$\require{mathtools}
\newcommand{\nc}{\newcommand}
%
%%% GENERIC MATH %%%
%
% Environments
\newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work
\renewcommand{\r}{\mathrm}
\renewcommand{\t}{\textrm}
%
% Delimiters
% (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing)
% .. one-part
\newcommand{\p}[1]{\mathopen{}\left( #1 \right)}
\renewcommand{\P}[1]{^{\p{#1}}}
\renewcommand{\b}[1]{\mathopen{}\left[ #1 \right]}
\newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}}
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\newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle}
\newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \strut \right\rVert}
\newcommand{\frob}[1]{\norm{#1}_\mathrm{F}}
\newcommand{\mix}[1]{\mathopen{}\left\lfloor #1 \right\rceil}
%% .. two-part
\newcommand{\inco}[2]{#1 \mathop{}\middle|\mathop{} #2}
\newcommand{\co}[2]{ {\left.\inco{#1}{#2}\right.}}
\newcommand{\cond}{\co} % deprecated
\newcommand{\pco}[2]{\p{\inco{#1}{#2}}}
\newcommand{\bco}[2]{\b{\inco{#1}{#2}}}
\newcommand{\setco}[2]{\set{\inco{#1}{#2}}}
\newcommand{\at}[2]{ {\left.#1\strut\right|_{#2}}}
\newcommand{\pat}[2]{\p{\at{#1}{#2}}}
\newcommand{\bat}[2]{\b{\at{#1}{#2}}}
\newcommand{\para}[2]{#1\strut \mathop{}\middle\|\mathop{} #2}
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\newcommand{\bffco}[4]{\bff{\cond{#1}{#2}}{\cond{#3}{#4}}}
%
% Greek
\newcommand{\eps}{\epsilon}
\newcommand{\veps}{\varepsilon}
\newcommand{\vpi}{\varpi}
% the following cause issues with real LaTeX tho :/ maybe consider naming it \fhi instead?
\let\fi\phi % because it looks like an f
\let\phi\varphi % because it looks like a p
\renewcommand{\th}{\theta}
\newcommand{\Th}{\Theta}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
%
% Miscellaneous
\newcommand{\LHS}{\mathrm{LHS}}
\newcommand{\RHS}{\mathrm{RHS}}
\DeclareMathOperator{\cst}{const}
% .. operators
\DeclareMathOperator{\poly}{poly}
\DeclareMathOperator{\polylog}{polylog}
\DeclareMathOperator{\quasipoly}{quasipoly}
\DeclareMathOperator{\negl}{negl}
\DeclareMathOperator*{\argmin}{arg\thinspace min}
\DeclareMathOperator*{\argmax}{arg\thinspace max}
% .. functions
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\err}{err}
\DeclareMathOperator{\ReLU}{ReLU}
% .. analysis
\let\d\undefined
\newcommand{\d}{\operatorname{d}\mathopen{}}
\newcommand{\dd}[1]{\operatorname{d}^{#1}\mathopen{}}
\newcommand{\df}[2]{ {\f{\d #1}{\d #2}}}
\newcommand{\ds}[2]{ {\sl{\d #1}{\d #2}}}
\newcommand{\ddf}[3]{ {\f{\dd{#1} #2}{\p{\d #3}^{#1}}}}
\newcommand{\dds}[3]{ {\sl{\dd{#1} #2}{\p{\d #3}^{#1}}}}
\renewcommand{\part}{\partial}
\newcommand{\partf}[2]{\f{\part #1}{\part #2}}
\newcommand{\parts}[2]{\sl{\part #1}{\part #2}}
\newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}}
% .. sets
\newcommand{\es}{\emptyset}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\zo}{\set{0,1}}
\newcommand{\pmo}{\set{\pm 1}}
\newcommand{\zpmo}{\set{0,\pm 1}}
% .... set operations
\newcommand{\sse}{\subseteq}
\newcommand{\out}{\not\in}
\newcommand{\minus}{\setminus}
\newcommand{\inc}[1]{\union \set{#1}} % "including"
\newcommand{\exc}[1]{\setminus \set{#1}} % "except"
% .. over and under
\renewcommand{\ss}[1]{_{\substack{#1}}}
\newcommand{\OB}{\overbrace}
\newcommand{\ob}[2]{\OB{#1}^\t{#2}}
\newcommand{\UB}{\underbrace}
\newcommand{\ub}[2]{\UB{#1}_\t{#2}}
\newcommand{\ol}{\overline}
\newcommand{\tld}{\widetilde} % deprecated
\renewcommand{\~}{\widetilde}
\newcommand{\HAT}{\widehat} % deprecated
\renewcommand{\^}{\widehat}
\newcommand{\rt}[1]{ {\sqrt{#1}}}
\newcommand{\for}[2]{_{#1=1}^{#2}}
\newcommand{\sfor}{\sum\for}
\newcommand{\pfor}{\prod\for}
% .... two-part
\newcommand{\f}{\frac}
\renewcommand{\sl}[2]{#1 /\mathopen{}#2}
