To illustrate the difference in peak memory consumption between Depth-First Search (DFS) and Breadth-First Search (BFS), let's consider the fundamental ways these algorithms operate and how they manage memory.
Memory Usage in DFS and BFS
DFS (Depth-First Search):
Memory Requirement: DFS uses a stack (or recursion, which implicitly uses the call stack) to keep track of the nodes. The memory required is proportional to the depth of the deepest node in the graph.
Space Complexity: The space complexity of DFS is $$O(\text{maxDepth})$$, where maxDepth is the maximum depth of the tree or graph. This is because DFS only needs to store the nodes along the current path from the root to the deepest node being explored[1][2][3][4].
BFS (Breadth-First Search):
Memory Requirement: BFS uses a queue to keep track of all the nodes at the current level before moving on to the next level. This means it needs to store all the nodes at the widest part of the graph.
Space Complexity: The space complexity of BFS is $$O(\text{maxWidth})$$, where maxWidth is the maximum number of nodes at any level in the graph. This can be significantly larger than the depth, especially in wide graphs[1][2][3][4].
Example to Illustrate the Difference
Consider a binary tree:
DFS: In a binary tree, DFS will only need to store nodes along a single path from the root to a leaf. For a balanced binary tree of height $$h$$, this means DFS will use $$O(h)$$ memory.
BFS: In the same binary tree, BFS will need to store all nodes at the current level. For a balanced binary tree of height $$h$$, the maximum number of nodes at any level is $$2^{h-1}$$. Therefore, BFS will use $$O(2^{h-1})$$ memory[2][3][4].
Practical Implications
DFS is generally more memory-efficient for deep but narrow graphs because it only needs to keep track of the current path.
BFS can consume much more memory for wide graphs because it needs to store all nodes at the current level.
Conclusion
The peak memory consumption of DFS is typically lower than that of BFS, especially in graphs where the width (number of nodes at the widest level) is much greater th...
