Dynamic Matchings in Convex Bipartite Graphs 1st Edition by Gerth Stolting Brodal, Loukas Georgiadis, Kristoffer Arnsfelt Hansen, Irit Katriel ISBN 9783540744566

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ISBN 10: 
ISBN 13: 9783540744566
Author: Gerth Stolting Brodal, Loukas Georgiadis, Kristoffer Arnsfelt Hansen, Irit Katriel

We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in O(log²|V|) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worst-case time. Our structure can report the mate of a matched vertex in the maximum matching in worst-case O( min { k log²|V| + log|V|, |V| log|V|}) time, where k is the number of update operations since the last query for the same pair of vertices was made. I

Dynamic Matchings in Convex Bipartite Graphs 1st Table of contents:

• Introduction
  1.1 Motivation and Background
  1.2 Problem Definition and Objectives
  1.3 Key Contributions
  1.4 Organization of the Paper

• Preliminaries
  2.1 Bipartite Graphs: Basic Concepts
  2.2 Dynamic Matchings in Graphs
  2.3 Convexity in Graph Theory
  2.4 Related Work and Challenges

• Convex Bipartite Graphs
  3.1 Definition of Convex Bipartite Graphs
  3.2 Properties and Structure of Convex Bipartite Graphs
  3.3 Applications of Convexity in Graph Theory
  3.4 Examples of Convex Bipartite Graphs

• Dynamic Matchings: Formal Definitions
  4.1 Definition of Dynamic Matching
  4.2 Problem Setup and Assumptions
  4.3 Dynamic Matching Algorithms
  4.4 Challenges in Dynamic Environments

• Algorithms for Dynamic Matchings in Convex Bipartite Graphs
  5.1 General Framework for Dynamic Matching
  5.2 Algorithmic Approaches for Convex Bipartite Graphs
  5.3 Update Operations in Dynamic Matching
  5.4 Efficiency and Optimality Conditions

• Analysis of Dynamic Matchings
  6.1 Time Complexity of Dynamic Algorithms
  6.2 Approximation Guarantees and Bounds
  6.3 Stability and Robustness of Matchings
  6.4 Comparison with Static Matchings

• Applications and Case Studies
  7.1 Dynamic Matching in Network Flow Problems
  7.2 Real-Time Allocation Problems
  7.3 Computational Complexity in Dynamic Systems
  7.4 Case Studies and Experimental Results

• Discussion and Future Directions
  8.1 Insights from Dynamic Matching in Convex Graphs
  8.2 Open Problems and Challenges
  8.3 Future Research Directions

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