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# Relabel-to-front Algorithm

The relabel-to-front algorithm is used to find the maximum flow in the network. The relabel-to-front algorithm is more efficient than the generic push-relabel method. In the push-relabel method, we can apply the basic operations of push and relabel in any order. The relabel-to-front algorithm chooses the order carefully and manages the network data structures efficiently.

First, we need to understand the basic operations i.e. push and relabel
Each vertex in the network has 2 variables associated with it which are height variable(h) and excess flow(e)

1. Push: If a vertex has excess flow and there is an adjacent node with lower height (in the residual graph) then we push flow from the vertex to the lower height node.
2. Relabel: If a vertex has excess flow and no adjacent node with lower height is available then we use relabel operation to increase the height of vertex so that it can perform push operation.

The relabel-to-front algorithm maintains a list of vertices in the network. It starts from the beginning of the list and repeatedly selects an overflowing vertex u and performs discharge operation on it.
Discharge operation is performing push and relabel operation until vertex u has no positive excess flow(e)
If a vertex is relabeled, it is moved to the front of the list, and the algorithm scans again.

Algorithm:

• Initialize the preflow and heights to the same values as in the generic push-relabel algorithm.
• Initialize list L which contains all vertices except source and sink.
• Initialize the current pointer of each vertex u to the first vertex in u’s neighbour list N. The neighbour list N contains those vertices for which there is a residual edge.
• While algorithm reaches the end of list L
• Select the vertex u from list L and perform discharge operation.
• If u were relabeled by discharge then move u to the front of the list.
• If u was moved to the front of the list, the vertex in the next iteration is the one following u in its new position in the list.

Example:
Consider the given flow network. On the right is shown the initial list L=(B, C) where initially u=B.

After the preflow initialization operation. Under each vertex in list L is its neighbor list N with the current neighbor circled.

Vertex B undergoes discharge operation as it has excess flow 3 (e=3). Vertex B has no node with lower height thus it performs relabel operation (h=1) and pushes flow 1 to vertex C.

Vertex B still has excess flow 2(e=2) thus it performs relabel operation (h=5)and pushes flow 2 to vertex A. Since vertex B is relabeled it remains in the front of the list. Now vertex C undergoes discharge operation as it has excess flow 1(e=1).

Vertex C performs relabel operation(h=1) and pushes flow 1 to node D.Since vertex C performed relabel operation it is moved to the front of the list.

Vertex B now follows vertex C in L but B has no excess flow. The RELABEL-TO-FRONT has reached the end of list L and terminates. There are no overflowing vertices, thus the preflow is a maximum flow. Here max flow is 1.

Time Complexity: Runs in O(V3) time on network G = (V, E). Thus it is more efficient than the generic push-relabel algorithm which runs in O(V2E) time.

Fast execution time: The RTF algorithm has a running time of O(V^3), where V is the number of vertices in the graph. Although this is not the fastest algorithm for maximum flow, it is fast enough for many practical applications.

Simplicity: The RTF algorithm is relatively easy to understand and implement compared to some other maximum flow algorithms such as the Push-Relabel algorithm.

Guaranteed optimality: The RTF algorithm always computes the maximum flow and can also compute the minimum cut of the graph.

Low memory requirements: The RTF algorithm has a low memory footprint, making it suitable for use in applications with limited memory resources.

Poor performance on some graphs: The RTF algorithm may perform poorly on some graphs, especially those with high density or graphs with a large difference between the maximum and minimum capacities.

Inefficient in handling large graphs: Although the RTF algorithm is fast enough for many practical applications, it may become inefficient when dealing with large graphs due to its O(V^3) running time.

The algorithm is not always easy to parallelize: The RTF algorithm is difficult to parallelize, which limits its usefulness in some applications that require high-performance computing.

Not suitable for dynamic graphs: The RTF algorithm is not suitable for dynamic graphs where the graph structure changes frequently since it requires significant computational overhead to maintain the data structures.

In summary, the RTF algorithm is a simple and efficient algorithm for solving the maximum flow problem in a directed graph, but its performance may suffer on some types of graphs, and it may not be suitable for dynamic graphs or parallel computing.