\newcommand{\ff}[2]{\mathchoice{\begin{smallmatrix}\displaystyle\vphantom{\p{#1}}#1\\[-0.05em]\hline\\[-0.05em]\hline\displaystyle\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}}
% .. arrows
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\DeclareMathOperator*{\<}{\!\;\longleftarrow\;\!}
\let\>\undefined
\DeclareMathOperator*{\>}{\!\;\longrightarrow\;\!}
\let\-\undefined
\DeclareMathOperator*{\-}{\!\;\longleftrightarrow\;\!}
\newcommand{\so}{\implies}
% .. operators and relations
\renewcommand{\*}{\cdot}
\newcommand{\x}{\times}
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\newcommand{\ce}{\coloneqq}
\newcommand{\ec}{\eqqcolon}
\newcommand{\ap}{\approx}
\newcommand{\ls}{\lesssim}
\newcommand{\gs}{\gtrsim}
% .. punctuation and spacing
\renewcommand{\.}[1]{#1\dots#1}
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%
% Levels of closeness
\newcommand{\scirc}[1]{\sr{\circ}{#1}}
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\newcommand{#2}{\mathrel{(#1)}}
\newcommand{#3}{\mathrel{#1}}
\newcommand{#4}{\mathrel{#1\!\!#1}}
\newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}}
\newcommand{#6}{\mathrel{(\sdot{#1})}}
\newcommand{#7}{\mathrel{(\slog{#1})}}
}
\let\lt\undefined
\let\gt\undefined
% .. vanilla versions (is it within a constant?)
\newcommand{\ez}{\scirc=}
\newcommand{\eq}{\simeq}
\newcommand{\eqq}{\mathrel{\eq\!\!\eq}}
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\newcommand{\lez}{\scirc\le}
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\newcommand{\gt}{\succ}
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% .. dotted versions (is it equal in the limit?)
\newcommand{\ed}{\sdot=}
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% .. log versions (is it equal up to log?)
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%
%
%%% SPECIALIZED MATH %%%
%
% Logic and bit operations
\newcommand{\fa}{\forall}
\newcommand{\ex}{\exists}
\renewcommand{\and}{\wedge}
\newcommand{\AND}{\bigwedge}
\renewcommand{\or}{\vee}
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\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\bigoplus}
\newcommand{\union}{\cup}
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\DeclareMathOperator{\One}{\mathbb{1}}
\DeclareMathOperator{\1}{\mathbb{1}} % use \mathbbm instead if using real LaTeX
\DeclareMathOperator{\LSB}{LSB}
%
% Linear algebra
\newcommand{\spn}{\mathrm{span}} % do NOT use \span because it causes misery with amsmath
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\Img}{Im}
\newcommand{\transp}{\mathsf{T}}
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% .. named tensors
\newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut
\newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}}
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%
% Probability
\newcommand{\tri}{\triangle}
\newcommand{\Normal}{\mathcal{N}}
% .. operators
\DeclareMathOperator{\supp}{supp}
\let\Pr\undefined
\DeclareMathOperator*{\Pr}{Pr}
\DeclareMathOperator*{\G}{\mathbb{G}}
\DeclareMathOperator*{\Odds}{Od}
\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
\DeclareMathOperator*{\Cov}{Cov}
\DeclareMathOperator*{\K}{K}
\DeclareMathOperator*{\corr}{corr}
\DeclareMathOperator*{\median}{median}
\DeclareMathOperator*{\maj}{maj}
% ... information theory
\let\H\undefined
\DeclareMathOperator*{\H}{H}
\DeclareMathOperator*{\I}{I}
\DeclareMathOperator*{\D}{D}
\DeclareMathOperator*{\KL}{KL}
% .. other divergences
\newcommand{\dTV}{d_{\mathrm{TV}}}
\newcommand{\dHel}{d_{\mathrm{Hel}}}
\newcommand{\dJS}{d_{\mathrm{JS}}}
%
%%% SPECIALIZED COMPUTER SCIENCE %%%
%
% Complexity classes
% .. classical
\newcommand{\Poly}{\mathsf{P}}
\newcommand{\NP}{\mathsf{NP}}
\newcommand{\PH}{\mathsf{PH}}
\newcommand{\PSPACE}{\mathsf{PSPACE}}
