Graph search algorithms guide the way we analyze connected data, from finding routes in traffic networks to detecting relationships in social platforms. While various techniques, including DFS, BFS, Dijkstra's, and A*, have unique methodologies, many of them fit into a broader family known as Best-First Search.

This post covers a wide range of methods under that umbrella. We’ll look at classic searches like DFS and BFS, explore how Dijkstra’s and A* extend the Best-First principle, and then move to iterative deepening, bidirectional searches, parallel approaches, and ways to handle changing graphs. By seeing how these methods connect, it becomes simpler to pick the right tool for each problem and recognize their shared foundations.

What is Graph Search Algorithms

A Short Introduction To Graphs

A graph is a collection of nodes (or vertices) connected by edges. These edges might be directed or undirected, weighted or unweighted, depending on the scenario. In road maps, cities act as nodes, roads act as edges, and distances can serve as weights. In social platforms, people are nodes, and friendships or follow links are edges. Graphs capture relationships in a structure that makes it possible to ask about paths, connectivity, and distances in a direct and flexible way.

Key Concepts In Graph Search

Graph search refers to algorithms that systematically explore or traverse a graph. Some searches, like DFS and BFS, focus on the order in which nodes are visited. Others, such as Dijkstra’s or A*, track and minimize costs to find the shortest route. These methods often share a common approach: they select a node, evaluate whether it meets the goal, and then move on to adjacent nodes based on certain rules. This repeats until the entire graph is explored or the desired path is found. Best-First Search provides a unifying lens for these algorithms by highlighting how each one balances known costs and heuristic estimates.

Applications In Various Domains

Graph search is more than an abstract concept. It appears in many fields to spot patterns, plan routes, and make sense of connections among data points. Below are some common examples:

1. Navigation and route planning
   We can represent road networks as graphs, where intersections and cities serve as nodes, and roads act as edges. Algorithms like Dijkstra’s and A* determine how to reach a destination in the shortest time.
2. Social network analysis
   Platforms such as Facebook or LinkedIn use graph based methods to recommend new contacts or find key influencers. By exploring clusters and relationships between users, these techniques reveal how communities form and interact.
3. Web search
   The internet is a huge graph of pages linked by hyperlinks. Search engines use variations of graph techniques to crawl websites, rank them, and return the most relevant content to users.
4. Network routing and load balancing
   Data moves through a network from node to node, with each connection having certain bandwidth or latency constraints. Pathfinding algorithms help administrators reduce bottlenecks and direct traffic more efficiently.
5. Game AI and pathfinding
   Many games map the environment onto nodes and edges so characters can navigate smoothly. A* is a popular choice for managing how non playable units move around obstacles.
6. Dependency resolution
   Most software projects rely on libraries that must be installed in a certain order. Graph search determines the correct sequence, which is critical for package managers and continuous integration systems.

Best-First Search

A variety of graph search methods have grown out of different needs. Some handle simple unweighted paths, others account for varying costs, and still others use heuristics to guide the search. Although these algorithms appear distinct, many share a common template: they select the next node to explore based on a mix of known costs and estimates. Best-First Search captures this idea, tying together techniques like Depth-First Search (DFS), Breadth-First Search (BFS), Dijkstra’s, Greedy Best-First, and A* within one overarching framework. By examining how these methods fit into Best-First Search, we can see why each algorithm excels in certain scenarios and where it may fall short.

Introduction to Best-First Search

Best-First Search provides a way to think about many pathfinding and traversal strategies under one umbrella. It’s based on a function \(f(n)=g(n)+h(n)\) where 

• \(g(n)\) is the cost from the start node to the current node \(n\),
• \(h(n)\) is a heuristic estimate of the cost from \(n\) to the goal.

Many well-known algorithms emerge from different ways of defining or emphasizing \(g(n)\) and \(h(n)\). This shared view makes it easier to compare methods, understand their strengths, and pick the most suitable one for a given problem.

DFS (Depth-First Search)

Depth-First Search explores a graph by venturing down one path as far as possible before backtracking. In practical terms, it’s often implemented using a stack:

1. Start by pushing the initial node onto the stack.
2. Pop the top node, process it, and push any unvisited neighbors.
3. Repeat until the stack is empty or the target is found.

