How Do BFS and DFS Differ in Traversal and Memory?

Breadth-First Search and Depth-First Search are two fundamental graph and tree traversal algorithms in classical computing. While both systematically visit all reachable vertices from a given starting node, they differ fundamentally in the order in which they explore vertices and the auxiliary data structures they use to track pending work.

Traversal Order

Breadth-First Search (BFS) explores a graph level by level. Starting from a source vertex \(s\), it visits all immediate neighbors of \(s\) at distance \(1\), then all unvisited vertices at distance \(2\), and so on. Because it expands outward in concentric layers, BFS guarantees finding the shortest path in terms of edge count in an unweighted graph.

Depth-First Search (DFS) explores as far as possible along each branch before backtracking. Starting from \(s\), it moves to an unvisited neighbor, then to an unvisited neighbor of that vertex, continuing until it reaches a dead end. At that point, it backtracks to the most recent vertex that still has unvisited neighbors and explores the next branch.

Auxiliary Memory Structures

The difference in traversal order is directly driven by the auxiliary data structure each algorithm uses to manage the frontier of discovered but unexplored vertices:

  • BFS uses a Queue (FIFO): A First-In, First-Out queue ensures that vertices discovered earlier are processed before vertices discovered later. When a vertex is dequeued, its unvisited neighbors are enqueued at the back, preserving the level-order progression.
  • DFS uses a Stack (LIFO): A Last-In, First-Out stack ensures that the most recently discovered vertex is processed next. DFS can be implemented iteratively using an explicit stack or recursively, where the programming language's call stack serves as the implicit LIFO structure.

Space and Time Complexity

For a graph \(G = (V, E)\) with \(|V|\) vertices and \(|E|\) edges, both algorithms have a time complexity of \(O(|V| + |E|)\) when using an adjacency list, because each vertex and edge is inspected at most a constant number of times.

Their auxiliary space requirements, however, behave differently depending on the shape of the graph:

  • BFS Space Complexity: \(O(w)\), where \(w\) is the maximum width of the graph. In a balanced binary tree of \(n\) nodes, the bottom level contains roughly \(n / 2\) nodes, so BFS requires \(O(n)\) space. In a wide, shallow graph, BFS consumes significantly more memory than DFS.
  • DFS Space Complexity: \(O(h)\), where \(h\) is the maximum depth (height) of the graph. In a balanced binary tree of \(n\) nodes, \(h = O(\log n)\), making DFS much more memory-efficient. However, in a degenerate graph shaped like a single long path, \(h = |V|\), so DFS requires \(O(|V|)\) space, whereas BFS would only need \(O(1)\) space.

Both algorithms also maintain a visited set of size \(O(|V|)\) in general graphs to prevent revisiting vertices and entering infinite loops around cycles.