From 126288502f64e1959e82bcab21c107f7fbee8f68 Mon Sep 17 00:00:00 2001
From: Camil Staps
Date: Tue, 15 Dec 2015 19:53:30 +0000
Subject: Finish report practical 2

---
 Practical2/report/notation.tex | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

(limited to 'Practical2/report/notation.tex')

diff --git a/Practical2/report/notation.tex b/Practical2/report/notation.tex
index cbbbc1d..0781524 100644
--- a/Practical2/report/notation.tex
+++ b/Practical2/report/notation.tex
@@ -1,9 +1,11 @@
 \section{Notation}
 \label{sec:notation}
 
-I will write $\bar a = \seq{a_i \mid i<n}$ for the sequence of numbers and $k$ for the number of substrings. I redefine $\round{x}$ to mean `the nearest multiple of $5$ to $x$'. We may define the `round profit' as $\PR(x)=x-\round{x}$.
+I will write $\bar a = \seq{a_i \mid i<n}$ for the sequence of numbers and $k+1$ for the number of substrings. I redefine $\round{x}$ to mean `the nearest multiple of $5$ to $x$'.
 
-Let $\PM_k(\bar a)$ be the minimal sum of rounded sums of $\bar a$ using $k$ partitions. I will use $\PP_k(\bar a)$ as the `maximum profit' of $\bar a$, that is, $\PP_k(\bar a) = \sum\bar a - \PM_k(\bar a)$. This leads to the term `profit' of a particular partitioning of $\bar a$ and of a particular (sub)string $\bar a'$, for which we need not introduce notation. The profit of a string $\bar a'$ is defined as $\PR(\sum\bar a)$. It is not difficult to see that the profit of a partition is equal to the some of the profits of its members.
+We may define the `rounding profit' of $x$ as $\PR(x)=x-\round{x}$. The rounding profit of a string $\bar a$ is defined as $\PR(\sum\bar a)$, and the rounding profit of a partition of $\bar a$ is the sum of the rounding profits of its elements.
 
-We define the gain $G^k_p(\bar a)$ as $\PP_k(\seq{a_i\in\bar a\mid i<p})$ (that is, the maximal profit on $\bar a$'s prefix with length $p$ using $k$ partitions). Lastly, let $S_i^j(\bar a)$ be the sum of all $p_x\in\bar a$ with $i\le x<j$.
+Let $\PM_k(\bar a)$ be the minimal sum of rounded sums of $\bar a$ using $k+1$ partitions. I will use $\PP_k(\bar a)$ as the `maximum profit' of $\bar a$ using $k+1$ partitions, that is, $\PP_k(\bar a) = \sum\bar a - \PM_k(\bar a)$.
+
+We define the gain $G^k_p(\bar a)$ as $\PP_k(\seq{a_i\in\bar a\mid i<p})$ (i.e., the maximal profit on $\bar a$'s prefix with length $p$ using $k+1$ partitions). Lastly, let $S_i^j(\bar a)$ be the sum of all $p_x\in\bar a$ with $i\le x<j$.
 
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