Suggest edit — STCONN

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title: "STCONN"
source: https://www.jemoka.com/posts/kbhstconn/
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\begin{equation} \text{STCONN} = \qty {\langle G, S, t \rangle : \text{$\exists^{?}$ a path from s $\to$ t $\in$ G}} \end{equation}
we solve this with BFS or DFS; but those algorithms&rsquo; working sets require linear space. Turns out, we can solve this algorithm $\text{STCONN} \in \text{SPACE}\qty(\log^{2}\qty(n))$
for directed $G$, we are not sure if its in L. for undirected $G$, Omer showed that its in L open problem: Savitch&rsquo;s Algorithm is really really slow; we are not sure if there is an algorithm in which we can solve it in better time.
Savitch&rsquo;s Algorithm Savitch&rsquo;s Algorithm is a middle-first search scheme for solving STCONN.
key observation: if there is a path form $s \to t$ of length $n$, then there must be a point $w$ such that $s\to w$ in $\frac{n}{2}$ steps and $w \to t$ in $\frac{n}{2}$ steps.
Goal: brute force search for this $w$. There is $n$ possibilities for $w$, we shall just try all of them! This engenders a more general problem named PATH: $\text{STCONN}\qty(s,t) = \text{PATH}\qty(s,t,\ceil{\log n})$
PATH \begin{equation} \text{PATH}\qty(x,y,k): \text{does there exist path of length $\leq 2^{k}$ from $x$ to $y$} \end{equation}
solution:
if $k = 0$, then return TRUE IFF $\qty(x,y) \in E \vee x = y$ (that $x$ and $y$ are adjacent or identical) else for each $w \in V$ if $\text{PATH}\qty(x,w,k-1) \wedge \text{PATH}\qty(w,y,k-1)$, return TRUE return FALSE space of this:
\begin{equation} S\qty(k) = O\qty(\log n) + S\qty(k-1) \end{equation}
because for every $w$, we need to track $\log n$ possible $w$, and then also need to recurse down to keep track the two $\text{PATH}$ calls; in particular, we recuse the two recurse PATH calls to prevent exponential growth. This gives $S(t) = O\qty(k \log n)$, so for $k = \log n$, we have $S \qty(\log n) = O \qty(\log^{2} n)$
time of this:
\begin{equation} T(k) = n 2 T(k-1) = \qty(2n)^{k} = n^{O(k)} \end{equation}
so for $k = \log n$ (n nodes), then $T \qty(\log n) = n^{O \qty(\log n)}$
because we can&rsquo;t recuse time, so the 2 times factor exsits.
STCONN is in NL see STCONN is in NL