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Graph partitioning involves dividing a graph into smaller, non-overlapping subgraphs while minimizing the number of edges between them, often to optimize parallel processing or improve computational efficiency. This technique is crucial in various fields such as computer science, network analysis, and scientific computing, where it helps in load balancing, reducing communication overhead, and enhancing data locality.
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Edge connectivity of a graph is the minimum number of edges that need to be removed to disconnect the graph. It provides insight into the robustness of a network, indicating how resilient it is to edge failures.
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Vertex connectivity of a graph is the minimum number of vertices that need to be removed to make the graph disconnected or reduce it to a trivial graph. It provides a measure of the graph's robustness against vertex removal and is an important parameter in understanding network resilience.
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The Max-Flow Min-Cut Theorem states that in a flow network, the maximum value of flow from a source to a sink is equal to the total weight of the edges in the smallest cut that separates the source and sink. This theorem provides a powerful way to analyze network flow problems, ensuring that the flow is optimal and no more flow can be achieved without increasing the capacity of the network or changing its structure.
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A Network Flow Algorithm is like a game where we try to move as much water as possible through pipes from a big water tank to another, without breaking the pipes. It helps us figure out the best way to send things from one place to another, like sending letters or cars, making sure everything goes smoothly and quickly.
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Disjoint sets are collections of sets that have no elements in common, meaning their intersection is an empty set. They are fundamental in various fields such as computer science, particularly in algorithms involving union-find operations and partitioning problems.
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Capacity constraint refers to the limitations that restrict the output or throughput of a system, process, or organization, often acting as a bottleneck that impacts overall efficiency and productivity. Identifying and managing these constraints is crucial for optimizing performance and achieving strategic objectives in various industries.
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The Maxflow-Mincut Theorem states that in a flow network, the maximum value of the feasible flow from a source to a sink is equal to the minimum capacity that, when removed, disconnects the source from the sink. This theorem bridges optimization and graph theory, providing a powerful tool for solving network flow problems.
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