8.6 Parallel Graph Algorithms

Graph algorithms present a significant challenge for parallel computing. Unlike the dense, regular data structures of matrices, graphs are fundamentally irregular, unstructured, and sparse. The performance of algorithms on these structures is often dictated not by arithmetic speed, but by memory access and network communication latency. This case study explores the difficulties of parallelizing graph algorithms and examines parallel approaches for three fundamental problems: breadth-first search (BFS), single-source shortest path (SSSP), and minimum spanning tree (MST).

The Challenge of Irregularity in Graphs

The difficulty in achieving high performance for parallel graph algorithms stems from several inherent characteristics:¹

Irregular Data Structures and Poor Memory Locality: Graphs are often represented using pointer-based structures like adjacency lists. Traversing a graph involves “pointer chasing,” which leads to random, unpredictable memory access patterns. This irregularity defeats hardware mechanisms—such as caching and prefetching—that modern processors rely on to hide memory latency.¹
Load Imbalance: The structure of real-world graphs is often highly non-uniform. For example, in social networks, some vertices may have millions of connections, while most have only a few. This variation in vertex degree makes it difficult to statically partition the graph in a way that gives each processor an equal amount of work.¹
High Communication-to-Computation Ratio: Graph algorithms are typically memory-bound rather than compute-bound. They perform few arithmetic operations for each vertex or edge accessed. Consequently, execution time is dominated by the cost of moving data, whether from main memory to the processor or between processors over a network.¹

Graph Traversal: Parallel Breadth-First Search (BFS)

Breadth-First Search (BFS) is a fundamental algorithm for exploring a graph layer by layer from a starting vertex. Its parallelization is a classic example of the level-synchronous approach.²

Methodology

The sequential BFS algorithm maintains a queue of vertices to visit. A parallel BFS algorithm replaces this single queue with a frontier, which is the set of all vertices at the current distance (or level) from the source. The algorithm proceeds in synchronous steps:²

Initialize the frontier to contain only the source vertex.
In parallel, each processor takes a subset of vertices from the current frontier.
For each vertex in its subset, the processor explores its neighbors. If a neighbor has not yet been visited, it is added to a local “next frontier.”
A barrier synchronization occurs. The local next frontiers from all processors are combined to form the global next frontier for the subsequent step.
Repeat from step 2 until the frontier is empty.

This process can be viewed as a sequence of sparse matrix-vector multiplications, where the matrix is the graph’s adjacency matrix and the vector represents the current frontier.³ The main challenge is the potential for severe load imbalance if some vertices in the frontier have vastly more neighbors than others. Optimization techniques focus on dynamic load balancing and reducing the cost of global synchronization between levels.²

Algorithm: ParallelBFS(Graph G, source vertex s, p processors)
Input: Graph $G = (V, E)$, source vertex $s$, $p$ processors
Output: Distance array dist[v] for all vertices $v$

// Initialize
parallel for all vertices $v \in V$ do
    dist[v] = $\infty$
    visited[v] = false
end parallel

dist[s] = 0
current_frontier = {s}
level = 0

// Level-synchronous BFS
while current_frontier is not empty do
    next_frontier = empty set

    // Partition frontier among processors
    frontier_per_processor = |current_frontier| / $p$

    parallel for each processor $i = 0$ to $p-1$ do
        local_next_frontier = empty set

        // Each processor processes its portion of the frontier
        for each vertex $v$ in current_frontier[i] do
            // Explore neighbors of $v$
            for each neighbor $u$ of $v$ do
                // Attempt to mark $u$ as visited
                if atomic_compare_and_swap(visited[u], false, true) then
                    dist[u] = level + 1
                    local_next_frontier.add(u)
                end
            end
        end

        // Collect local next frontiers
        contribute(local_next_frontier, next_frontier)
    end parallel

    // Barrier synchronization - wait for all processors
    barrier()

    // Gather all local next frontiers into global next frontier
    current_frontier = gather(next_frontier)
    level = level + 1
end

return dist

Time Complexity per level: $O((|V| + |E|)/p)$ Total levels: $O(\text{diameter of graph})$ Synchronization: $O(\text{diameter})$ barriers

Key challenges:

Load imbalance due to irregular vertex degrees
Contention when marking vertices as visited
Cost of global synchronization at each level

Single-Source Shortest Path (SSSP): Parallelizing Dijkstra’s Algorithm

Dijkstra’s algorithm finds the shortest paths from a single source to all other vertices in a weighted graph with non-negative edge weights. Its standard implementation, however, is inherently sequential.⁴

The Sequential Bottleneck

The core of Dijkstra’s algorithm is a greedy strategy: in each step, it extracts the one vertex from a priority queue that has the globally minimum tentative distance from the source.⁵ This requirement to find a single global minimum creates a dependency that prevents the straightforward parallel processing of multiple vertices.

Parallel Approaches

Parallelizing Dijkstra’s requires relaxing this strict greedy constraint.

