Merge branch 'master' of gitlab.odin.cse.buffalo.edu:ahuber/SketchingWorlds

mult_distinct_p.tex

@@ -12,7 +12,7 @@ In this section, we will prove that computing $\expct\limits_{\vct{W} \sim \pd}\
 \subsection{Preliminaries}
 
 Our hardness results are based on (exactly) counting the number of occurrences of a fixed graph $H$ as a subgraph in $G$. Let $\numocc{G}{H}$ denote the number of occurrences of pattern $H$ in graph $G$. %, where, for example, $\numocc{G}{\ed}$ means the number of single edges in $G$.
-In particular, we will consider the problems of computing the following counts (given $G$ as an input in its adjacency list representation): $\numocc{G}{\tri}$ (the number of triangles), $\numocc{G}{\threepath}$ (the number of $3$-paths),  $\numocc{G}{\threedis}$ (the number of $3$-matchings or collection of three node-disjoint edges) and its generalization $\numocc{G}{\kmatch}$ (the number of $k$-matchings or collections of $k$ node-disjoint edges).
+In particular, we will consider the problems of computing the following counts (given $G$ as an input and its adjacency list representation): $\numocc{G}{\tri}$ (the number of triangles), $\numocc{G}{\threepath}$ (the number of $3$-paths),  $\numocc{G}{\threedis}$ (the number of $3$-matchings or collection of three node-disjoint edges) and its generalization $\numocc{G}{\kmatch}$ (the number of $k$-matchings or collections of $k$ node-disjoint edges).
 %
 Our hardness result in \Cref{sec:multiple-p} is based on the following result:
 
@@ -35,7 +35,7 @@ There exists a constant $\eps_0>0$ such that given an undirected graph $G=(V,E)$
 \end{hypo}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %
-Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection whether $G$ has a triangle or not takes time $\Omega\inparen{|E|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$. 
+Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection of whether $G$ has a triangle or not takes time $\Omega\inparen{|E|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$. 
 %The current best known algorithm to count the number of $3$-matchings, to
 %\AR{Need to add something about 3-paths and 3-matchings as well.}
 
@@ -73,7 +73,7 @@ Computing $\rpoly_G^\kElem(\prob_i,\dots,\prob_i)$ for arbitrary $G$ and any $(2
 %
 We will prove the above result by reduction from the problem of computing the number of $k$-matchings in $G$. Given the current best-known algorithm for this counting problem, our results imply that unless the state-of-the-art $k$-matching algorithms are improved, we cannot hope to solve our problem in time better than $\Omega_k\inparen{m^{k/2}}$, which is only quadratically faster than expanding $\poly_{G}^\kElem(\vct{X})$ into its \abbrSMB form and then using \Cref{cor:expct-sop}. By contrast the approximation algorithm we present in \Cref{sec:algo} has runtime $O_k\inparen{m}$ for  this query (since it runs in linear-time on all lineage polynomials).
 
-Here, we present a reduction from the problem of couting $\kElem$-matchings in a graph to our problem:
+Here, we present a reduction from the problem of counting $\kElem$-matchings in a graph to our problem:
 \begin{Lemma}\label{lem:qEk-multi-p}
 Let $\prob_0,\ldots, \prob_{2\kElem}$ be distinct values in $(0, 1]$.  Then given the values $\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$ for $0\leq i\leq 2\kElem$, the number of $\kElem$-matchings in $G$ can be computed in $O\inparen{\kElem^3}$ time.
 \end{Lemma}