This approach can also be written in a recursive style, where the system’s call stack plays the same role. Although DFS does not specifically seek the shortest path, you can frame it as a Best-First Search variant by defining:

\[g(n)=\text{depth}(n),h(n)=0\]

where \(\text{depth}(n)\) is the node's depth in the search tree. As a result, \(f(n) =g(n)=\text{depth}(n)\) which aligns with DFS’s behavior of going deeper before expanding other branches.

DFS proves useful in:

• Cycle detection, since it can track back edges to see if a loop exists.
• Topological sorting on directed acyclic graphs (DAGs).
• Backtracking algorithms for puzzle solving or configuration checks.

Because DFS typically retains only the path from the start node to the current node plus a visited set, it can be memory efficient, making it helpful when exploring large graphs where storing all nodes in a queue (as in BFS) might be too costly.

BFS (Breadth-First Search)

Breadth-First Search explores a graph level by level. In practice, it’s often implemented using a queue:

1. Enqueue the initial node.
2. Dequeue the front node, process it, and enqueue any unvisited neighbors.
3. Repeat until the queue is empty or the target is found.

This method guarantees the shortest path in an unweighted graph. You can view BFS as a Best-First Search variant by defining:

\[g(n)=\text{distance}(n),h(n)=0\]

where \(\text{distance}(n)\) is the number of edges from the start node to \(n\) in an unweighted graph. As a result, \(f(n)=g(n)=\text{distance}(n)\) which means BFS systematically explores all nodes one edge away from the start, then two edges away, and so on.

BFS proves useful in:

• Shortest path searches in unweighted graphs, where each edge is treated equally.
• Level-order tree traversal, where all nodes at a given depth must be processed before moving deeper.

Because BFS stores every node at the current level (and potentially the next) in a queue, it can consume significant memory for large or dense graphs. Despite this, it remains the go-to option when a guaranteed shortest path is needed in an unweighted setting.

Dijkstra’s Algorithm

Dijkstra’s Algorithm extends BFS to handle weighted graphs where all edge costs are nonnegative. It keeps track of the path cost from the start node to each node and always expands the node with the smallest current cost:

1. Assign a tentative cost of 0 to the start node and infinity to all others.
2. Use a priority queue to pick the node \(u\) with the smallest cost so far.
3. For each neighbor \(v\) of \(u\), compute the cost of reaching \(v\) through \(u\). If it’s lower than \(v\)’s current cost, update \(v\)’s cost and add \(v\) to the queue.
4. Repeat until the queue is empty or all reachable nodes have been processed.

You can see this as a Best-First Search variant similar to BFS, with \(g(n)=\text{distance}(n),h(n)=0\) so \(f(n)=g(n)=\text{distance}(n).\)

For an unweighted graph (or one where all edges have the same cost), Dijkstra’s works like BFS, except BFS uses a standard queue while Dijkstra’s uses a priority queue (min-heap). In practice, priority queues are well-suited for sparse graphs, where the number of edges is relatively small. For dense graphs, the overhead of updating and reordering a large priority queue can outweigh its benefits, making simpler approaches potentially faster.

Dijkstra’s Algorithm is helpful in:

• Navigation services that must account for varying distances or travel times.
• Network routing where different links have different bandwidth or latency costs.
• Scheduling or resource allocation whenever tasks or edges have unique costs to process.

When a suitable heuristic is available, you might consider A* instead, since it can often find the same path more efficiently by focusing the search toward the goal.

Greedy Best-First Search

Greedy Best-First Search focuses on moving directly toward the goal, effectively ignoring the path cost so far. It typically uses a priority queue to pick the node that appears closest to the goal:

1. Define a heuristic \(\text{heuristic}(n)\) that estimates how far \(n\) is from the goal.
2. Enqueue the start node with priority based on \(\text{heuristic}(n)\).
3. Repeatedly dequeue the node \(u\) with the smallest heuristic value.
4. If \(u\) is the goal, stop. Otherwise, enqueue each unvisited neighbor \(v\) and prioritize by \(\text{heuristic}(v)\).
5. Continue until the goal is reached or the queue is empty.

