Heaps: Intro

01-prelim/prelim.tex

@@ -3,14 +3,6 @@
 \singlechapter{1}
 \fi
 
-\def\optable#1{$$
-\def\cr{\crcr\noalign{\smallskip}}
-\vbox{\halign{
-\hbox to 9em{##\hfil}&\vtop{\hsize=0.65\hsize\parindent=0pt\strut ##\strut}\crcr
-#1
-\noalign{\vskip-\smallskipamount}
-}}$$}
-
 \chapter[prelim]{Preliminaries}
 
 Generally, a~data structure is a~``black box'', which contains some data and

04-heaps/Makefile

+TOP=..
+PICS=
+
+include ../Makerules

04-heaps/heaps.tex

+\ifx\chapter\undefined
+\input adsmac.tex
+\singlechapter{4}
+\fi
+
+\chapter[heaps]{Heaps}
+
+\df{Heaps} are a~family of data structures for dynamic selection of a~minimum.
+They maintain a~set of items, each equipped with a~\em{priority.} The priorities
+are usually numeric, but we will assume that they come from some abstract universe
+and they can be only compared. Besides the priority, the items can carry arbitrary
+other data, usually called the \em{value} of the item.
+
+Heaps typically support the following operations. Since a~heap cannot efficiently
+find an~item by neither its priority nor value, operations which modify items get
+an~abstract identifier of the item. The identifiers are assigned on insertion;
+internally, they are usually pointers.
+\optable{
+$\opdf{Insert}(p,v)$	& Create a~new item with priority~$p$ and value~$v$ and
+			  return its identifier. \cr
+\opdf{Min}		& Find an~item with minimum priority. If there are
+			  multiple such items, pick any. If the heap is empty, return~$\emptyset$. \cr
+\opdf{ExtractMin}	& Find an~item with minimum priority and remove it
+			  from the heap. If the heap is empty, return~$\emptyset$. \cr
+$\opdf{Decrease}(\<id>,p)$
+			& Decrease priority of the item identified by~\<id> to~$p$. \cr
+$\opdf{Increase}(\<id>,p)$
+			& Increase priority of the item identified by~\<id> to~$p$. \cr
+$\opdf{Delete}(\<id>)$
+			& Remove the given item from the heap.
+			  It is usually simulated by $\alg{Decrease}(\<id>,-\infty)$
+			  followed by \alg{ExtractMin}. \cr
+$\opdf{Build}((p_1,v_1),\ldots)$
+			& Create a~new heap containing the given items.
+			  It is equivalent to a~sequence of \alg{Inserts} on an empty
+			  heap, but it can be often implemented more efficiently. \cr
+}
+By reversing the order on priorities, we can obtain the a~maximal heap, which maintains
+the maximum instead of the minimum.
+
+\obs{
+We can sort a~sequence of $n$~items by inserting them to a~heap and then calling
+\alg{ExtractMin} $n$~times. The standard lower bound on comparison-based sorting implies
+that at least one of \alg{Insert} and \alg{ExtractMin} must take $\Omega(\log n)$ time
+amortized.
+}
+
+We can implement the heap interface using a~search tree, which yields $\Theta(\log n)$
+time complexity of all operations except \alg{Build}. However, specialized constructions
+presented in this chapter
+will achieve $\Theta(\log n)$ amortized time for \alg{ExtractMin}, $\Theta(n)$ for \alg{Build},
+and $\Theta(1)$ amortized for all other operations.
+
+\section{Regular heaps}
+
+\section{Binomial heaps}
+
+\section{Lazy binomial heaps}
+
+\section{Fibonacci heaps}
+
+\endchapter

tex/adsmac.tex

@@ -269,6 +269,15 @@
 % C++
 \def\Cpp{C{\tt ++}}
 
+% Tabulka operací datové struktury
+\def\optable#1{$$
+\def\cr{\crcr\noalign{\smallskip}}
+\vbox{\halign{
+\hbox to 9em{##\hfil}&\vtop{\hsize=0.65\hsize\parindent=0pt\strut ##\unskip\strut}\crcr
+#1
+\noalign{\vskip-\smallskipamount}
+}}$$}
+
 %%% Fonty %%%
 
 \def\chapfont{\setfonts[LMSansDC/24]}