From 590bc9cbd9696f1ee79ebe2ae498c7b7da96c91c Mon Sep 17 00:00:00 2001 From: Atri Rudra Date: Fri, 31 Jul 2020 21:12:48 -0400 Subject: [PATCH] starting my pass on Lemma 5. --- poly-form.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/poly-form.tex b/poly-form.tex index 5c1e6d2..de408f0 100644 --- a/poly-form.tex +++ b/poly-form.tex @@ -181,8 +181,9 @@ If we can compute $\poly(\vct{X})$ in $T(m)$ time for $O(1)$ distinct values of \AR{Follows from the fact that the corresponding coefficient matrix is the so called Vandermonde matrix, which has full rank.} \AH{This Vandermonde matrix I need to research. Right now, the last sentences are just parrotting Atri.} +\AR{Jul 31: Did not make a pass on anything above this.} \begin{Lemma}\label{lem:const-p} -If we can compute $\poly(\vct{X})$ in T(m) time for $\wElem_1 =\cdots= \wElem_\numTup = \prob$, then we can count the number of triangles, 3-paths, and 3-matchings in $G$ in $T(m) + O(m)$ time. +If we can compute $\poly(\vct{X})$ in T(m) time for $\wElem_1 =\cdots= \wElem_\numTup = \prob$,\AR{Nope, this is not correct. Either use $\tilde{Q}(p,\dots,p)$ or $\mathbb{E}_{\vct{W}\sim P^{(\vct{p})}}[Q(\vct{W})]$ for $\vct{p}=(p,\dots,p)$. Also you have to have $G$ in the notation $Q$ as the statement as the claim is you can do the computation for {\em every} $G$ with $m$ edges in $T(m)$ then you can compute the number of triangles etc. for arbitrary $G$ as well.} then we can count the number of triangles, 3-paths, and 3-matchings in $G$ in $T(m) + O(m)$ time. \end{Lemma} @@ -540,4 +541,4 @@ Further algebraic manipulations result in The roots for \cref{eq:det-combine} are $p = 0, p = 1$, and $p = i$. Thus, we have proved the lemma for fixed $p \in (0, 1)$. \end{proof} -\qed \ No newline at end of file +\qed