Delta-Stepping Algorithm: This is a widely used parallel SSSP algorithm. Instead of processing one vertex at a time, it processes vertices in “buckets,” where each bucket corresponds to a range of distances of size $\Delta$ . In each phase, all “light” edges (those with weight $\leq \Delta$ ) connected to vertices in the current lowest-indexed non-empty bucket can be relaxed in parallel. This may introduce redundant work, but it exposes significant parallelism by allowing many vertices to be processed concurrently.⁶
Data Parallelism: A simpler but less scalable approach is to use data parallelism within each step of the sequential algorithm, for instance, by parallelizing the search for the minimum-distance vertex or the relaxation of edges using frameworks like OpenMP.⁷

Algorithm: DeltaStepping_SSSP(Graph G, source s, $\Delta$, p processors)
Input: Weighted graph $G = (V, E)$, source $s$, parameter $\Delta$, $p$ processors
Output: Shortest distances dist[v] from $s$ to all vertices $v$

// Initialize distances
parallel for all vertices $v \in V$ do
    dist[v] = $\infty$
end parallel
dist[s] = 0

// Buckets organize vertices by distance ranges $[i \cdot \Delta, (i+1) \cdot \Delta)$
buckets = array of empty sets
buckets[0].add(s)
current_bucket = 0

while exists non-empty bucket do
    // Find lowest non-empty bucket
    while buckets[current_bucket] is empty do
        current_bucket = current_bucket + 1
    end

    // Process current bucket until empty
    while buckets[current_bucket] is not empty do
        current_set = buckets[current_bucket]
        buckets[current_bucket] = empty

        // Relax light edges (weight $\leq \Delta$) in parallel
        parallel for each vertex $v$ in current_set do
            for each edge $(v, u)$ with weight$(v, u) \leq \Delta$ do
                new_dist = dist[v] + weight(v, u)
                if new_dist < dist[u] then
                    old_dist = atomic_min(dist[u], new_dist)
                    if old_dist > new_dist then
                        move u from bucket[floor(old_dist/$\Delta$)] to bucket[floor(new_dist/$\Delta$)]
                    end
                end
            end
        end parallel
        barrier()
    end

    // Relax heavy edges (weight $> \Delta$) once per bucket
    parallel for each vertex $v$ that was in current bucket do
        for each edge $(v, u)$ with weight$(v, u) > \Delta$ do
            new_dist = dist[v] + weight(v, u)
            if new_dist < dist[u] then
                old_dist = atomic_min(dist[u], new_dist)
                if old_dist > new_dist then
                    move u from bucket[floor(old_dist/$\Delta$)] to bucket[floor(new_dist/$\Delta$)]
                end
            end
        end
    end parallel
    barrier()
end

return dist

Time Complexity: $O((|V| + |E|/\Delta) \times \log |V|)$ expected Parameter $\Delta$ : Trade-off between parallelism and redundant work

Small $\Delta$ : More buckets, less redundant work, less parallelism
Large $\Delta$ : Fewer buckets, more redundant work, more parallelism

Minimum Spanning Tree (MST)

An MST of a weighted, undirected graph is a subgraph that connects all vertices with the minimum possible total edge weight. The parallelizability of MST algorithms depends on their underlying greedy strategy.

Prim’s Algorithm: Prim’s algorithm is structurally similar to Dijkstra’s. It grows a single MST by iteratively adding the cheapest edge that connects a vertex inside the growing tree to a vertex outside of it.⁸ This reliance on a single, global greedy choice makes it inherently sequential and difficult to parallelize effectively.⁹
Kruskal’s Algorithm: Kruskal’s algorithm is more amenable to parallelism. It sorts all edges by weight and then adds edges from the sorted list if they do not form a cycle.¹⁰ The initial edge sorting is highly parallelizable. The main sequential bottleneck is the cycle detection step, which typically uses a union-find data structure. While this step is challenging to parallelize, variants like Filter-Kruskal have been developed to improve performance by partitioning edges and filtering out those guaranteed to be in the same component.¹¹
Boruvka’s Algorithm: This algorithm is often considered the most naturally parallel MST algorithm. In each step, every vertex (or component) simultaneously finds the cheapest edge connecting it to another vertex (or component), and all these edges are added to the MST. This process of finding local minimums and contracting components is repeated until a single tree remains. The ability for all components to make independent greedy choices in each step exposes massive parallelism.⁸

Algorithm: ParallelBoruvka_MST(Graph G, p processors)
Input: Weighted undirected graph $G = (V, E)$, $p$ processors
Output: Minimum Spanning Tree edges

// Initialize: each vertex is its own component
parallel for each vertex $v \in V$ do
    component[v] = v
    cheapest_edge[v] = null
end parallel

MST_edges = empty set
num_components = $|V|$

while num_components > 1 do
    // Phase 1: Each component finds its minimum-weight outgoing edge
    parallel for each vertex $v \in V$ using $p$ processors do
        // Find cheapest edge leaving v's component
        min_edge = null
        min_weight = $\infty$

        for each edge $(v, u)$ incident to $v$ do
            if component[v] $\neq$ component[u] and weight$(v, u)$ < min_weight then
                min_edge = $(v, u)$
                min_weight = weight$(v, u)$
            end
        end

