1069 lines
47 KiB
TeX
1069 lines
47 KiB
TeX
\documentclass[landscape, 8pt, a4paper, oneside]{extarticle}
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% \documentclass[portrait, 8pt, a4paper, oneside, twocolumn]{extarticle}
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\usepackage{teamnote}
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\usepackage[hangul]{kotex}
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\usepackage[most]{tcolorbox}
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\usepackage{multirow}
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\usepackage{adjustbox}
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\usepackage{subfiles}
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\usepackage{amsmath}
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\usepackage{etoolbox}
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\AtBeginEnvironment{align*}{%
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\setlength{\abovedisplayskip}{0pt}
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\setlength{\belowdisplayskip}{0pt}
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\setlength{\abovedisplayshortskip}{0pt}
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\setlength{\belowdisplayshortskip}{0pt}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{enumitem}
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\setlistdepth{9}
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\newlist{IdeaNote}{enumerate}{6}
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\setlist[IdeaNote]{topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex}
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\setlist[IdeaNote, 1]{label= \Roman*. }
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\setlist[IdeaNote, 2]{label= \Alph*. }
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\setlist[IdeaNote, 3]{label= \arabic*. }
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\setlist[IdeaNote, 4]{label= \roman*) }
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\setlist[IdeaNote, 5]{label= \alph*) }
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\setlist[IdeaNote, 6]{label= \arabic*) }
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{multicol}
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\setlength\columnseprule{0.5pt}
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\teamnote{POSTECH}{ConSpirito}{}{ICPC Seoul Regional}
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\ShowUsage
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\ShowComplexity
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\HideAuthor
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\begin{document}
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\begin{multicols}{3}
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\maketitlepage
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\par\noindent\rule{\columnwidth}{0.4pt}
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\section{Have you...}
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\subsection{tried...}
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\begin{itemize}
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\item \textcolor{red}{\textbf{Reading the problem once more?}}
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\item doubting ``obvious" things?
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\item writing obivous things?
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\item radical greedy approach?
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\item thinking in reverse direction?
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\item a greedy algorithm?
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\item network flow when your greedy algorithms stuck?
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\item a dynamic programming?
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\item checking the range of answer?
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\item random algorithm?
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\item graph modeling using states?
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\item inverting state only on odd indexes?
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\item calculating error bound on a real number usage?
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\end{itemize}
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\subsection{checked...}
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\begin{itemize}
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\item \textcolor{red}{\textbf{you have read the statement correctly?}}
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\item typo copying the team note?
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\item initialization on multiple test case problem?
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\item additional information from the problem?
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\item undefined behavior?
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\item overflow?
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\item function without return value?
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\item real number error?
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\item implicit conversion?
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\item comparison between signed and unsigned integer?
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\end{itemize}
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\subsection{Geometry?}
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\begin{itemize}
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\item 1 point? 2 points? line? 3 or more points on a line?
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\item parallel line? points on line? circle?
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\item degenerate case? endpoints of segment?
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\end{itemize}
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% \pagebreak
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Algorithmic Idea Note}
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When you don't have any ideas, please bruteforce it.
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\begin{tcolorbox}[breakable, enhanced, sharp corners, colback=white, colframe=black, boxrule=1pt, left=0pt]
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\begin{IdeaNote}
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\item Complete Search: Backtracking \& Pruning
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\item Math
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\begin{IdeaNote}
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\item Number Theory
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\begin{IdeaNote}
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\item Prime Number
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\begin{IdeaNote}
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\item Sieve of Eratosthenes, Prime Factorization
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\item Fast Prime Verdict; Millar-Rabin
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\item Fast Prime Factorization; Pollad Rho
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\item Primitive Root
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\end{IdeaNote}
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\item Extended Euclidean Algorithm; Diophantos Equation
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\item Chinese Remainder Theorem
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\item Harmonic Lemma
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\item Floor Sum (Sum of Rational Arithmetic Sequence)
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\item Several Sieves
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\end{IdeaNote}
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\item Linear Programming
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\begin{IdeaNote}
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\item Solve (some) LP with Shortest Path
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\end{IdeaNote}
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\item FFT \& Polynomials
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\begin{IdeaNote}
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\item FFT : Convolution
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\begin{IdeaNote}
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\item High precision FFT with modulo 1e9+7
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\end{IdeaNote}
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\item NTT : Number Theoretic Tranform
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\item Quotient Ring (Formal Power Series)
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\begin{IdeaNote}
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\item Multiplication
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\item FPS : Inverse / Division
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\item Integration / Differentiation
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\item FPS : Logarithm / Exponentiation
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\item FPS : Power of Polynomial
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\item Division - Quotient \& Remainder
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\item Polynomial Taylor Shift
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\item Multipoint Evaluation
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\end{IdeaNote}
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\end{IdeaNote}
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\item Combinatorics
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\begin{IdeaNote}
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\item Labeled Combinatorial Target
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\item The Twelvefold Way (12정도)
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\item Generating Function
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\begin{IdeaNote}
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\item OGF(Ordinary Generating Function)
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\item EGF(Exponentional Generating Function)
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\item DGF?(Dirichlet Genetration Function) : Number Theory
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\end{IdeaNote}
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\item Umbral Calculus
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\end{IdeaNote}
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\end{IdeaNote}
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\item Linear Algebra
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\item Geometry
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\begin{IdeaNote}
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\item Basic Tools
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\begin{IdeaNote}
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\item Outer Product (CCW)
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\item Sorting by Polar
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\item Segment Intersection
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\item Closest Point
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\item Furthest Point
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\end{IdeaNote}
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\item Convex Polygon (Convex Hull)
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\begin{IdeaNote}
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\item Convex Hull Construction
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\item Convex Layer
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\item Rotating Calipers
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\item Point Containment
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\item Tangent to convex polygon
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\item Inner and Outer Tangent of two Convex's
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\end{IdeaNote}
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\item General Polygon
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\item Half Plane Intersection
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\item Bulldozer Trick
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\item Delaunay Triangulation : Voronoi diagram
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\end{IdeaNote}
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\item Greedy
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\begin{IdeaNote}
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\item Rearrangement Inequality
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\end{IdeaNote}
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\item DP
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\begin{IdeaNote}
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\item DP Optimization
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\begin{IdeaNote}
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\item Convex Hull Trick
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\item Alien's Trick (Lagrangian Relaxation)
