From b2eb347c5b4b5b8d835f5b4be607065454d36f88 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?V=C3=A1clav=20Kon=C4=8Dick=C3=BD?=
<koncicky@kam.mff.cuni.cz>
Date: Sun, 11 Apr 2021 19:14:21 +0200
Subject: [PATCH] Dynamization: Full dynamization
---
vk-dynamic/dynamic.tex | 34 ++++++++++++++++++++++++++++++++++
1 file changed, 34 insertions(+)
diff --git a/vk-dynamic/dynamic.tex b/vk-dynamic/dynamic.tex
index 4753d1b..21314a7 100644
--- a/vk-dynamic/dynamic.tex
+++ b/vk-dynamic/dynamic.tex
@@ -377,4 +377,38 @@ construction steps, each taking $B_S(2^i)/2^i$ time. Summing this
together, we get the required upper bound.
\qed
+\subsection{Full dynamization}
+
+For our definition of search problems, it is not easy to delete elements, as
+anytime we wished to delete an element we would need to take apart and split a
+structure into a few smaller ones. This could never be able to amortize to
+decent deletion time.
+
+Instead of that, we will want the underlying static structure to have an
+ability to cross out elements. These elements will no longer participate in
+queries, but they will count towards the structure size and complexity.
+
+Once we have ability to cross out elements, we can upgrade the semidynamic data
+structure to support deletion. We add a binary search tree or another set
+structure which maps each element to a block it lives in. For each element we
+keep a pointer on its instance in the BST. When we build a new block, we can
+update all its current elements in the tree in constant time (and insert the
+new one in logarithmic time).
+
+Insertion time complexity then will always take at least logaritmic time and
+space requirements increase by the BST.
+
+Deletion then finds an element in the BST, locates it in the corresponding
+block and crosses it out. We also keep the count of crossed out elements. If
+this count becomes a certain fraction of all elements, we rebuild the structure
+completely.
+
+Before having to rebuild the whole structure, we cross-out at least $\Theta(n)$
+elements, so the deletion time can be amortized and it will result in same time
+complexity as insertion.
+
+There also exists an worst-case version of full dynamization which carefully
+manipulates with blocks, splitting and merging them as required. The details
+are somewhat complicated, so we will not look at them further.
+
\endchapter
--
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