Does reachability belong to P?

Question

Reachability is defined as follows: a digraph $G = (V, E)$ and two vertices $v,w \in V$. Is there a directed path from $v$ to $w$ in $G$?

Is it possible to write a polynomial time algorithm for it?

I asked this question on mathematics and got no answer by far.

Answer 1

Although you already know from the other answers that the question is solvable in polynomial time, I thought I would expand on the computational complexity of reachability since you used complexity terminology in your question.

Reachability (or st-connectivity) in digraphs is the prototypical $NL$-complete problem where $NL$ stands for non-deterministic log space and we use deterministic log-space reductions (although I think it remains complete for $NC^1$ reductions, too).

• To see why it is in $NL$, notice that you can guess a next vertex at every step and verify that it is connected to the previous vertex. A series of correct guesses exists if and only if there is a path from $s$ to $t$.
• To see why is $NL$-hard, notice that the behavior of a non-deterministic Turing machine can be represented by a configuration graph. The nondeterministic machine accepts only if there exists a path from the start configuration to an accept configuration, and if the machine only uses $S(n)$ tape entries then the configuration graph is of size $O(|\Gamma|^{S(n)})$ where $\Gamma$ is the tape alphabet. If $S(n)$ is logarithmic, then the resulting graph is of polynomial size and the result follows.

But I don't have access to a non-deterministic machine, so why should I care? Well, we know lots of things about $NL$; one of those is that is in $P$, which you know from the other answers. However, here are tighter facts that can be useful:

• From Savitch's theorem we know that $NL \subseteq \text{DSPACE}((\log n)^2)$: even on a deterministic machine you don't need that much space to solve the question.

• We know that $NL \subseteq NC^2$: this means that in the circuit model, your question can be solved by a polynomial sized circuit of depth $O((\log n)^2)$. In a more "heuristic" sense, this means that the problem is parallelizable since Nick's Class captures the idea of quick solutions on a parallel computer.

• We know that $NL \subseteq \text{LOGCFL}$ which means that it is not harder (up to log-space reductions) than membership checking in context-free languages which can be a good source of intuition.

Finally, the directed nature of the graph is essential. If the graph is undirected then we believe the question is significantly easier. In particular, undirected st-connectivity is complete for $L$ (deterministic log space) under first-order reductions (Reingold 2004; pdf).

Answer 2

Yes, this problem is solveable in linear time, $O(|V|+|E|)$ to be precise.

The two classic solutions to this are Breadth-First search and Depth-First search.

The algorithms basically look like this:

current = v
while (current has an edge to an unmarked vertex)
    if current == w
        return true
    mark current as visited
    for each u where (v,u) is in E
        add u to the Open List
    current = a vertex from the Open List
return false

BFS uses a queue as the open list, adding to the back and taking from the front. DFS uses a stack. In any implementation like this, each node and each edge are visited at most once, so the algorithm runs in linear time.

Answer 3

Yes, it is in P.

The natural algorithm for is very simple, so simple it doesn't really serve as a learning experience to state it here (it's readily available in almost any text or on the web).

To put you on the right path:

1. What algorithms do you know for exploring a graph?
2. If you were at the start vertex $v$, what's the obvious way of trying to find $w$?