Semi-streaming algorithm for $s$-$t$ connectivity

Question

Let $G=(V,E)$ be an undirected graph. Given a pair of vertices $s,t \in V$, how can we construct a semi-streaming algorithm which determines is $s$ and $t$ are connected? Is there any way to construct such an algorithm which scans the input stream only once?

Note that a semi-streaming algorithm is presented an arbitrary order of the edges of $G$ as a stream, and the algorithm can only access this input sequentially in the order it is given; it might process the input stream several times. The algorithm has a working memory consisting of $O(n\log^{O(1)}n)$ bits.

Answer 1

You cannot do it in a single pass. Consider the set of all graphs of the following form: the vertices are $\{s,t\} \cup A \cup B$, where $|A|=|B|=n/2-1$. The only allowed edges are between $s$ and $A$, between $A$ and $B$, and between $B$ and $t$.

Suppose that the algorithm is first presented with the $(A,B)$ edges, and then with the $(s,A),(B,t)$ edges. After reading the $(A,B)$ edges, it must "remember" all of them, since there could be only one $(s,A)$ edge and only one $(B,t)$ edge, in which case the answer depends on whether the corresponding vertices in $A$ and $B$ are connected. Since there are $2^{(n/2-1)^2}$ choices for the $(A,B)$ edges, the algorithm must have a memory of at least $(n/2-1)^2$ bits (since after reading the $(A,B)$ edges, it could be in any of $2^{(n/2-1)^2}$ possible states).

Answer 2

I just wanted to add a comment to Yuval's answer but I don't have 50 reputation yet.

We don't have to distinguish all different $2^{(n/2-1)^2}$ subgraphs since for some subgraphs their answer to $s$-$t$ connectivity would be the same. I believe that we can use a disjoint set union data structure that solves this problem... (imagine that you are running Kruskal's algorithm on a unweighted graph, the goal is to come up with a connectivity witness shows that $s$ and $t$ are connected, you only need $O(n\log n)$ bits of information so store a tree.)