Edmonds Karp algorithm in C++
Algorithms are essential in computer science and graph theory for effectively tackling challenging challenges. The Edmonds-Karp algorithm, included in the broader category of network flow algorithms, is an outstanding one. The modeling of situations including transportation systems, communication networks, and even social interactions frequently uses network flows. The maximum flow problem in these networks is explicitly addressed by the Edmonds-Karp algorithm, which is essential for maximizing the distribution of resources, data, or products.
Learning about Network Flow and the Maximum Flow Issue
Consider a network as a set of linked nodes, each representing a place of interest, and the edges that connect them reflect the flow channels. The transfer of a certain amount (such as data, products, or traffic) along these edges from a source node to a sink node is represented by the network flow. The goal is to find the best technique to move this amount while considering the edges' limitations. Finding the maximum quantity of flow that can be sent from the source to the washbasin without exceeding the limitations of the edges is where the maximum flow problem comes into play.
The Edmonds-Karp Algorithm
A particular use of the Ford-Fulkerson method, a fundamental computational strategy for resolving the maximum flow problem, is the Edmonds-Karp algorithm. Edmonds-Karp is unique in that it efficiently finds augmenting paths—paths that can carry more flow while abiding by capacity restrictions—using breadth-first search (BFS).
![Edmonds Karp algorithm in C++](https://static.tutorialandexample.com/cpp/edmonds-karp-algorithm-in-cpp1.png)
Walkthrough of an algorithm
The BFS method is the core of the Edmonds-Karp algorithm. The algorithm begins at the source node and moves across the network in stages, evaluating all potential routes to the sink node methodically. The shortest augmenting pathways are always found first thanks to BFS, which has a big impact on the algorithm's convergence and time complexity.
The algorithm keeps track of the path travelled while traversing the BFS in a "parent" array. For the eventual reconstruction of the augmenting routes, this array is crucial. The method recognizes edges with remaining capacity during the BFS and updates the parent array appropriately.
Enhancing paths and bottleneck capacity
The augmenting pathways that link the source and the sink are found by BFS as they traverse the graph. The minimal capacity of the edges along an augmenting path is the bottleneck capacity of that path. How much more flow can be sent down the path depends on the bottleneck capacity.
The method changes the capacities of the edges along the path and their corresponding reverse edges after an augmenting path has been identified (to preserve the balance of flow). This step is essential to ensure that the flow stays within the bounds of the capacities and that the residual graph—the graph with updated capacities—is generated correctly.
Iterative Refinement and Ending
In an iterative process, the Edmonds-Karp algorithm updates the flow and finds augmenting pathways until no more can be identified. The maximum flow is thus attained at the point where no augmenting path can be discovered using BFS.
The fact that the Edmonds-Karp algorithm ensures termination is one of its most important benefits. Since the method increases the flow along augmenting paths by the bottleneck capacity, the flow can only increase a finite number of times. This is because BFS always finds the shortest augmenting paths.
Time Research is Complicated
There is an intriguing temporal complexity to the Edmonds-Karp algorithm. Ford-Fulkerson uses exponential time complexity in some cases, while Edmonds-Karp uses BFS to ensure that augmenting paths are identified in a specific order. This path length restriction to O(VE2) greatly reduces the worst-case time complexity, where V is the number of nodes and E is the number of edges. Despite not always being the best method, this strategy has a termination guarantee and often performs better in many real-world situations, making it a helpful tool for optimizing network traffic.
Applications in Practical Situations
The Edmonds-Karp method is useful in many situations that arise in real life. One of the most notable examples is in logistics and transportation. Imagine a situation where commodities need to be transported with varying capacities on varying routes from many suppliers to multiple consumers. The distribution of goods can be improved with the Edmonds-Karp algorithm to increase effectiveness and cut costs.
Data packets are routed through various pathways in communication networks, which is another application. Utilizing the technique will ensure balanced data transfer across network paths, avoiding congestion and ensuring effective data flow.
Limitations and Extensions
The Edmonds-Karp algorithm has some restrictions even though it is a powerful tool for solving the maximum flow problem. It assumes that capacities are fixed and do not fluctuate over time, hence it cannot handle cases with real-number capabilities.
