From b4eeffa93a0ac7aadb6a85b8af0231f6b9ab16d7 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Tue, 9 Feb 2021 13:03:21 +0100
Subject: [PATCH] Linear probing: Forgotten constant in the proof
---
06-hash/hash.tex | 6 +++---
1 file changed, 3 insertions(+), 3 deletions(-)
diff --git a/06-hash/hash.tex b/06-hash/hash.tex
index e9c4fe4..8d6b704 100644
--- a/06-hash/hash.tex
+++ b/06-hash/hash.tex
@@ -806,9 +806,9 @@ $$
We bound the former probability simply by~1 and use the previous corrolary to bound
the latter probability:
$$
-\E[\vert R\vert] \le 3 + \sum_{\ell\ge 0} 2^{\ell+3} \cdot q^{2^\ell}
- = 3 + 8 \cdot \sum_{\ell\ge 0} 2^\ell \cdot q^{2^\ell}
- \le 3 + 8 \cdot \sum_{i\ge 1} i\cdot q^i.
+\E[\vert R\vert] \le 3 + \sum_{\ell\ge 0} 2^{\ell+3} \cdot 12 \cdot q^{2^\ell}
+ = 3 + 8 \cdot 12 \cdot \sum_{\ell\ge 0} 2^\ell \cdot q^{2^\ell}
+ \le 3 + 96 \cdot \sum_{i\ge 1} i\cdot q^i.
$$
Since the last sum converges for an arbitrary $q\in(0,1)$, the expectation of~$\vert R\vert$
is at most a~constant. This concludes the proof of the theorem.
--
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