FIFO Push Relabel Algorithm in C++

The FIFO Push-Relabel algorithm is an efficient method for solving the maximum flow problem in network flow optimization. This algorithm is a variant of the Push-Relabel algorithm, which is designed to determine the maximum amount of flow that can be sent through a network from a designated source node to a target (sink) node. The term "Push-Relabel" refers to the two main operations involved in the algorithm: pushing excess flow along edges and relabeling nodes to ensure a feasible flow.

Introduction:

In the context of the FIFO Push-Relabel algorithm, the "FIFO" (First-In-First-Out) aspect comes into play when choosing which excess flow to push. It ensures that the algorithm maintains a fair order in pushing excess flow out of nodes, which can contribute to improved performance.

The basic idea behind the Push-Relabel algorithms, including FIFO Push-Relabel, is to maintain a preflow, an intermediate flow assignment that satisfies certain constraints, and iteratively refine it until it becomes a valid maximum flow. During each iteration, the algorithm pushes excess flow from nodes with surplus to neighboring nodes and relabels nodes to maintain feasibility.

Push-Relabel Algorithm:

• Push Operation: In the Push-Relabel algorithm, the order in which excess flow is pushed is not strictly defined. It can be based on various rules, such as choosing the node with the highest height or using a "first-in-first-out" (FIFO) order.
• Pivoting Rules: Push-Relabel algorithms typically involve selecting a node with excess flow and pushing flow to one of its neighboring nodes. The selection of the node to push from and the destination node can vary based on pivoting rules.
• Heuristic Rules: Various heuristics and rules can be applied to decide when to push flow and when to relabel nodes. These rules influence the efficiency and convergence of the algorithm.

FIFO Push-Relabel Algorithm:

• FIFO Queue: The key distinction in FIFO Push-Relabel is the use of a First-In-First-Out (FIFO) queue to determine the order in which nodes with excess flow are processed during each iteration.
• Push Operation Order: Excess flow is pushed in the order in which nodes entered the queue, ensuring a fair distribution of pushing operations among nodes with excess flow.
• Fairness Aspect: The FIFO ordering adds a fairness aspect to the algorithm, potentially reducing the likelihood of certain nodes being starved of excess flow.

Program:

Let us take an example to illustrate the use of FIFO Push Relabel Algorithm in C++.

Output:

Enter the number of nodes and edges: 4 5
Enter the edges in the format (from to capacity):
0 1 3
0 2 2
1 2 1
1 3 1
2 3 3
Enter the source and sink nodes: 0 3
Maximum Flow: 4

Explanation:

• Graph Representation:

The algorithm works on a directed graph that models a flow network. Nodes represent entities, and edges represent connections with a capacity indicating the maximum flow that can traverse the edge.

• Initialization:

The algorithm initializes various data structures:

excess: It tracks the excess flow at each node.

height: It represents the height (distance from the source) of each node.

seen: It keeps track of the current neighbor being considered during the discharge operation.

activeNodes: A queue for maintaining nodes with excess flow that need to be processed.

• Push Operation:

The main operation is "pushing" excess flow from a node with surplus to its neighboring nodes. It is done along edges that have available capacity. The flow is adjusted both on the forward and reverse edges.

• Relabel Operation:

When pushing is not possible (due to height restrictions), a "relabel" operation is performed on the node. It increases its height to a level that allows for potential future pushes.

• Discharge Operation:

The discharge operation is the core of the algorithm. It repeatedly attempts to push excess flow out of a node until the excess becomes zero. It involves choosing a neighbor to push flow to or relabeling the current node.

• FIFO Queue:

The FIFO aspect ensures that nodes are processed in a First-In-First-Out order. It helps maintain fairness in dealing with excess flow at nodes, potentially improving the overall efficiency of the algorithm.

• Maximum Flow Calculation:

The algorithm starts by initializing the source node with excess flow and height. After that, it iteratively performs the discharge operation on active nodes in the queue until no more nodes remain.

• Result:

After the algorithm completes, the maximum flow is determined by checking the excess flow at the sink node.

Time Complexity:

• Worst-Case Time Complexity: The worst-case time complexity of the Push-Relabel algorithm, including FIFO Push-Relabel, is generally considered to be O(V^3), where V is the number of vertices in the graph. This worst-case complexity arises when the algorithm needs to perform a significant number of relabeling operations.
• Average-Case Time Complexity: In practice, the average-case time complexity is often better than the worst-case, and the algorithm can perform well on various instances. The complexity depends on factors like the structure of the graph, the choice of pivoting rules, and the initial heights assigned to nodes.

Space Complexity:

• Memory Usage: The space complexity of the algorithm is primarily determined by the data structures used to store information about the graph and the algorithm's progress. In the provided code, the space complexity is dominated by:
• The graph representation, requiring O(V + E) space, where V is the number of vertices and E is the number of edges.
• Arrays to store excess flow (excess), node heights (height), and seen nodes (seen), each requiring O(V)
• The FIFO queue (activeNodes), which can take up to O(V) space in the worst case.
• Total Space Complexity: Combining these factors, the total space complexity of the algorithm is O(V + E).