        // Update component's cheapest edge atomically
        comp_id = component[v]
        atomic_update_min(cheapest_edge[comp_id], min_edge, min_weight)
    end parallel

    barrier()

    // Phase 2: Add selected edges to MST (avoiding duplicates)
    selected_edges = empty set

    parallel for each component $c$ do
        if cheapest_edge[c] $\neq$ null then
            edge = cheapest_edge[c]
            // Add edge if not already added by other endpoint
            if edge not in selected_edges then
                selected_edges.add(edge)
            end
        end
    end parallel

    MST_edges = MST_edges $\cup$ selected_edges
    barrier()

    // Phase 3: Merge components (Union-Find)
    parallel for each edge $(u, v)$ in selected_edges do
        // Union operation: merge component[u] and component[v]
        root_u = find_root(u, component)
        root_v = find_root(v, component)

        if root_u $\neq$ root_v then
            union(root_u, root_v, component)
            atomic_decrement(num_components)
        end
    end parallel

    barrier()

    // Phase 4: Path compression for efficiency
    parallel for each vertex $v$ do
        component[v] = find_root(v, component)
        cheapest_edge[component[v]] = null  // Reset for next iteration
    end parallel

    barrier()
end

return MST_edges

Time Complexity: $O(\log |V|)$ iterations $\times$ $O(|E|/p)$ per iteration Parallelism: High - all vertices/components work independently each round Key advantage: Naturally parallel, no single sequential bottleneck

The challenges in parallelizing these graph algorithms reveal a deeper pattern: the degree of parallelism is often inversely related to the “greediness” of the sequential algorithm. Algorithms like Dijkstra’s and Prim’s, which rely on a single, globally optimal choice at each step, create sequential bottlenecks. In contrast, algorithms like BFS, which explores all options at a given level, or Boruvka’s, which allows for multiple local greedy choices to be made simultaneously, are far more parallelizable. To parallelize a strongly greedy algorithm often requires relaxing its strict criteria, as seen in the delta-stepping approach, trading sequential algorithmic elegance for parallel performance.

This leads to a fundamental paradigm shift for irregular problems. For regular problems like matrix multiplication, the goal of parallel algorithm design is often static optimization: creating a balanced decomposition of a predictable workload. For graphs, this is often impossible. The most effective parallel graph algorithms are therefore designed not to eliminate irregularity, but to tolerate and adapt to it. They shift the focus from static data decomposition to dynamic adaptation, employing mechanisms like work-stealing, asynchronous communication, and dynamic tasking to manage an unpredictable and evolving workload efficiently.

References

(PDF) Challenges in Parallel Graph Processing. - ResearchGate, accessed October 6, 2025, https://www.researchgate.net/publication/220439595_Challenges_in_Parallel_Graph_Processing ↩ ↩² ↩³ ↩⁴
Parallel breadth-first search - Wikipedia, accessed October 6, 2025, https://en.wikipedia.org/wiki/Parallel_breadth-first_search ↩ ↩² ↩³
Parallel Breadth-First Search on Distributed Memory Systems - People @EECS, accessed October 6, 2025, https://people.eecs.berkeley.edu/~aydin/sc11_bfs.pdf ↩
A Parallelization of Dijkstra’s Shortest Path Algorithm - People, accessed October 6, 2025, https://people.mpi-inf.mpg.de/~mehlhorn/ftp/ParallelizationDijkstra.pdf ↩
Dijkstra’s algorithm - Wikipedia, accessed October 6, 2025, https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm ↩
Parallel single-source shortest path algorithm - Wikipedia, accessed October 6, 2025, https://en.wikipedia.org/wiki/Parallel_single-source_shortest_path_algorithm ↩
Parallel Dijkstra’s Algorithm: SSSP in Parallel - GeeksforGeeks, accessed October 6, 2025, https://www.geeksforgeeks.org/dsa/parallel-dijkstras-algorithm-sssp-in-parallel/ ↩
Implementing Kruskal’s and Prim’s Algorithms: A Comprehensive Guide - AlgoCademy, accessed October 6, 2025, https://algocademy.com/blog/implementing-kruskals-and-prims-algorithms-a-comprehensive-guide/ ↩ ↩²
Difference between Prim’s and Kruskal’s algorithm for MST - GeeksforGeeks, accessed October 6, 2025, https://www.geeksforgeeks.org/dsa/difference-between-prims-and-kruskals-algorithm-for-mst/ ↩
Kruskal’s algorithm - Wikipedia, accessed October 6, 2025, https://en.wikipedia.org/wiki/Kruskal%27s_algorithm ↩
Parallel algorithms for minimum spanning trees - Wikipedia, accessed October 6, 2025, https://en.wikipedia.org/wiki/Parallel_algorithms_for_minimum_spanning_trees ↩