We can see this as a Best-First Search variant by defining:

\[g(n)=0,h(n)=heuristic(n)\]

where \(\text{heuristic}(n)\) estimates the cost from node \(n\) to the goal. As a result, \(f(n)=\text{heuristic}(n)\).

Because it always selects the node with the smallest heuristic value, Greedy Best-First Search may quickly find a path, but it does not guarantee finding the shortest path unless the heuristic satisfies strict conditions (such as admissibility, and even then, additional checks are usually needed).

This approach can be appealing when:

• Speed is more critical than guaranteed optimality.
• Heuristics provide a decent guess of proximity to the goal, and the cost of overshooting isn’t high.

For scenarios requiring a reliable shortest path while still leveraging a heuristic, consider A*, which combines the actual path cost and the heuristic estimate to systematically ensure optimality under the right conditions.

A* Search

A* (pronounced “A star”) balances the strengths of Dijkstra’s and Greedy Best-First Search by combining both actual cost so far and a heuristic:

1. Define a cost-so-far function, \(\text{costFromStart}(n)\), to track the total cost of reaching \(n\).
2. Define a heuristic, \(\text{heuristic}(n)\), that estimates how far \(n\) is from the goal.
3. Use a priority queue to pick the node \(u\) with the smallest sum of these two values.

In the Best-First Search framework, A* sets

\[g(n)=\text{costFromStart}(n),h(n)=\text{heuristic}(n),\]

so \(f(n)=g(n)+h(n)=\text{costFromStart}(n)+\text{heuristic}(n)\).

As long as the heuristic is admissible (never overestimates the true cost to the goal) and the graph has nonnegative edge costs, A* guarantees the shortest path. Its efficiency advantage over Dijkstra’s often comes from focusing exploration around the goal instead of blindly exploring in all directions.

A* is widely adopted in:

• Navigation and route finding, where good heuristics can guide the search quickly.
• Game AI, to move characters or units in a realistic and efficient way.
• Puzzle solvers (like the 8-puzzle or 15-puzzle), where heuristics approximate the number of moves left.

A* offers a sweet spot: it can be more direct than Dijkstra’s when a reliable heuristic is available, yet still provide optimal solutions (unlike pure Greedy Best-First). This makes it a standard choice in many shortest-path scenarios.

A Note On Admissibility

In a Best First approach, a heuristic h(n) is admissible if it never goes above the true cost of reaching the goal from n. For A*, this condition, along with nonnegative edge costs, ensures the algorithm finds the shortest path. The search order respects the real costs and does not discard any path that could be better. If the heuristic rises above the true cost, A* might skip a path that was actually cheaper. The closer your heuristic is to the actual cost, without exceeding it, the more efficiently A* will move toward the goal.

For instance, choosing \(h(n)=0\) is admissible because it never overestimates the cost. However, it offers no guidance, so A* behaves like Dijkstra's. That still yields the shortest path, but without the search-direction advantage a good heuristic can provide.

Other Techniques: Classic Variations and Extensions

Some well-established methods lie outside the direct Best First framework yet handle many practical cases efficiently. Here, we look at Iterative Deepening DFS (IDDFS) and Bidirectional Search, which tackle large or infinite spaces and can cut down the number of nodes explored.

Iterative Deepening DFS (IDDFS)

Iterative Deepening DFS (IDDFS) combines the depth exploration of DFS with the layered approach of BFS. It proceeds in rounds, each time running a DFS limited to a certain depth. After finishing one round, it increases the depth limit and repeats:

1. Set a depth limit of 0 (or 1, depending on the convention).
2. Run DFS up to that depth limit.
3. Increase the limit by one.
4. Repeat until the goal is found or you reach a predefined cutoff.

IDDFS offers the completeness of BFS (it will eventually reach the goal node at the shallowest depth) combined with the memory efficiency of DFS (it mostly stores just one path in memory at a time). This makes it a strong choice in large or infinite state spaces, such as puzzle searches, where a traditional BFS might become too large to handle.

Bidirectional Search

Bidirectional Search aims to reduce the search space by running two simultaneous searches: one starting from the initial node and another from the goal node. The two frontiers advance until they meet:

1. Front Search: Begin from the start node.
2. Reverse Search: Begin from the goal node.
3. Alternate or parallelize expansions in each direction.
4. Stop when a node in the front search appears in the reverse search frontier (or vice versa).

