Linear probing: Forgotten constant in the proof

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.