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\item Slope Trick
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\end{IdeaNote}
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\end{IdeaNote}
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\item String
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\begin{IdeaNote}
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\item KMP(Knuth-Morris-Pratt), Z, Manacher Algorithm
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\item Trie
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\item Aho-Corasick
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\item Suffix Array \& LCP Array
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\item Eertree
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\item Wavelet Tree
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\end{IdeaNote}
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\item Graph
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\begin{IdeaNote}
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\item Searching : DFS/BFS
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\item DAG(Directed Acyclic Graph) : Topological Sorting
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\item MST(Minimum Spanning Tree)
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\begin{IdeaNote}
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\item Kruskal Algorithm
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\item Prim Algorithm
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\item Euclidian MST
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\end{IdeaNote}
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\item Shortest Path
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\begin{IdeaNote}
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\item Dijkstra Algorithm
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\item Bellman-Ford Algorithm
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\item Floyd-Warshall Algorithm
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\item Shortest Path DAG
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\end{IdeaNote}
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\item Connectivity
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\begin{IdeaNote}
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\item Offline Dynamic Connectivity (Odc)
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\item Online Dynamic Connectivity
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\begin{IdeaNote}
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\item Euler Tour Tree
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\item Top Tree
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\end{IdeaNote}
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\end{IdeaNote}
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\item DFS tree
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\begin{IdeaNote}
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\item SCC(Strongly Connected Component)
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\begin{IdeaNote}
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\item Graph Compression
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\item 2-SAT Problem
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\item Offline Incremental SCC
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\end{IdeaNote}
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\item BCC (BiConnected Component)
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\begin{IdeaNote}
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\item Block Cut Tree
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\item Cactus Graph
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\end{IdeaNote}
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\item Articulation Points and Bridges
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\end{IdeaNote}
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\item Network Flow
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\begin{IdeaNote}
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\item Bipartite Matching
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\begin{IdeaNote}
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\item Hall's Marrage Theorem
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\item Minimum Vertex Cover on Bipartite
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\item Maximum Independent Set on Bipartite
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\item Minimum Path Cover on DAG
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\item Maximum Antichain on DAG
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\end{IdeaNote}
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\item Ford-Fulkerson/Edmonds-Karp Algorithm
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\item Dinic's Algorithm
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\item Push-Relabel Algorithm
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\item MCMF(Minimum Cost Maximum Flow)
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\item Minimum s-t Cut = Maximum Flow
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\item Global Cut(Stoer-Wagner Algorithm)
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\item Circulation
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\item General Matching
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\end{IdeaNote}
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\item Treewidth
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\end{IdeaNote}
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\item Tree
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\begin{IdeaNote}
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\item LCA(Lowest Common Ancestor)
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\item Heavy-Light Decomposition
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\item Centroid Decomposition
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\item Link-Cut Tree
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\end{IdeaNote}
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\item Data Structure
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\begin{IdeaNote}
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\item C++ Standard Library
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\begin{IdeaNote}
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\item Stack, Queue, List, Vector, Deque
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\item Priority Queue; Heap
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\item Set, Map : Binary Search Tree
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\item Unordered Set, Unordered Map : Hashing
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\item PBDS(Policy-Based Data Structure)
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\item Rope (Cord)
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\end{IdeaNote}
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\item Disjoint Set (Unoin-Find structure)
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\begin{IdeaNote}
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\item Union by Rank / Path Compression
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\item UF with LCA (Making Root)
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\item UF with Edge Weight
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\item UF with Unjoining
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\begin{IdeaNote}
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\item Unjoin from latest (Stack undoing)
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\item Unjoin from earliest (Queue undoing)
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\item Unjoin by Priority (Priority undoing)
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\end{IdeaNote}
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\end{IdeaNote}
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\item Sparse Table
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\item Range Query Structure
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\begin{IdeaNote}
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\item Square Root Decomposition
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\item Fenwick Tree
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\item Segment Tree
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\begin{IdeaNote}
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\item Lazy Propagation \& Generalization
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\item 금광 ST (Maximum Adjacent Sum of Given Range)
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\item PST (Persistent Segment Tree)
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\item MST (Merge Sort Tree)
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\item Segment Tree on Tree (HLD)
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\item Li-Chao Tree (Segment Add Get Min)
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\item ST Beats
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\item Kinetic ST
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\end{IdeaNote}
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\item Splay Tree
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\begin{IdeaNote}
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\item Range Reverse / Range Shift
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\end{IdeaNote}
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\end{IdeaNote}
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\end{IdeaNote}
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\item Sorting \& Searching
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\begin{IdeaNote}
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\item Sorting
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\item Searching
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\begin{IdeaNote}
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\item Binary Search : Monotone Sequence / function
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\begin{IdeaNote}
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\item Lower bount / Upper bound
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\item LIS (Longest Increasing Subsequence)
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\item PBS (Parallel Binary Search)
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\end{IdeaNote}
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\item Ternary Search : Unimodal Sequence / function
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\begin{IdeaNote}
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\item Fibonacci Search (Golden Ratio Search)
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\end{IdeaNote}
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\end{IdeaNote}
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\end{IdeaNote}
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\item Numerical Analysis
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\begin{IdeaNote}
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\item Numerical Differentiation
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\item Gradient Descent
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\end{IdeaNote}
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\item Technic
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\begin{IdeaNote}
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\item Coordinate Compression
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\item Two Pointer/Sliding Window
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\item Sweeping
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\item Meet in the Middle
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\item Bitmasking
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\item Small to Large
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\item Randomization
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\begin{IdeaNote}
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\item Verifying Matrix Multiplication
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\end{IdeaNote}
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\item Query Technic
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\begin{IdeaNote}
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\item Offline Query
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\begin{IdeaNote}
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\item Mo's Algorithm
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\end{IdeaNote}
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\end{IdeaNote}
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\end{IdeaNote}
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\end{IdeaNote}
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\end{tcolorbox}
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\Algorithm{POROGOD}{}{}{cpp}{source/poro.cpp}
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\pagebreak
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% \Algorithm{astilate}{}{}{cpp}{source/Fundemental.cpp}
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\subsection{Tips for Inequality with Rational Number}
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Let $A$, $B$, $x$, $n$ be integer, and operator `/' means floor division(quotient).