\renewcommand{\L}{\mathsf{L}}
% .. probabilistic
\newcommand{\formost}{\mathsf{Я}}
\newcommand{\RP}{\mathsf{RP}}
\newcommand{\BPP}{\mathsf{BPP}}
\newcommand{\MA}{\mathsf{MA}}
\newcommand{\AM}{\mathsf{AM}}
\newcommand{\IP}{\mathsf{IP}}
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% .. circuits
\newcommand{\NC}{\mathsf{NC}}
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\newcommand{\ThrC}{\mathsf{TC}}
\newcommand{\Ppoly}{\mathsf{P}/\poly}
\newcommand{\Lpoly}{\mathsf{L}/\poly}
% .. resources
\newcommand{\TIME}{\mathsf{TIME}}
\newcommand{\SPACE}{\mathsf{SPACE}}
\newcommand{\TISP}{\mathsf{TISP}}
\newcommand{\SIZE}{\mathsf{SIZE}}
% .. keywords
\newcommand{\coclass}{\mathsf{co}}
\newcommand{\Prom}{\mathsf{Promise}}
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% Boolean analysis
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\DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree
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% .. functions (small caps sadly doesn't work)
\DeclareMathOperator{\Par}{\mathrm{Par}}
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\DeclareMathOperator{\HW}{\mathrm{HW}}
\DeclareMathOperator{\Thr}{\mathrm{Thr}}
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\DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}}
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\DeclareMathOperator{\SAT}{\mathrm{SAT}}
\DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}}
%
% Dynamic optimality
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%
% Alignment
\DeclareMathOperator{\Amp}{\mathrm{Amp}}
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%%% TYPESETTING %%%
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% In "text"
\newcommand{\heart}{\heartsuit}
\newcommand{\nth}{^\t{th}}
\newcommand{\degree}{^\circ}
% remove these last two if using real LaTeX
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\newcommand{\qedhere}{\tag*{$\blacksquare$}}
%
% Fonts
% .. bold
\newcommand{\BA}{\boldsymbol{A}}
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% .. calligraphic
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% .. typewriter
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\newcommand{\TV}{\mathtt{V}}
\newcommand{\TW}{\mathtt{W}}
\newcommand{\TX}{\mathtt{X}}
\newcommand{\TY}{\mathtt{Y}}
\newcommand{\TZ}{\mathtt{Z}}$
Adapted from a talk by Prasanna Ramakrishnan at Stanford’s teach-us-anything seminar.
You probably know the drill about $s$-$t$ connectivity:
- DFS/BFS gets you $\TISP(m,n)$,
- Savitch’s gets you $\TISP(n^{O(\log n)}, \log^2 n)$.
Can you get the best of both worlds? Some believe $\TISP(\poly(n), \polylog(n))$ is possible. But for now, the best we can get in polynomial time is $n / 2^{\Omega(\sqrt{\log n})}$ space.
Idea
Pras claims that if you’re given one week and the hint “find something in the middle of BFS and Savitch’s”, you could come up with it. I doubt I could.
The idea is to split the problem into two parts:
- one BFS-like part that “splits the path from $s$ to $t$ into $n/\lambda$ segments of length $\lambda$”;
- one Savitch-like part that can check whether two nodes are at distance $\leq \lambda$ (as long as $\lambda$ is small).
The BFS-like part makes a lot of concessions in time in order to cut space. The Savitch-like part makes a lot of concessions in space in order to cut time.
BFS-like part: cut it up
BFS requires a lot of space to store all the “equidistance shells”: for each $d$, it stores the set$V_d$ of all vertices at distance $d$ from node $s$. One natural way to save on this is to only store one out of $\lambda$ of those shells: there must be some remainder $r$ such that the shells $V_r, V_{\lambda+r}, V_{2\lambda+r}, \ldots$ contain only $O(n/\lambda)$ nodes.