Because each search only expands a portion of the full distance (roughly half in many cases), the combined effort can be much smaller than a single search covering the entire distance. This approach works best when it is easy to reverse the problem or define a clear way to move from goal to start (e.g., an undirected graph or a directed graph where edges can be reversed logically). Bidirectional Search is especially useful in pathfinding tasks where the branching factor is large and the distance between start and goal is not trivial.

Advanced Techniques: Parallel and Dynamic Approaches

Graph searches in large-scale or changing environments can strain traditional methods. Parallelization helps spread the workload across multiple processors or machines, while dynamic algorithms handle changing graphs without restarting from scratch. These advanced techniques shine in the following areas.

Parallel Graph Search Methods

As graph datasets expand, running searches on a single core or machine can become a bottleneck. Parallel graph search addresses this by distributing tasks across several processors or nodes in a cluster. Each processor handles a part of the search, and results are merged as the algorithm proceeds:

• Shared-Memory Parallelism: Multiple threads work concurrently on a shared graph data structure. Algorithms like parallel Breadth-First Search (BFS) or Dijkstra's algorithm can be adapted for shared-memory environments. Careful synchronization using locks or atomic operations is crucial to avoid conflicts when different threads try to read or update the same nodes or edges.
• Distributed Parallelism: The graph is partitioned across different machines, each running its own search process. Algorithms like distributed BFS or the Δ-stepping algorithm are designed for this setting. Efficient communication between machines is essential to exchange partial results and maintain consistency. Frameworks like Apache Spark's GraphX provide tools for distributed graph processing.

When implementing parallel graph algorithms, several key challenges and trade-offs must be considered. Workload balancing is essential for optimal performance, requiring the use of graph partitioning techniques to create balanced partitions in distributed settings. In shared-memory systems, performance can degrade due to excessive synchronization causing contention between threads. Additionally, communication between machines in distributed systems can become a significant bottleneck, making it crucial to minimize data transfer and implement efficient communication patterns.

Dynamic Graph Search Methods

In many real-world scenarios, graphs are not fixed. Edges and nodes appear, disappear, or change weight, and rerunning a search from scratch every time would be inefficient. Dynamic graph search methods update previous results incrementally, providing mechanisms for real-time updates, essential in scenarios like live traffic navigation or evolving social networks.

• Real-Time Updates for Dynamic Graphs: Dynamic algorithms enable real-time adaptation to changes in edge weights, and node/edge additions or removals. For example, in live traffic navigation, these algorithms continuously adjust the best routes based on the most recent congestion data, ensuring drivers receive up-to-date guidance without recalculating the entire graph structure.
• Handling Edge Weight Changes: When the cost of an edge changes (e.g., increased travel time due to traffic), dynamic algorithms like Dynamic SWSF-FP efficiently recalculate shortest paths that utilize that edge, reflecting the updated distances without a full recomputation.
• Managing Structural Graph Changes: Algorithms such as St-Dynamic A* are designed to handle the removal or addition of nodes and edges. When a link fails (edge removal) or a new connection is established (edge addition), only the paths directly affected by these structural changes are rebuilt.

Dynamic methods offer compelling advantages in graph processing. Their efficiency stems from focusing on incremental changes, allowing them to perform significantly faster than approaches that recompute solutions from scratch, while their responsiveness enables them to provide up-to-date results in rapidly changing environments, making them particularly valuable for applications that require real-time or near-real-time responses.

Conclusion

Graph search algorithms help us discover paths, relationships, and patterns in data. Many of these methods align with a Best First view, balancing cost information and heuristic estimates. This unites approaches from DFS and BFS to Dijkstra’s, Greedy Best-First, and A*. Other techniques, such as IDDFS and Bidirectional Search, address large or infinite searches, and parallel or dynamic methods offer scaling and adaptability in changing environments. By understanding their core similarities and differences, you can choose the right algorithm for each task and see how they work together in practice.

PuppyGraph offers a comprehensive suite of graph algorithms that operate directly on your relational data. Want to experience the power of various distributed graph algorithms? Download the forever free PuppyGraph Developer Edition, or book a demo with our graph expert team.