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\begin{align*}
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Ax \leqslant B & \Leftrightarrow x \leqslant B/A \\
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Ax < B & \Leftrightarrow x \leqslant (B-1)/A \\
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Ax \geqslant B & \Leftrightarrow x \geqslant (B+A-1)/A \\
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Ax > B & \Leftrightarrow x \geqslant B/A+1 \\ \\
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x < n & \Leftrightarrow x+1\leqslant n
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\end{align*}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Math}
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\subfile{source/Math/math}
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\subsection{Prime Number}
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\subsubsection{Distribution of Prime Number}
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\begin{center}
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\begin{tabular}{|
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>{\columncolor[HTML]{BFBFBF}}c |c|
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>{\columncolor[HTML]{BFBFBF}}c |c|
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>{\columncolor[HTML]{BFBFBF}}c |c|}
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\hline
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1e2 & 25 & 1e6 & 78,498 & 1e10 & \textless 5e8 \\ \hline
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1e3 & 168 & 1e7 & 664,579 & 1e11 & \textless 5e9 \\ \hline
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1e4 & 1,229 & 1e8 & \textless 6e6 & 1e12 & \textless 4e10 \\ \hline
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1e5 & 9,592 & 1e9 & \textless 6e7 & 1e13 & \textless 4e11 \\ \hline
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\end{tabular}
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\end{center}
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\subsubsection{Prime Gap}
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\begin{align*}
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2 \cdot 10^{5} \text{ 이하의 소수 간극 } &\le 100\\
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2^{32} \text{ 이하의 소수 간극 } &\le 464 \\
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2^{64} \text{ 이하의 소수 간극 } &\le 1550
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\end{align*}
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\Algorithm{Miller-Rabin Algorithm}
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{\texttt{is\_p(X)} : returns true if $X$ is prime, otherwise false.
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\begin{align*}
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\text{ When } X \le 2^{32}, D &= \{2, 7, 61\} \text{ is sufficient; } \\
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X \le 2^{64}, D &= \{p | p \text{ is prime}, p \le 37 \} \text{ is sufficient.}
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\end{align*}}
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{$\mathcal O (\log^3 X)$}
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{cpp}
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{source/Math/MillerRabin.cpp}
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\Algorithm{Pollad Rho Algorithm}
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{ \texttt{po\_rho(N)} : returns array of prime factors of X.}
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{$\mathcal O (N^{1/4})$}
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{cpp}
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{source/Math/PolladRho.cpp}
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\Algorithm{Primitive Root}
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{Calculate one of the primitive roots of given prime.}
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{}{cpp}
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{source/Math/PrimitiveRoot.cpp}
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\Algorithm{Diopantos Equation(Extended Euclidian Algorithm)}
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{\texttt{diophantos(a, b)} : return one integer solution of $ax+by=1$, satisfying $0 \le x < b$. }
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{$\mathcal O \left(\log \left( \max(a, b) \right) \right)$}
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{cpp}
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{source/Math/Diophantos.cpp}
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\Algorithm{Chinese Remainder Theorem}
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{\texttt{crt(pll p, pll q)} : return \texttt{pll r}, satisfying follows:
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\begin{align*}
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x &\equiv \texttt{p.fi} \mod \texttt{p.se} \\
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\text{and } x &\equiv \texttt{q.fi} \mod \texttt{q.se} \\
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\leftrightarrow x &\equiv \texttt{r.fi} \mod \texttt{r.se}
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\end{align*}
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If there's no such \texttt{r}, return $\{-1, -1\}$.
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}
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{$\mathcal O(\log A)$}
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{cpp}
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{source/Math/CRT.cpp}
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\Algorithm{Harmonic Lemma}
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{\texttt{f(N)} : return the value
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$$\sum_{i=1}^{N} \left\lfloor \frac{N}{i} \right\rfloor = \left\lfloor \frac{N}{1}\right\rfloor + \left\lfloor \frac{N}{2} \right\rfloor + \cdots + \left\lfloor \frac{N}{N}\right\rfloor $$
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Using the fact that $\left\lfloor \frac{N}{x} \right\rfloor$ has $\mathcal O(\sqrt N)$ different values.}
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{$\mathcal O(\sqrt N )$}
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{cpp}
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{source/Math/Harmonic.cpp}
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\Algorithm{Floor Sum (Sum of Floor of Rational Arithmetic Sequence)}
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{\texttt{floor\_sum(A, B, C, N)} : return the value
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$$\sum_{x=0}^{N-1} \left\lfloor \frac{Ax+B}{C} \right\rfloor $$}
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{$\mathcal O(\log N)$}
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{cpp}
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{source/Math/FloorSum.cpp}
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\Algorithm{FFT - Convolution}
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{}
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{$\mathcal O (N\log N)$}
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{cpp}
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{source/Math/FFTConv.cpp}
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\Algorithm{NTT - Number Theoretic Transform}
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{helloworld}
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{}
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{cpp}
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{source/Math/NTT.cpp}
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\subsubsection{Good prime numbers to run NTT}
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\begin{center}
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\begin{adjustbox}{width=0.8\columnwidth}
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\begin{tabular}{|c|c|
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>{\columncolor[HTML]{C0C0C0}}l |}
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\hline
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595 591 169 & 71\textless{}\textless{}23|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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645 922 817 & 77\textless{}\textless{}23|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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897 581 057 & 107\textless{}\textless{}23|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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998 244 353 & 119\textless{}\textless{}23|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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1 300 234 241 & 155\textless{}\textless{}23|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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1 224 736 769 & 73\textless{}\textless{}24|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
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2 130 706 433 & 127\textless{}\textless{}24|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
|
|
167 772 161 & 5\textless{}\textless{}25|1 & \cellcolor[HTML]{C0C0C0} \\ \cline{1-2}
|
|
469 762 049 & 7\textless{}\textless{}26|1 & \multirow{-9}{*}{\cellcolor[HTML]{C0C0C0}$\omega$ = 3} \\ \hline
|
|
\end{tabular}
|
|
\end{adjustbox}
|
|
\end{center}
|
|
|
|
\Algorithm{Polynomial (Formal Power Series)}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Math/Polynomial.cpp}
|
|
|
|
\subsection{Combinatorics}
|
|
\subsubsection{Labeled Combinatorial Target}
|
|
\begin{itemize}
|
|
\item Permutation, Combination (with, w/o repetition)
|
|
\begin{itemize}
|
|
\item Permutation $_nP_r = \frac{n!}{r!(n-r)!}$
|
|
\item Combination $_nC_r = \binom n k = \frac{n!}{r!}$
|
|
\item Permutation with repetition $_n\Pi_r = n^r$
|
|
\item Combination with repetition $_nH_r = _{n+r-1}C_r$
|
|
\end{itemize}
|
|
\item Catalan Number $C_n$ : the number of regular bracket string of length $2n$.
|
|
\begin{itemize}
|
|
\item $C_n = \frac{1}{n+1}\binom {2n}n$
|
|
\item $C_n : 1, 1, 2, 5, 14, 42, 132,429,1430,\cdots$ (from index 0)
|
|
\item OGF of $C_n$ : $c(x) = 1+xc(x)^2, c(c) = \frac{1-\sqrt{1-4x}}{2x}$
|
|
\item Catalan's triangle $C(n, k)$ : the number of string with $n$ X's, $k$ Y's, where the number of Y's is not greater than the number of X's when written down the string.
|
|
\begin{itemize}
|
|
\item i.e. $C(4, 3)$ : XXXXYYY, XXYXYYX are OK, but XXYYYXX is not OK.
|
|
\item $C(n, k) = \binom{n+k}{k} - \binom{n+k}{k-1} = \frac{n-k+1}{n+1}\binom{n+k}{k}$
|
|
\item We can give the condition relax : $C_m(n, k)$ : the number of string with $n$ X's, $k$ Y's, where the number of Y's is not greater than (the number of X's) + $m$ when written down the string.
|
|
\item $C_m(n, k) = \binom{n+k}{k}-\binom{n+k}{k-m}$ when $m\leqslant k \leqslant n+m-1$.
|
|
\item $C_1(n, k) = C(n, k)$
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\item Derangement(교란순열) $D_n$ : the number of permutation of length $n$ which $\forall i\;p_i \neq i$.
|
|
\begin{itemize}
|
|
\item $D_n : 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496,1334961 \cdots$ (from index 0)
|
|
\item EGF of $D_n$ : $D(x) = \sum_{n=0} \frac{D_n}{n!}x^n = \frac{e^{-x}}{1-x}$
|
|
\end{itemize}
|
|
\item Stirling Number of the 1st/2nd kind.
|
|
\begin{itemize}
|
|
\item Unsigned Stirling Number of the 1st kind $c(n, k) = \genfrac{[}{]}{0pt}{}{n}{k}$ : the number of permutation of length n with k disjoint cycles.
|
|
\begin{itemize}
|
|
\item $c(n+1, k) = n \times c(n, k) + c(n, k-1)$
|
|
\item with boundary condition $c(0, 0) = 1, c(n, 0) = c(0, k) = 0 \; \forall n, k>0$
|
|
\item $c(n, k)$
|
|
\begin{itemize}
|
|
\item $c(0, 0) = 1$; $c(x_{\neq0}, 0) = 0$; $c(x, x) = 1$
|
|
\item $c(x, x) = 1$
|
|
\item $c(5, *) = \{0, 24, 50, 35, 10 1\}$
|
|
\item $c(6, *) = \{0, 120, 274, 225, 85, 15, 1\}$
|
|
\end{itemize}
|
|
% \begin{tabular}{|
|
|
% >{\columncolor[HTML]{ECECEC}}r |
|
|
% >{\columncolor[HTML]{F8F9FA}}r |r|r|r|l|l|l|}
|
|
% \hline
|
|
% \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} \textit{n\textbackslash{}k}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{0}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{1}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{2}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{3}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{4}}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{5}}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{6}}} \\ \hline
|
|
% {\color[HTML]{202122} \textbf{0}} & {\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{1}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{2}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{3}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 2} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 3} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{4}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 6} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 11} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 6} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{5}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 24} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 50} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 35} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 10}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{6}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 120} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 274} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 225} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 85}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 15}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} \\ \hline
|
|
% \end{tabular}
|
|
\item OGF of $\genfrac{[}{]}{0pt}{}{n}{k}$ : for fixed $n$, $\sum_{k=0}^n \genfrac{[}{]}{0pt}{}{n}{k} x^k = x(x+1)(x+2)\cdots(x+n-1)$ : we can calculate it by polynomial shift \& divide and conquer
|
|
\item EGF of $\genfrac{[}{]}{0pt}{}{n}{k}$ : for fixed $k$, $\sum_{n=k}^\infty \genfrac{[}{]}{0pt}{}{n}{k} \frac{x^n}{n!} = \frac{(-\log(1-z))^k}{k!} $
|
|
\item Signed Stirling Number of the 1st kind $s(n, k) = (-1)^{n-k}c(n, k)$
|
|
\item EGF of $s(n, k)$ : for fixed $k$, $\sum_{n=k}^\infty s(n, k) \frac{x^n}{n!} = \frac{(\log(1+z))^k}{k!} $
|
|
\end{itemize}
|
|
\item Stirling Number of the 2nd kind $S(n, k) = \genfrac{\{}{\}}{0pt}{}{n}{k}$ : the number of partition of $n$ labeled objects into $k$ nonempty unlabelled subsets.
|
|
\begin{itemize}
|
|
\item $S(n, k) = \frac{1}{k!}\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}i^n$
|
|
\item $S(n,k)=S(n - 1,k - 1)+k\cdot S(n - 1,k)$
|
|
\item $S(n, k)$
|
|
\begin{itemize}
|
|
\item $S(0, 0) = 1$; $S(x_{\neq 0}, 0) = 0$; $S(x, x) = 1$
|
|
\item $S(5, *) = \{0, 1, 15, 25, 10, 1\}$
|
|
\item $S(6, *) = \{0, 1, 31, 90, 65, 15, 1\}$
|
|
\end{itemize}
|
|
% \begin{tabular}{|
|
|
% >{\columncolor[HTML]{ECECEC}}r |
|
|
% >{\columncolor[HTML]{F8F9FA}}r |r|r|r|l|l|l|}
|
|
% \hline
|
|
% \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} \textit{\textbf{n\textbackslash{}k}}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{0}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{1}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{2}} & \cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{3}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{4}}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{5}}} & \multicolumn{1}{r|}{\cellcolor[HTML]{ECECEC}{\color[HTML]{202122} \textbf{6}}} \\ \hline
|
|
% {\color[HTML]{202122} \textbf{0}} & {\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{1}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{2}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \multicolumn{1}{l|}{} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{3}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 3} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{4}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 7} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 6} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} & & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{5}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 15} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 25} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 10}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} & \\ \hline
|
|
% {\color[HTML]{202122} \textbf{6}} & {\color[HTML]{202122} 0} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 31} & \cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 90} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 65}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 15}} & \multicolumn{1}{r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} 1}} \\ \hline
|
|
% \end{tabular}
|
|
\item OGF of $\genfrac{\{}{\}}{0pt}{}{n}{k}$ : for fixed $n$, $\sum_{k=0}^n\genfrac{\{}{\}}{0pt}{}{n}{k}x^k=T_n(x)$, where $T_n(x) = e^{-x}\sum_{k=0}^{\infty}\frac{k^n}{k!}x^k$ is Touchard Polynomials.
|
|
\item EGF of $\genfrac{\{}{\}}{0pt}{}{n}{k}$ : for fixed $k$, $\sum_{n=k}^\infty\genfrac{\{}{\}}{0pt}{}{n}{k}\frac{x^n}{n!}=\frac{(e^x-1)^k}{k!}$
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\item Bell Number $B_n$ : the number of partition of set with size $n$.
|
|
\begin{itemize}
|
|
\item i.e. partitionize $\{a, b, c\}$ is $\{\{a\}, \{b\}, \{c\}\}, \{\{a , b\}, \{c\}\}$, $\{\{b, c\}, \{a\}\}, \{\{c, a\}, \{b\}\}, \{\{a, b, c\}\}$. Therefore, $B_3 = 5$.
|
|
\item $B_n : 1,1,2,5,15,52,203,877,4140,\cdots$ (from index 0)
|
|
\item $B_n = \sum_{k=0}^n\genfrac{\{}{\}}{0pt}{}{n}{k}$ (2nd kind of stirling number)
|
|
\item EGF of $B_n$ : $B(x) = \sum_{n=0}\frac{B_n}{n!}x^n=e^{e^x-1}$
|
|
\item Ordered Bell Number(Fubini Number) $a_n$ : the number of distinct weak ordering on a set of n elements.
|
|
\begin{itemize}
|
|
\item $a_n : 1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, \cdots$
|
|
\item $a_n = \sum_{k=0}^nk!\genfrac{\{}{\}}{0pt}{}{n}{k}$ (stirling number of the 2nd kind)
|
|
\item EGF of $a_n$ : $a(x) = \sum_{n=0}^\infty \frac{a_n}{n!}x^n = \frac{1}{2-e^x}$
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\item Integer Partition $P(n, k)$ : the number of partitions of $n$ into exactly $k$ parts.
|
|
\begin{itemize}
|
|
\item i.e. $P(4, 2) = 2$ since $4 = 1+3 = 2+2$
|
|
\item we also say $P(n) = \sum_{k=1}^{n}P(n, k)$
|
|
\item $P(n, k)$
|
|
\begin{itemize}
|
|
\item $P(x, 1) = P(x, x) = 1$
|
|
\item $P(8, *) = \{1, 4, 5, 5, 3, 2, 1, 1\}$
|
|
\item $P(9, *) = \{1, 4, 7, 6, 5, 3, 2, 1, 1\}$
|
|
\end{itemize}
|
|
\item $P(n) = \{1, 2, 3, 5, 7, 11, 15, 22, 30, 42\}$
|
|
|
|
% \begin{tabular}{|
|
|
% >{\columncolor[HTML]{EAECF0}}c |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |
|
|
% >{\columncolor[HTML]{F8F9FA}}l |}
|
|
% \hline
|
|
% \multicolumn{1}{|r|}{\cellcolor[HTML]{F8F9FA}{\color[HTML]{202122} \textit{\textbf{n\textbackslash{}k}}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{1}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{2}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{3}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{4}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{5}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{6}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{7}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{8}}} & \multicolumn{1}{c|}{\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{9}}} \\ \hline
|
|
% {\color[HTML]{202122} \textbf{1}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{2}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{3}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{4}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{5}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{6}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{7}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 4} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{8}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 4} & {\color[HTML]{202122} 5} & {\color[HTML]{202122} 5} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} } \\ \hline
|
|
% {\color[HTML]{202122} \textbf{9}} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 4} & {\color[HTML]{202122} 7} & {\color[HTML]{202122} 6} & {\color[HTML]{202122} 5} & {\color[HTML]{202122} 3} & {\color[HTML]{202122} 2} & {\color[HTML]{202122} 1} & {\color[HTML]{202122} 1} \\ \hline
|
|
% \end{tabular}
|
|
\end{itemize}
|
|
\item 12정도(the Twelvefold Way) : the number of functions \\
|
|
(Domain $|N| = n$, Codomain $|X| = x$)\\
|
|
\begin{tabular}{|
|
|
>{\columncolor[HTML]{F8F9FA}}c |c|c|c|}
|
|
\hline
|
|
\cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textit{\textbf{$f$-class}}} & \cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{Any $f$}} & \cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{Injective}} & \cellcolor[HTML]{EAECF0}{\color[HTML]{202122} \textbf{Surjective}} \\ \hline
|
|
{\color[HTML]{202122} \textbf{Distinct $f$}} & \textit{$_n\Pi_x$} & \textit{$_xP_n$} & $x!\genfrac{\{}{\}}{0pt}{}{n}{k}$ \\ \hline
|
|
{\color[HTML]{202122} \textit{\textbf{\begin{tabular}[c]{@{}c@{}}$S_n$ orbits\\ $f \circ S_n$\end{tabular}}}} & \textit{$_xH_n$} & \textit{$_xC_n$} & $_{n-1}C_{n-x}$ \\ \hline
|
|
{\color[HTML]{202122} \textit{\textbf{\begin{tabular}[c]{@{}c@{}}$S_x$ orbits\\ $S_x \circ f$\end{tabular}}}} & $\sum_{k=0}^x \genfrac{\{}{\}}{0pt}{}{n}{k}$ & $[n \leqslant x]$ & $\genfrac{\{}{\}}{0pt}{}{n}{k}$ \\ \hline
|
|
{\color[HTML]{202122} \textit{\textbf{\begin{tabular}[c]{@{}c@{}}$S_n\times S_x$ orbits\\ $S_x \circ f \circ S_n$\end{tabular}}}} & $P(n+x, x)$ & $[n \leqslant x]$ & $P(n, x)$ \\ \hline
|
|
\end{tabular}
|
|
\end{itemize}
|
|
|
|
% \Algorithm{General Lucas Comb}
|
|
% {}
|
|
% {}
|
|
% {cpp}
|
|
% {source/Math/general_lucas.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Linear Algebra}
|
|
|
|
\Algorithm{Matrix}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{"source/Linear Algebra/Matrix.cpp"}
|
|
|
|
\Algorithm{Berlekamp Massey}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{"source/Linear Algebra/BerlekampMassey.cpp"}
|
|
|
|
\Algorithm{XOR Basis}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{"source/Linear Algebra/XORBasis.cpp"}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Geometry}
|
|
\Algorithm{Mindset}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Geometry/Mindset.cpp}
|
|
|
|
\Algorithm{Point in Convex Polygon}
|
|
{\texttt{isunder(upper\_hull[], (length of upper\_hull), x)} : return true if $x$(point) is under upper\_hull, otherwise false;
|
|
|
|
\texttt{isunder(lower\_hull[], (length of lower\_hull), x, true)} : return true if $x$(point) is over lower\_hull, otherwise false.
|
|
|
|
Point in Polygon : isunder(upper, u, x) \&\& isupper(lower, u, x, true)}
|
|
{}
|
|
{cpp}
|
|
{source/Geometry/PointInConvexPolygon.cpp}
|
|
|
|
\Algorithm{Geo}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Geometry/Geo.cpp}
|
|
|
|
\Algorithm{Half-Plane Intersection}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Geometry/HPI.cpp}
|
|
|
|
\Algorithm{Bulldozer}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Geometry/Bulldozer.cpp}
|
|
|
|
|
|
% \Algorithm{kactl templete}
|
|
% {}
|
|
% {}
|
|
% {cpp}
|
|
% {source/Geometry/geometry_kaere.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Greedy}
|
|
\subsection{Rearrange Inequality - Extensions}
|
|
Let $A_i, B_i$ be non-decreasing sequence of length $n$, and $p$ be some permutation, and let $inc := (1, 2, \dots, n-1, n), \;dec := (n, n-1, \dots, 2, 1)$.