Advantages:

There are several advantages of the FIFO Push Relabel Algorithm in C++. Some main advantages of the FIFO Push Relabel Algorithm are as follows:

1. Efficiency: The FIFO Push-Relabel algorithm is known for its efficiency in finding the maximum flow in a network. By maintaining a First-In-First-Out order in the queue during the pushing phase, the algorithm often achieves good performance in practice.
2. Adaptability: The algorithm is adaptable to various types of networks and can handle graphs with irregular structures. Its versatility makes it applicable to a wide range of real-world problems, including those with dynamic characteristics.
3. Fairness: The use of a FIFO queue in the pushing phase introduces a fairness aspect to the algorithm. It helps distribute the pushing of excess flow among nodes more evenly, potentially avoiding the starvation of certain nodes.
4. Simplicity of Implementation: The FIFO Push-Relabel algorithm is relatively straightforward to implement compared to some other maximum flow algorithms. The key operations (pushing, relabeling, and discharging) are conceptually clear, making it accessible for implementation and understanding.
5. Parallelization: The Push-Relabel algorithms, including FIFO Push-Relabel, can be parallelized efficiently. It makes them suitable for implementation on parallel computing architectures, taking advantage of multi-core processors or distributed computing environments.

Applications:

There are several applications of the FIFO Push Relabel Algorithm in C++. Some main applications of the FIFO Push Relabel Algorithm are as follows:

1. Network Flow Optimization: The primary application of FIFO Pushing Relabel algorithm is optimizing the flow of resources through networks. It includes transportation networks, communication networks, and supply chain management. The algorithm helps find the most efficient way to move goods, information, or other resources from a source to a destination.
2. Circulation Problems: In circulation problems, the goal is to find a feasible flow in a network that satisfies certain constraints. It can be applied in the design and optimization of systems like traffic circulation in road networks or the circulation of goods in a manufacturing process.
3. Image Segmentation: Image segmentation involves partitioning an image into regions or objects. The maximum flow algorithm can be applied to solve certain image segmentation problems, where the goal is to identify and separate different regions based on certain criteria.
4. Task Assignment: In project management or task assignment scenarios, the algorithm can be used to optimize the allocation of resources or tasks to different individuals or teams, ensuring the efficient completion of a project.
5. Telecommunication Network Design: The algorithm can be employed in designing and optimizing telecommunication networks, determining the most efficient way to transmit data between different network nodes while considering capacity constraints.
6. Airline Scheduling: In airline scheduling, the maximum flow algorithm can help optimize flight schedules, considering factors such as aircraft capacity, airport capacities, and the demand for air travel between different locations.
7. Water Flow in Irrigation Systems: The algorithm can be applied to optimize water flow in irrigation systems, ensuring that water is distributed efficiently to different fields while respecting capacity constraints and minimizing wastage.
8. Biomedical Applications: In biomedical research, the algorithm can be used in the analysis of biological networks, such as metabolic pathways or gene regulatory networks. It can help model and optimize the flow of substances or information within biological systems.
9. Resource Allocation in Cloud Computing: In cloud computing environments, the algorithm can assist in optimizing the allocation of computing resources to different tasks or services, considering capacity constraints and workload variations.

Disadvantages:

There are several disadvantages of the FIFO Push Relabel Algorithm in C++. Some main disadvantages of the FIFO Push Relabel Algorithm are as follows:

1. Worst-Case Time Complexity: The worst-case time complexity of the FIFO Push-Relabel algorithm can be relatively high, particularly on certain types of graphs. In practice, the algorithm performs well on average, but worst-case scenarios can be a concern in critical applications.
2. Sensitivity to Initial Heights: The algorithm's performance can be sensitive to the initial heights assigned to nodes. Poor choices in height initialization may lead to increased relabeling operations, affecting the overall efficiency of the algorithm.
3. Choice of Pivoting Rule: The performance of the algorithm is influenced by the choice of pivoting rules for selecting nodes during the pushing and relabeling operations. Different rules may lead to different convergence rates and overall efficiency.
4. Memory Requirements: The algorithm may require significant memory, especially in scenarios with large graphs. The storage of excess flows, heights, and other data structures can lead to increased memory usage, which may be a concern in resource-constrained environments.
5. Not Always the Fastest Algorithm: While the FIFO Push-Relabel algorithm is efficient in many cases, it may not always be the fastest algorithm for maximum flow problems. Depending on the characteristics of the network, other algorithms like the Edmonds-Karp algorithm or the Dinic's algorithm might outperform it.
6. Complexity in Dynamic Networks: Adapting the algorithm to dynamic networks, where capacities or the network structure change over time, can add complexity. Maintaining the preflow and adjusting the algorithm dynamically might require additional considerations.
7. Potential for Starvation: In certain situations, the FIFO Push-Relabel algorithm may exhibit a phenomenon called "starvation", where certain nodes may repeatedly receive excess flow, limiting the overall progress of the algorithm.
8. Not Always Suitable for Sparse Graphs: The algorithm's performance may not be optimal on sparse graphs where only a small fraction of edges are active during each iteration. Other algorithms, like the Ford-Fulkerson algorithm with suitable augmenting paths, might be more efficient in such cases.

Conclusion:

In conclusion, the FIFO Push-Relabel algorithm is a powerful and efficient method for solving the maximum flow problem in network optimization. Its key advantages include its adaptability to various network structures, fairness in the distribution of excess flow, simplicity of implementation, and practical performance in real-world applications.

The algorithm's efficiency, adaptability to dynamic networks, and parallelization potential make it a valuable tool in solving complex optimization problems. Its application extends to a wide range of domains, including network design, transportation planning, resource allocation, and beyond. The guaranteed convergence to an optimal solution adds to its reliability.

In short, the FIFO Push-Relabel algorithm stands out as a versatile and effective tool for solving maximum flow problems, contributing to advancements in fields where efficient resource allocation and flow optimization are critical.