Relabel-to-front Algorithm
Last Updated :
11 Jul, 2025
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).
- 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.
- 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.
Advantages:
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.
Disadvantages:
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.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem