From 6973081a7d00bda0f7b617f902029d9dd8033119 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Thu, 23 Feb 2017 11:25:05 +0200 Subject: [PATCH 01/39] Typo fix --- chapter02.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapter02.tex b/chapter02.tex index 875c181..15141ee 100644 --- a/chapter02.tex +++ b/chapter02.tex @@ -478,7 +478,7 @@ the maximum sum of a subarray that ends at that position. After this, the answer for the problem is the maximum of those sums. -Condider the subproblem of finding the maximum-sum subarray +Consider the subproblem of finding the maximum-sum subarray that ends at position $k$. There are two possibilities: \begin{enumerate} From 8868ff10abdbac74bb09a59c915e53552ba70df4 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Thu, 23 Feb 2017 11:31:29 +0200 Subject: [PATCH 02/39] Remove unused reference --- list.tex | 5 ----- 1 file changed, 5 deletions(-) diff --git a/list.tex b/list.tex index 6cd1b06..600f3d0 100644 --- a/list.tex +++ b/list.tex @@ -69,11 +69,6 @@ Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE. In \emph{Annual Symposium on Combinatorial Pattern Matching}, 36--48, 2006. -\bibitem{fis11} - J. Fischer and V. Heun. - Space-efficient preprocessing schemes for range minimum queries on static arrays. - \emph{SIAM Journal on Computing}, 40(2):465--492, 2011. - \bibitem{flo62} R. W. Floyd Algorithm 97: shortest path. From 93bb7d1ed928a8e206b914e2f4d0b15b84bb4d82 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Fri, 24 Feb 2017 00:38:24 +0200 Subject: [PATCH 03/39] Better table of estimates --- chapter02.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/chapter02.tex b/chapter02.tex index 15141ee..b686b13 100644 --- a/chapter02.tex +++ b/chapter02.tex @@ -309,15 +309,15 @@ assuming that the time limit is one second. \begin{center} \begin{tabular}{ll} -input size ($n$) & required time complexity \\ +input size & required time complexity \\ \hline -$n \le 10^{18}$ & $O(1)$ or $O(\log n)$ \\ -$n \le 10^{12}$ & $O(\sqrt n)$ \\ -$n \le 10^6$ & $O(n)$ or $O(n \log n)$ \\ -$n \le 5000$ & $O(n^2)$ \\ -$n \le 500$ & $O(n^3)$ \\ -$n \le 25$ & $O(2^n)$ \\ -$n \le 10$ & $O(n!)$ \\ +$n \approx 10$ & $O(n!)$ \\ +$n \approx 20$ & $O(2^n)$ \\ +$n \approx 100$ & $O(n^3)$ \\ +$n \approx 1000$ & $O(n^2)$ \\ +$n \approx 10^5$ & $O(n \log n)$ \\ +$n \approx 10^6$ & $O(n)$ \\ +$n$ is large & $O(1)$ or $O(\log n)$ \\ \end{tabular} \end{center} From a8723ccfb875b7d1d6167078b8773043fba2efe7 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Fri, 24 Feb 2017 00:52:49 +0200 Subject: [PATCH 04/39] Better description for Andrew's algorithm --- chapter30.tex | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/chapter30.tex b/chapter30.tex index c582e2a..640f587 100644 --- a/chapter30.tex +++ b/chapter30.tex @@ -396,21 +396,20 @@ an easy way to construct the convex hull for a set of points in $O(n \log n)$ time. The algorithm constructs the convex hull -in two steps: +in two parts: first the upper hull and then the lower hull. -Both steps are similar, so we can focus on +Both parts are similar, so we can focus on constructing the upper hull. -We sort the points primarily according to +First, we sort the points primarily according to x coordinates and secondarily according to y coordinates. -After this, we go through the points and always -add the new point to the hull. -After adding a point we check using cross products -whether the tree last point in the hull turn left. -If this holds, we remove the middle point from the hull. -After this we keep checking the three last points -and removing points, until the three last points -do not turn left. +After this, we go through the points and +add each point to the hull. +Always after adding a point to the hull, +we make sure that the last line segment +in the hull does not turn left. +As long as this holds, we repeatedly remove the +second last point from the hull. The following pictures show how Andrew's algorithm works: From b8cd47058a3b7830382087faf337771b8c530802 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Fri, 24 Feb 2017 00:55:16 +0200 Subject: [PATCH 05/39] Better reference style --- list.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/list.tex b/list.tex index 600f3d0..9952315 100644 --- a/list.tex +++ b/list.tex @@ -138,7 +138,7 @@ \bibitem{pac13} J. Pachocki and J. Radoszweski. Where to use and how not to use polynomial string hashing. - \emph{Olympiads in Informatics}, 2013. + \emph{Olympiads in Informatics}, 7(1):90--100, 2013. \bibitem{pri57} R. C. Prim. 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