On some kinds of networks, algorithms that are more sophisticated perform better than Edmonds-Karp, such as the Push-Relabel algorithm and Dinic's algorithm. To achieve quicker convergence and better time complexity in particular situations, these algorithms make use of increasingly complicated techniques and data structures.
![Edmonds Karp algorithm in C++](https://static.tutorialandexample.com/cpp/edmonds-karp-algorithm-in-cpp2.png)
Code implementation of the Ford-Fulkerson algorithm, specifically the Edmonds-Karp algorithm, for finding the maximum flow in a network flow problem.
#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;
const int INF = 1e9;
// Representation of the graph using an adjacency matrix
vector<vector<int>> graph;
vector<int> parent;
// Breadth-First Search to find augmenting paths
bool bfs(int source, int sink) {
fill(parent.begin(), parent.end(), -1);
parent[source] = source;
queue<pair<int, int>> q;
q.push({source, INF});
while (!q.empty()) {
int cur = q.front().first;
int flow = q.front().second;
q.pop();
for (int next = 0; next < graph.size(); ++next) {
if (parent[next] == -1 && graph[cur][next] > 0) {
parent[next] = cur;
int new_flow = min(flow, graph[cur][next]);
if (next == sink)
return new_flow;
q.push({next, new_flow});
}
}
}
return 0;
}
int edmondsKarp(int source, int sink) {
int maxFlow = 0;
int newFlow;
while (newFlow = bfs(source, sink)) {
maxFlow += newFlow;
int cur = sink;
while (cur != source) {
int prev = parent[cur];
graph[prev][cur] -= newFlow;
graph[cur][prev] += newFlow;
cur = prev;
}
}
return maxFlow;
}
int main() {
int numNodes, numEdges;
cin >> numNodes >> numEdges;
graph.assign(numNodes, vector<int>(numNodes, 0));
parent.assign(numNodes, -1);
for (int i = 0; i < numEdges; ++i) {
int from, to, capacity;
cin >> from >> to >> capacity;
graph[from][to] = capacity;
}
int source, sink;
cin >> source >> sink;
int maxFlow = edmondsKarp(source, sink);
cout << "Maximum Flow: " << maxFlow << endl;
return 0;
}
Explanation of the code
Step 1: Include the necessary headers: stream for input/output, vector for representing graphs, queue for BFS implementation, and algorithm for various utility functions are the common C++ libraries that are included.
Step 2: Define Constants: Choosing a value for the upper bound that symbolizes infinity. It serves as a starting value for enhancing path flow as well as for initializing the flow values during BFS.
Step 3: Define data structures: During BFS traversal, these vectors are utilized to represent the graph and keep track of parent nodes.
Step 4: Breadth-First Search (BFS): BFS function that looks for paths with additional information. It begins at the source node and layers the graph as it examines it. It tracks the path followed using the parent array. If it discovers a way to the bottleneck capacity of the path, which is the lowest capacity of the edges, is returned by the sink node.
Step 5: Implementing the Edmonds-Karp Algorithm: Edmonds-Karp algorithm implementation is done via this function. The bfs function is repeatedly invoked to discover augmenting pathways, and the graph's flow values are updated by the bottleneck capacity. It keeps going until no other enhancing pathways are discovered.
Step 6: Primary Purpose: Input values for the number of nodes, edges, edge information, source, and sink are read by the main function. It invokes the edmondsKarp function, initializes the graph and parent vectors, and outputs the algorithm's estimated maximum flow.
Conclusion
The Edmonds-Karp method is a testament to how well complex optimization issues can be solved when basic ideas like BFS are combined with algorithmic knowledge. Its capacity to determine the maximum flow in network situations has broad ramifications, affecting industries as diverse as logistics and transportation as well as communication networks and beyond. The termination guarantee and practical applicability of the method make it a valuable weapon in the toolbox of any computer scientist or engineer tasked with resolving network flow optimization issues, even though it might not always be the most effective solution for all cases.