Hopcroft–Karp algorithm time complexity

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Understanding the Hopcroft–Karp Algorithm's Time Complexity

Abstract representation of a bipartite graph with edges connecting two sets of nodes, symbolizing a matching process.

Delve into the intricacies of the Hopcroft–Karp algorithm, a cornerstone for finding maximum cardinality matchings in bipartite graphs, and unravel the factors contributing to its efficient O(E√V) time complexity.

The Hopcroft–Karp algorithm is a highly efficient algorithm used to find a maximum cardinality matching in a bipartite graph. A matching is a set of edges where no two edges share a common vertex. A maximum cardinality matching is a matching that contains the largest possible number of edges. Understanding its time complexity is crucial for appreciating its performance in various applications, from resource allocation to network flow problems.

Core Concepts and Algorithm Overview

Before diving into the complexity analysis, let's briefly review the fundamental ideas behind Hopcroft–Karp. The algorithm operates in phases, iteratively finding augmenting paths to increase the size of the current matching. An augmenting path is a path that starts and ends with unmatched vertices and alternates between edges not in the matching and edges in the matching. Finding such a path allows us to flip the matching status of its edges, thereby increasing the total number of matched edges by one.

flowchart TD
    A[Start with empty matching M] --> B{Find shortest augmenting paths using BFS}
    B -->|No augmenting paths| C[Algorithm terminates: M is maximum matching]
    B -->|Augmenting paths found| D[Identify maximal set of vertex-disjoint shortest augmenting paths]
    D --> E[Augment M along these paths]
    E --> B

High-level flow of the Hopcroft–Karp algorithm

The key innovation of Hopcroft–Karp lies in its ability to find a maximal set of vertex-disjoint shortest augmenting paths in each phase. This is achieved by using a Breadth-First Search (BFS) to layer the graph and identify shortest augmenting paths, followed by a Depth-First Search (DFS) to find a maximal set of such paths. This multi-path augmentation strategy is what gives the algorithm its superior performance compared to simpler augmenting path algorithms like Ford-Fulkerson on general graphs.

Detailed Time Complexity Analysis

The time complexity of the Hopcroft–Karp algorithm is O(E√V), where E is the number of edges and V is the number of vertices in the bipartite graph. Let's break down how this complexity is derived.

Each phase of the algorithm consists of two main steps:

BFS for Layering: A BFS is performed to construct a layered graph, identifying all shortest augmenting paths from unmatched vertices in one partition to unmatched vertices in the other. This step takes O(E) time, as it explores each edge and vertex at most once.
DFS for Augmentation: A series of DFS traversals are then performed on the layered graph to find a maximal set of vertex-disjoint shortest augmenting paths. Each DFS traversal finds one augmenting path. While a single DFS takes O(E) time in the worst case, the crucial insight is that after an edge is used in an augmenting path, it's either removed or its direction is reversed, preventing it from being traversed again in the same phase in a way that would lead to a non-shortest path. The total time for all DFS traversals in a single phase is also O(E).

Therefore, each phase of the algorithm takes O(E) time.

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The 'maximal set' of augmenting paths means that once a path is found and used for augmentation, the vertices involved in that path are considered 'used' for the current phase, preventing other paths from using them. This ensures vertex-disjointness within a phase.

The critical part of the analysis is determining the number of phases. It has been proven that the Hopcroft–Karp algorithm requires at most O(√V) phases. This is because after O(√V) phases, the length of the shortest augmenting path must increase. When the shortest augmenting path length increases, it implies that the current matching is already quite large, and further augmentations become less frequent. Since each phase increases the matching size by at least one, and there are at most V/2 edges in a maximum matching, the total number of phases is bounded by O(√V).

Combining the O(E) time per phase with the O(√V) phases, we arrive at the total time complexity of O(E√V).

Comparison with Other Matching Algorithms

To put the O(E√V) complexity into perspective, let's compare it with other common algorithms for maximum matching:

Ford-Fulkerson (using BFS for augmenting paths, i.e., Edmonds-Karp): When applied to a bipartite matching problem by constructing a flow network, Edmonds-Karp has a complexity of O(VE^2) or O(V^2E) in general graphs. For bipartite matching, it can be O(VE) if capacities are unit. Hopcroft–Karp is generally faster.
Simple Augmenting Path Algorithms: Algorithms that find one augmenting path at a time (e.g., using DFS) would have a worst-case complexity of O(V * E) because there can be up to V/2 augmenting paths, and each path search takes O(E) time. Hopcroft–Karp's ability to find multiple paths per phase significantly improves this.

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While O(E√V) is the tightest known bound for general bipartite graphs, for dense graphs where E is close to V^2, the complexity approaches O(V^2.5). For sparse graphs, it's closer to O(V^1.5).

The Hopcroft–Karp algorithm's efficiency makes it the preferred choice for finding maximum cardinality matchings in bipartite graphs in many practical scenarios. Its clever use of BFS for layering and DFS for multi-path augmentation ensures that it performs significantly better than simpler approaches, especially for larger graphs.