|
|
|
|
\begin{align*}
|
|
S(p) &= \sum_{i=1}^n A_i B_{p_i}
|
|
&& \text{ maximize } S &&&\Rightarrow p : inc&&&&\\
|
|
& && \text{ minimize } S &&&\Rightarrow p:dec&&&&\\
|
|
P_{max}(p) &= \max_i(A_iB_{p_i})
|
|
&& \text{ minimize } P_{max} &&&\Rightarrow p : dec&&&&\\
|
|
P_{min}(p) &= \min_i(A_iB_{p_i})
|
|
&& \text{ maximize } P_{min} &&&\Rightarrow p:dec&&&&\\
|
|
A_{max}(p) &= \max_i(A_i+B_{p_i}) && \text
|
|
{ minimize } A_{max} &&& \Rightarrow p:dec&&&&\\
|
|
A_{min}(p) &= \min_i(A_i+B_{p_i}) && \text{ maximize } A_{min} &&& \Rightarrow p:dec&&&&\\
|
|
D_{max}(p) &= \max_i|A_i-B_{p_i}| && \text{ minimize } D_{max} &&& \Rightarrow p:inc&&&&\\
|
|
D_{min}(p) &= \min_i|A_i-B_{p_i}| && \text{ maximize } D_{min} &&& \Rightarrow p:?&&&&\\
|
|
\end{align*}
|
|
\begin{align*}
|
|
\text{ Permutate } A \text{ s.t.} \text{ maximize } & \sum_{i=1}^n {A_i A_{i+1}} \;(\text{Let } A_{n+1} = A_1) \\ &\Rightarrow \text{ Pendulum Arrangement }
|
|
\end{align*}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{DP}
|
|
|
|
\Algorithm{LIS}
|
|
{}{}{cpp}{source/DP/LIS.cpp}
|
|
|
|
\subsection{DP Optimization}
|
|
|
|
\subsubsection{Convex Hull Optimization}
|
|
\begin{align*}
|
|
\text{ Recurrence : }& D[i] = \min_{j<i} { \left(B[j] \times A[i] + D[j] \right)}\\
|
|
\text{ Complexity : }& \mathcal O(N^2) \rightarrow \mathcal O(N\log N)
|
|
\end{align*}
|
|
|
|
\subsubsection{Divide and Conquer Optimization}
|
|
\begin{align*}
|
|
\text{ Recurrence : }& D[i][j] = \min_{k<i} { \left(D[i-1][k] + C[k][j]\right)}\\
|
|
\text{ Condition : }& C[i][j] \text{ is Monge } \\
|
|
(\text{ if }a \leqslant b \leqslant c \leqslant d, &\text{ then } C[a][c] + C[b][d] \leqslant C[a][d] + C[b][c] )\\
|
|
\text{ Complexity : }& \mathcal O(KN^2) \rightarrow \mathcal O(KN\log N)
|
|
\end{align*}
|
|
|
|
\inputminted[]{cpp}{source/DP/DnC.cpp}
|
|
|
|
\subsubsection{Monotone Queue Optimization}
|
|
\begin{align*}
|
|
\text{ Recurrence : }& D[i] = \min_{j<i} { \left(D[j] + C[j][i]\right)}\\
|
|
\text{ Condition : }& C[i][j] \text{ is Monge } \\
|
|
\text{ Complexity : }& \mathcal O(N^2) \rightarrow \mathcal O(N\log N)
|
|
\end{align*}
|
|
|
|
\subsubsection{Knuth's Optimization}
|
|
\begin{align*}
|
|
\text{ Recurrence : }& D[i][j] = \min_{i\leqslant k < i} { \left(D[i][k] + D[k+1][j] \right) + C[i][j] } \\
|
|
\text{ Condition : }& C[i][j] \text{ is Monge } \& \\ & C[a][d] \geqslant C[b][c] \text{ for } a \leqslant b \leqslant c \leqslant d \\
|
|
\text{ Complexity : }& \mathcal O(N^3) \rightarrow \mathcal O(N^2)
|
|
\end{align*}
|
|
|
|
구간에 대해 동적 계획법(DP)을 수행할 때, 다음과 같은 점화식이 있다고 가정합니다:
|
|
\[ a[i][j] = \min_{i < k < j} \big( a[i][k] + a[k][j] \big) + f(i, j) \]
|
|
여기서 (최소화된) 최적의 \( k \)가 \( i \)와 \( j \) 모두에 대해 증가한다고 하면, 구간의 길이에 따라 DP를 계산하며 \( a[i][j] \)에 대해 \( k = p[i][j] \)를 \( p[i][j-1] \)부터 \( p[i+1][j] \) 사이에서만 탐색하면 됩니다.
