to design dedicated memory managers that provide better performance than the general-purpose memory management of the operating system. Though this problem seems similar to the dynamic storage allocation problem, there are fundamental differences. We make a connection between max-coloring and online graph coloring and use this to devise a simple 2-approximation algorithm for max-coloring on interval graphs. We also show that a simple first-fit strategy, that is a natural choice for this problem, yields an 8-approximation algorithm. We show this result by proving that the first-fit algorithm for online coloring an interval graph G uses no more than 8 ċ χ( G ) colors, significantly improving the bound of 26 ċ χ( G ) by Kierstead and Qin [1995]. We also show that the max-coloring problem is NP-hard. The problem of online coloring of intervals with bandwidths is a simultaneous generalization of online interval coloring and online bin packing. The input is a set I of intervals, each interval i ∈ I having an associated bandwidth b ( i ) ∈ (0, 1]. We seek an online algorithm that produces a coloring of the intervals such that for any color c and any real r , the sum of the bandwidths of intervals containing r and colored c is at most 1. Motivated by resource allocation problems, Adamy and Erlebach [2003] consider this problem and present an algorithm that uses at most 195 times the number of colors used by an optimal offline algorithm. Using the new analysis of first-fit coloring of interval graphs, we show that the Adamy-Erlebach algorithm is 35-competitive. Finally, we generalize the Adamy-Erlebach algorithm to a class of algorithms and show that a different instance from this class is 30-competitive.

Many challenging scheduling, planning, and resource allocation problems come with real-world input data and hard problem constraints, and reduce to optimizing a cost function over a combinatorially defined feasible set, such as colorings of a graph. Toward tackling such problems with quantum computers using quantum approximate optimization algorithms, we present novel efficient quantum alternating operator ansatz (QAOA) constructions for optimization problems over proper colorings of chordal graphs. As our primary application, we consider the flight-gate assignment problem , where flights are assigned to airport gates as to minimize the total transit time of all passengers, and feasible assignments correspond to proper graph colorings of a conflict graph derived instancewise from the input data. We leverage ideas from classical algorithms and graph theory to show our constructions have the desirable properties of restricting quantum state evolution to the feasible subspace, and satisfying a particular reachability condition for most problem parameter regimes. Using classical preprocessing we show that we can always find and construct a suitable initial quantum (superposition) state efficiently. We show our constructions in detail, including explicit decompositions to a universal set of basic quantum gates, which we use to bound the required resource scaling as low-degree polynomials of the input parameters. In particular, we derive novel QAOA mixing operators and show that their implementation cost is commensurate with that of the QAOA phase operator for flight-gate assignment. A number of quantum circuit diagrams are included such that our constructions may be used as a template toward development and implementation of quantum gate-model approaches for a wider variety of potentially impactful real-world applications.

In computer science, dynamic storage allocation refers to the mechanisms for managing primary and secondary storage under a fluctuating demand. In this paper we identify three principal sources of inefficient use of storage: partition fragmentation, checkerboarding and internal fragmentation. The first arises when a set of records, to be stored in a set of independent storage units, cannot be partitioned into subsets whose cumulative sizes are equal to the capacities of the storage units. Checkerboarding refers to the alternating holes and occupied regions of a storage device which occur as records come and go unpredictably. Finally, internal fragmentation occurs in systems that allocate storage only in certain prescribed amounts, e.g. the nearest power of two no smaller than the amount requested. In connection with these types of fragmentation, a number of algorithm and storage design questions arise. Our basic goal is to review analyses of combinatorial models of these problems. The major theme of the results is the characterization of approximation algorithms in terms of asymptotic bounds on their performance relative to optimization rules. Our presentation is distinguished by its attempt to bring together for the first time many disparate research efforts in dynamic storage allocation, classifying them by the types of fragmentation studied.

Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The condition which we follow in graph coloring is that the incident vertices/edges have not the same color. There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and others are online algorithm. Where offline means the graph is known in advance and the online means that the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is added to the graph and the main issue is that we want to minimize the number of colors. We cannot change the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors online in multiple rounds.

First-Fit coloring of bounded tolerance graphs

Dynamic storage allocation with limited compaction - complexity and some practical implications

Coloring and domination in graphs are two well explored areas of research in graph theory. Blending these notions, the dominator coloring of graphs was introduced in the literature; following which several variants of domination related coloring patterns have been defined and studied, based on different types of coloring and domination in graphs. A vertex coloring of a graph that demands the existence of a path in which every internal vertex between two vertices has a unique color is called a rainbow vertex coloring of the graph. In this article, we investigate the rainbow dominator coloring of graphs; a vertex coloring that combines the concepts of rainbow vertex coloring and dominator coloring of graphs. We discuss some properties of the rainbow dominator coloring of graphs and determine the rainbow dominator chromatic number of certain classes of graphs and their complements.

We consider a single machine scheduling problem with multiple maintenance activities, where the maintenance duration function is of the linear form f(t) = a+bt with a ≥ 0 and b > 1. We propose an approximation algorithm named FFD-LS2I with a worst-case bound of 2 for problem. We also show that there is no polynomial time approximation algorithm with a worst-case bound less than 2 for the problem with b ≥ 0 unless P = NP, which implies that the FFD-LS2I algorithm is the best possible algorithm for the case b > 1 and that the FFD-LS algorithm, which is proposed in the literature, is the best possible algorithm for the case b ≤ 1 both from the worst-case bound point of view.

Given a graph G = (V,E) and a set R ⊆ V×V of requests, we consider to assign a set of edges to each node in G so that for every request (u, v) in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each node. In this paper, we give an advanced investigation about the difficulty of MCD by focusing on the relationship between its (in)approximability and request structures. We first show that MCD with general R has Θ(logn) lower and upper bounds on approximation ratio under the assumption P ≠ NP, where n is the number of nodes in G. We then assume R forms a clique structure, called Subset-Full, which is a natural setting in the context of the application. Interestingly, under this natural setting, MCD becomes to be 2-approximable, though it has still no polynomial time approximation algorithm whose factor better than 677/676 unless P = NP. Finally, we show that this approximation ratio can be improved to 3/2 for undirected variant of MCD with Subset-Full.