Graph Search
Internal
Overview
Searching a graph is a fundamental problem in computer science. To search a graph means systematically following the edges of a graph so we visit all the vertices of that graph. Techniques for searching a graph lie at the heart of the field of graph algorithms. There are many methods to search graphs, and the most known and used are breadth-first search and depth-first search.
Both breadth-first search and depth-first search algorithms are very efficient, in that they walk the entire graph in linear time of the number of vertices and edges O(n + m). A key implementation idea is to mark the nodes that have been visited already so the algorithm does not look at them twice, thus saving unnecessary work.
Generic Graph Search Algorithm
All graph search methods share the generic algorithm described in this section. Graph search subroutines are passed as input a start vertex, also called source vertex, from which the search starts. The algorithm has two goals:
- Goal 1. Find everything that is "findable" starting from the source vertex - all findable vertices. By findable we mean that there's a path from the source vertex to the findable vertex. Directed and undirected graphs behave slightly differently, in that for a directed graph, the edge can only be followed in the "forward" direction.
- Goal 2. Do it efficiently, by avoiding to explor anything twice. If the graph has n vertices and m arcs, we look at a running time O(n + m).
A generic algorithm should be able to maintain a boolean per node that says whether we have "seen" the node or not. Everything is initially set to "unexplored", except the start node. It is useful to think of nodes seen so far as "territory" conquered by the algorithm. There is going to be a frontier between the conquered and unconquered territory. The goal of the generic algorithm is to supplement the conquered territory at each step by one node, assuming there's a node adjacent to the territory already conquered. The algorithm consists in a while loop where we iterate as many times as we can, we look for an edge with one endpoint that we already explored and one endpoint we did not explore yet, until we don't find such an edge anymore. That exits the while loop and stops the algorithm. If we can find such an edge, we "suck" v in the conquered territory and repeat.
The claim is that this generic algorithm does what we want - finds all nodes and does not look at anything twice. This can be proven through contradiction and induction, respectively.
There are different strategies in picking the next nodes to look at, while iterating, and these strategies lead to different algorithms with different properties.
Graph Search Time Complexity
Both BFS and DFS are variants of the generic algorithm presented above. A key characteristic of the algorithm is that it iterates over all nodes in the graph, so it is Ω(n). For each node, the algorithm looks at a subset of edges, and this is how it discover adjacent unexplored node. Another key characteristic is that seen nodes are marked as "visited", so the algorithm looks at each node at maximum one time and at each edge maximum two times. I consequence the upper bound of the running time is O(m + n).
Breadth-First Search (BFS)
External: CLRS page 594.
The algorithm systematically explores the nodes in "layers", using a cautious and tentative approach. The starting node will be in its own layer. The neighbors of s will constitute layer 1, and so on. This is a cautions and tentative exploration of the graph. There is a close correspondence between these layers and shortest path distances. The algorithm is implemented using a queue (first in first out).
The algorithm is linear time O(n + m) with small constants.
BFS can be used to:
- compute shortest paths
- compute connected components of an undirected graph
BFS Algorithm Pseudocode
The general idea is to start from a node, mark it as explored and put it in a queue. Then, in a while loop as long as the queue has elements, remove the head of the queue, which is always explored, determine its adjacent unexplored nodes - in the same layer, mark them as explored and put them at the back of the queue. Ignore the adjacent explored node, to avoid doing unnecessary work The queue always contains explored nodes at the frontier between the explored area and the unexplored area. Iterate until the queue is empty and the whole connected component is explored.
BFS(graph G, start vertex s)
# All nodes are assumed unexplored
initialize a Queue Q (FIFO)
mark s as explored
place s in Q
while Q has elements
remove the head of the queue v
for each edge (v, w):
if w unexplored:
mark w as explored
add w to Q
When we "mark as explored" we can actually annotate each node marked as explored with a value of a distance from s that indicates the layers the nodes are in. This is how to count shortest path in graphs.
BFS Applications
Find the Shortest Path in a Graph
Find Connected Components in an Undirected Graphs
Depth-First Search (DFS)
External: CLRS page 603.
Depth-first search performs a more aggressive search than BFS. It explores the graph as deep as it can while backtracking only when necessary. It uses a different data structure, a stack (last in first out). There is a recursive implementation when the stack structure is implicitly implemented by the recursive calls. The algorithm is linear time O(n + m) with small constants.
DFS Iterative Algorithm Pseudocode
The general idea is to start from a node, mark it as explored and put it at the top of a stack (LIFO). Then, in a while loop as long as the stack has elements, pop the top of the stack, which is always explored, determine its adjacent unexplored nodes, choose one, mark it as explored and put it at the top of the stack. Ignore the adjacent explored node, to avoid doing unnecessary work The stack always contains explored nodes at the frontier between the explored area and the unexplored area. Iterate until the stack is empty and the whole connected component is explored. The "aggressiveness" of the algorithm comes from the fact that it chooses an unexplored node that is farther and farther away from the start node, aways biased towards the most recently discovered explored nodes, and backtracks when it runs out of options, as opposite to BFS that favors nodes arranged in layers at the same distance from the start node.
Iterative_DFS(graph G, start vertex s)
# All nodes are assumed unexplored
initialize a Stack S (LIFO)
mark s as explored
push s at the top of S
while S has elements
pop the top of the stack v
for each edge (v, w):
if w unexplored:
mark w as explored
push w at the top of S
DFS Recursive Algorithm Pseudocode
The explicit use of a stack can be naturally replaced with a recursive call:
Recursive_DFS(graph G, start vertex s)
mark s as explored
for every edge (s, v):
if v unexplored:
Recursive_DFS(G, v)