Push-relabel algorithm

Definition

The push-relabel algorithm is a method used for solving the maximum flow problem in a flow network by maintaining a preflow condition and adjusting the flow through local operations. Instead of relying solely on augmenting paths, it uses two main operations: pushing excess flow from a vertex to its neighbors and relabeling vertices to change their heights, allowing for more efficient flow adjustments. This algorithm is particularly advantageous in networks where the number of vertices is large and has been widely applied due to its efficient handling of complex flow scenarios.

5 Must Know Facts For Your Next Test

1. The push-relabel algorithm was developed by Andrew V. Goldberg and Robert E. Tarjan in 1986 and has since become one of the primary methods for solving maximum flow problems.
2. The algorithm's efficiency comes from local adjustments of flow rather than global searches for augmenting paths, making it particularly effective in dense graphs.
3. Push-relabel operates in two phases: the initialization phase where the heights of vertices are set, and the main phase where pushes and relabels are executed until no more pushes are possible.
4. The time complexity of the push-relabel algorithm can be as low as O(V^3) or O(V^2E) depending on the implementation, where V is the number of vertices and E is the number of edges.
5. Unlike augmenting path algorithms, which may require examining many paths, push-relabel often quickly finds an optimal flow with fewer overall operations.

Review Questions

• How does the push-relabel algorithm differ from traditional augmenting path algorithms in solving maximum flow problems?
  • The push-relabel algorithm differs from traditional augmenting path algorithms by focusing on local adjustments rather than searching for global paths. While augmenting path methods like Ford-Fulkerson rely on finding paths with available capacity, push-relabel maintains a preflow condition and allows excess flows at vertices to be pushed to neighbors. This results in potentially faster convergence, especially in dense networks where numerous paths may exist.
• In what situations would you prefer using the push-relabel algorithm over other maximum flow algorithms?
  • The push-relabel algorithm is preferred in scenarios involving dense graphs or networks with many vertices due to its ability to efficiently manage flow through localized operations. Additionally, when the capacity of edges varies significantly or when multiple paths exist between source and sink, push-relabel can quickly optimize flows without exhaustively searching for every possible path. Its effectiveness in these situations makes it a popular choice among various maximum flow solutions.
• Evaluate how the implementation of height functions influences the efficiency of the push-relabel algorithm in determining flow.
  • The implementation of height functions in the push-relabel algorithm significantly enhances its efficiency by guiding how flows are adjusted between vertices. By assigning heights based on distances from the sink, it helps direct pushes toward lower-height neighbors, minimizing unnecessary operations. This strategy not only optimizes each step but also reduces overall iterations required to reach maximum flow, making it particularly effective in complex networks where traditional methods might struggle.

Related terms

Max Flow Min Cut Theorem: A fundamental theorem that states that the maximum value of flow from a source to a sink in a flow network is equal to the capacity of the minimum cut separating the source and sink.

Preflow: A condition in a flow network where the flow into a vertex can exceed the flow out, leading to excess flow that can be pushed to neighboring vertices.

Height Function: A function that assigns a height to each vertex in a flow network, used to determine how far away a vertex is from the sink and influencing the push-relabel operations.