How do you compute $V_{(i+1)\lambda+r}$ from the previous ones though? The idea is that $v \in V_{(i+1)\lambda+r}$ if and only if:
- it’s at distance exactly $\lambda$ from $V_{i\lambda+r}$;
- it’s not in any of the earlier shells $V_0, \ldots, V_{i\lambda+r-1}$.
The first part is easy to check using two small-distance queries from each $w \in V_{i\lambda+r}$. And you can check the second part simply by checking that $v$ is not a distance $<\lambda$ from the previous shells we memorized $V_{r}, \ldots, V_{(i-1)\lambda + r}$. That’s a lot of queries, but we’re fine with polynomial blow-ups in time!
Overall, we only make a polynomial number of queries, and we need $O(\frac{n \log n}{\lambda})$ bits of memory to store all the shells $V_r, V_{\lambda+r}, \ldots$. So if there was some $\TISP(t(n,\lambda),s(n,\lambda))$ way to check whether two points are at distance at most $\lambda$, we could solve STCON in
\[\TISP(\poly(n)\cdot t(n,\lambda), \frac{n \log n}{\lambda} + s(n,\lambda)).\]
This means that to get poly-time sublinear-space connectivity, we need
- $\lambda \geq \omega(\log n)$;
- $t(n, \lambda) \leq \poly(n)$;
- $s(n, \lambda) \leq o(n)$.
Savitch-like part
Let’s figure what time-space trade-off we can get from Savitch assuming $s$ and $t$ are at distance $\leq \lambda$. Out of the box, it gets you $n^{O(\log \lambda)}$ time and $O(\log n \log \lambda)$ space. Since $\lambda$ is superconstant, this is definitely not going to be polynomial-time, but this is already something nontrivial: for example, setting $\lambda = (\log n)^{100}$ gets you $\TISP(n^{O(\log \log n)}, n/(\log n)^{99})$, which is incomparable with BFS and Savitch’s.
But really, out-of-the-box Savitch’s cares way too much about optimizing space. We only want sublinear space (we have this annoying $\frac{n \log n}{\lambda}$ anyway), so we should be able to do much better!
The trick is to reduce Savitch’s branching factor at the expense of space. Instead of choosing a specific middle vertex among all $n$ possible vertices, we’ll choose a collection of possible middle vertices among $k$ collections of $n/k$ vertices (decided arbitrarily, say by their remainder modulo $k$).
The specification of one recursion of Savitch’s then becomes:
$\mathrm{Savitch}(d, i \in [k],j \in [k],x \in \zo^{n/k})$: returns a vector $y \in \zo^{n/k}$ representing which nodes from collection $j$ are reachable within distance $\leq d$ from at least one of the nodes in collection $i$ that are indicated by a $1$ in $x$.
To solve this recursive problem, first initialize $y \gets 0^{n/k}$, then for each possible “middle collection” $l \in [k]$:
- make a recursive call to find out the vector $z \in \zo^{n/k}$ of each possible node in collection $l$ that is reachable within distance $\leq d/2$ from at least one of the nodes in collection $i$ indicated by $x$;
- make a second recursive call to find out the vector $w \in \zo^{n/k}$ of each possible node in collection $j$ that is reachable within distance $\leq d/2$ from at least one of the nodes in collection $l$ indicated by $z$;
- set $y \gets y \vee w$ (component-wise).
Overall, there are $\log \lambda$ recursion levels, the branching factor is $k$, and we need $\poly(n)$ time and $O(n/k + \log k)$ memory at each recursion level, so this gives
\[\TISP\left(\poly(n) \cdot k^{O(\log \lambda)}, (\log \lambda)(n/k + \log k)\right).\]
Choosing parameters
Let’s ignore the $+ \log k$, it will end up being small anyway. We’re left with
\[\TISP\left(\poly(n) \cdot k^{O(\log \lambda)}, \frac{n \log n}{\lambda} + \frac{n \log \lambda}{k}\right).\]
The log factors in the right-hand side are probably not going to be very different, so a reasonable first step is to equalize the denominators by setting $k \coloneqq \lambda$, and get
\[\TISP\left(\poly(n) \cdot \lambda^{O(\log \lambda)}, \frac{n \log n}{\lambda}\right).\]
Then, it only remains to solve for $\lambda^{O(\log \lambda)} \leq\poly(n)$, which holds as long as $\lambda \le 2^{O(\sqrt{\log n})}$. So the best bound we can get for polynomial time is
\[
\TISP\left(\poly(n), \frac{n}{2^{\Omega\p{\sqrt{\log n}}}}\right).
\]