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Graph theory with applications

01 Jan 1976-
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Citations
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Journal ArticleDOI
TL;DR: When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput /spl lambda/(n) obtainable by each node for a randomly chosen destination is /spl Theta/(W//spl radic/(nlogn)) bits persecond under a noninterference protocol.
Abstract: When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput /spl lambda/(n) obtainable by each node for a randomly chosen destination is /spl Theta/(W//spl radic/(nlogn)) bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is /spl Theta/(W/spl radic/An) bit-meters per second. Thus even under optimal circumstances, the throughput is only /spl Theta/(W//spl radic/n) bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.

9,008 citations

Journal ArticleDOI
TL;DR: This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.
Abstract: We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least.87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

3,932 citations

Journal Article
TL;DR: In this survey I have collected everything I could find on graph labelings techniques that have appeared in journals that are not widely available.
Abstract: A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

2,367 citations

Book
20 May 2005
TL;DR: In this paper, the mathematical underpinnings of robot motion are discussed and a text that makes the low-level details of implementation to high-level algorithmic concepts is presented.
Abstract: A text that makes the mathematical underpinnings of robot motion accessible and relates low-level details of implementation to high-level algorithmic concepts. Robot motion planning has become a major focus of robotics. Research findings can be applied not only to robotics but to planning routes on circuit boards, directing digital actors in computer graphics, robot-assisted surgery and medicine, and in novel areas such as drug design and protein folding. This text reflects the great advances that have taken place in the last ten years, including sensor-based planning, probabalistic planning, localization and mapping, and motion planning for dynamic and nonholonomic systems. Its presentation makes the mathematical underpinnings of robot motion accessible to students of computer science and engineering, rleating low-level implementation details to high-level algorithmic concepts.

1,811 citations

Proceedings ArticleDOI
01 Aug 1999
TL;DR: The first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination are described.
Abstract: We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.

1,537 citations

References
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Journal ArticleDOI
TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

22,704 citations

Book
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher: With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter. 0201000296B04062001

9,262 citations

Journal ArticleDOI
01 Feb 1956
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409

5,104 citations

Journal ArticleDOI
TL;DR: In this paper, algorithms for the solution of the general assignment and transportation problems are presen, and the algorithm is generalized to one for the transportation problem.
Abstract: In this paper we presen algorithms for the solution of the general assignment and transportation problems. In Section 1, a statement of the algorithm for the assignment problem appears, along with a proof for the correctness of the algorithm. The remarks which constitute the proof are incorporated parenthetically into the statement of the algorithm. Following this appears a discussion of certain theoretical aspects of the problem. In Section 2, the algorithm is generalized to one for the transportation problem. The algorithm of that section is stated as concisely as possible, with theoretical remarks omitted.

3,918 citations

Book ChapterDOI
TL;DR: In this paper, the problem of finding a maximal flow from one given city to another is formulated as follows: "Consider a rail network connecting two cities by way of a number of intermediate cities, where each link has a number assigned to it representing its capacity".
Abstract: Introduction. The problem discussed in this paper was formulated by T. Harris as follows: “Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”

2,731 citations