Push-relabel algorithm

Definition

The push-relabel algorithm is a method used to solve the maximum flow problem in networks. It works by maintaining a preflow, where excess flow can be pushed from nodes to their neighbors, and uses a relabeling technique to adjust the heights of nodes, enabling the movement of flow through the network. This algorithm is particularly efficient for dense networks and can find optimal solutions by iteratively pushing flow until no more pushes can be made, ensuring that all flow is maximized while satisfying capacity constraints.

5 Must Know Facts For Your Next Test

1. The push-relabel algorithm operates with two main operations: 'push', which transfers excess flow from one vertex to an adjacent vertex, and 'relabel', which increases the height of a vertex when no more pushes can occur.
2. This algorithm can handle graphs with both directed and undirected edges, making it versatile for different types of network flows.
3. It typically performs better than other algorithms like Ford-Fulkerson in dense graphs due to its ability to handle large amounts of flow efficiently.
4. The termination condition of the push-relabel algorithm is when there are no active vertices left with excess flow, indicating that all possible flows have been pushed to the sink.
5. The complexity of the push-relabel algorithm is O(V^2 * E) for general graphs, where V is the number of vertices and E is the number of edges.

Review Questions

• How does the push-relabel algorithm ensure that it finds the maximum flow in a network?
  • The push-relabel algorithm ensures that it finds the maximum flow by maintaining a preflow and continuously pushing excess flow from vertices to neighboring vertices until no more pushes can be made. By utilizing the relabel operation, it adjusts the heights of nodes dynamically, allowing for better routing of flows through the network. The algorithm terminates when there are no active vertices with excess flow, confirming that all possible flows have been maximized within capacity constraints.
• Discuss the advantages of using the push-relabel algorithm over other maximum flow algorithms, particularly in dense networks.
  • One significant advantage of the push-relabel algorithm over others like Ford-Fulkerson is its efficiency in dense networks. In scenarios where there are many edges relative to vertices, push-relabel can achieve better performance due to its ability to process multiple flows simultaneously through relabeling and pushing operations. Additionally, since it doesn't rely on augmenting paths like Ford-Fulkerson does, it can reduce the time complexity significantly in cases where many paths exist, making it a preferred choice for high-density graph problems.
• Evaluate how modifying the push-relabel algorithm could improve its performance in specific applications or network types.
  • Modifying the push-relabel algorithm could enhance its performance by incorporating heuristics or priority rules that dictate which vertices should be processed first based on their excess flow levels or heights. By prioritizing nodes with higher excess flows or lower heights for relabeling, one can potentially minimize unnecessary operations and speed up convergence to optimal solutions. Additionally, adapting this algorithm for specialized network types, such as bipartite graphs or transportation networks, could leverage unique properties of these structures to further streamline computations and reduce complexity.

Related terms

Maximum Flow Problem: A problem in network flow theory that seeks to find the greatest possible flow from a source node to a sink node while respecting capacity limits on the edges.

Preflow: A flow configuration where the flow into a node may exceed the flow out of it, allowing nodes to accumulate excess flow temporarily until it can be pushed to other nodes.

Network Capacity: The maximum amount of flow that can pass through an edge in a network, which serves as a constraint in flow problems.