|
|
|
|
\subsubsection{Aliens Trick (Lagrangian relaxation)}
|
|
\begin{align*}
|
|
\text{ Recurrence : }& D[k][i] = \min_{j<i} \left( D[k-1][j] + C[j+1][i]\right)\\
|
|
\text{ Condition : }& D[x][N] \text{ is convex } ( \text{ is implied when } C[i][j] \text{ is Monge } ) \\
|
|
& \left( f(x+1)-f(x) \leqslant f(x+2)-f(x+1) \right)\\
|
|
\text{ Complexity : }& \mathcal O(KN^2) \rightarrow \mathcal O(N^2 \log |W|)
|
|
\end{align*}
|
|
\subsubsection{Slope Trick}
|
|
13323 BOJ 수열 1/2. 수열 A가 주어진다. 증가수열 B에 대해, $\sum_{i=1}^N |A_i - B_i|$를 최소화하고, 그 B를 찾아라.
|
|
\inputminted[]{cpp}{source/DP/SlopeTrick.cpp}
|
|
|
|
\Algorithm{LineContainer}
|
|
{}
|
|
{$\mathcal O(\log N)$}
|
|
{cpp}
|
|
{source/DP/LineContainer.cpp}
|
|
|
|
% \Algorithm{Line Container-GMS}
|
|
% {}
|
|
% {}
|
|
% {cpp}
|
|
% {source/DP/LineContainerGMS.cpp}
|
|
|
|
\Algorithm{SoS(Some over Subsets)}
|
|
{}
|
|
{$\mathcal O(N\cdot2^N)$}
|
|
{cpp}
|
|
{source/DP/SoS.cpp}
|
|
|
|
% \Algorithm{}
|
|
% {}
|
|
% {$\mathcal O(N?)$}
|
|
% {cpp}
|
|
% {source/}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{String}
|
|
|
|
% \Algorithm{std::string}
|
|
% {}
|
|
% {}
|
|
% {}
|
|
% {source/String/std::string.cpp}
|
|
|
|
\Algorithm{KMP}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/String/KMP.cpp}
|
|
|
|
\Algorithm{F, Z, M, SA(Suffix Array), LCP(Longest Common Prefix)}
|
|
{For string s(1-indexed) of length N;
|
|
\begin{align*}
|
|
\texttt{F[i] = }& \text{maximum } k<i && \text{ s.t. } s[1\dots k] = s[i-k+1 \dots i]\\
|
|
\texttt{Z[i] = }& \text{maximum } k && \text{ s.t. } s[1\dots k] = s[i \dots i+k-1]\\
|
|
\texttt{M[i] = }& \text{maximum } k && \text{ s.t. } s[i-k+1 \dots i+k-1] \text{ is palindrom.}\\
|
|
\texttt{SA[i] = }& k && \text{ s.t. } s[k\dots N] \text{ is the } i^{th} \text{ smallest of } \\
|
|
& && \{s[1\dots N],\; s[2\dots N],\; \cdots,\; s[N\dots N]\}\\
|
|
\texttt{LCP[i] = }& \text{maximum } k && \text{ s.t. } s[SA[i-1]\dots SA[i-1]+k-1]\\
|
|
& && = s[SA[i] \dots SA[i]+k-1]
|
|
\end{align*}
|
|
}
|
|
{$\mathcal O(N),\mathcal O(N),\mathcal O(N),\mathcal O(N\log N),\mathcal O(N)$, respectively}
|
|
{cpp}
|
|
{source/String/F_Z_M_SA_LCP.cpp}
|
|
|
|
\Algorithm{Aho-Corasick}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/String/AhoCorasick.cpp}
|
|
|
|
\Algorithm{Eertree}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/String/Eertree.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Graph}
|
|
|
|
\Algorithm{SCC - Tarjan Algorithm}
|
|
{\texttt{scn[i]} : SCC number of node $i$, \texttt{nscc} : the number of SCCs}
|
|
{}
|
|
{cpp}
|
|
{source/Graph/TarjanSCC.cpp}
|
|
|
|
\SubAlgorithm{2-SAT}
|
|
{For all $(x||y)$ clause, make the edge from $!x$ to $y$, and from $!y$ to $x$, and then find the SCC from graph. Check whether for all $x$, $x$ and $!x$ are not in the same SCC. To reconstruct the 2-SAT, make it false for $x$ or make it true for $!x$ if we can, in decreasing order of scn.}
|
|
{}
|
|
{cpp}
|
|
{source/Graph/2SAT.cpp}
|
|
|
|
\Algorithm{Bipartite Matching - with DFS}
|
|
{Let's say that graph is bipartite. And Let's say that one group is $A$, and the other graph is $B$. $|A|=N$, $|B|=M$. \texttt{matching(c = s)} : add one matching from $s \in A$. If successfully matched, return true; otherwise return false. \texttt{selby[i] = } store $s\in A$, s.t. $i\in B$ is matched with $s$. \\ (e.g.) \texttt{forr(i, n) ans += matching(c=i);}}
|
|
{$\mathcal O(VE)$}
|
|
{cpp}
|
|
{source/Graph/BipartiteMatching.cpp}
|
|
|
|
\subsubsection{Hall's Marriage Theorem}
|
|
For some bipartite graph $G = (A \cup B, E)$, $G$ has perfect matching if and only if for every $S \subseteq A$, $|N_G(S)| \geqslant |S|$ holds.
|
|
\subsubsection{Minimum Vertex Cover on Bipartite Graph(Kőnig's Therorem)}
|
|
On bipartite graph, $$|\texttt{Minimum Vertex Cover}| = |\texttt{Maximum Matching}|$$
|
|
|
|
To find Minimum Vertex Cover, ( \added )
|
|
|
|
\subsubsection{Maximum Independent Set on Bipartite Graph}
|
|
On bipartite graph, $$|\texttt{Maximum Independent Set}|=|V|-|\texttt{Maximum Matching}|$$
|
|
|
|
* Note : Complement of the Vertex Cover is the Independent Set.
|
|
|
|
\subsubsection{Minimum Path Cover on DAG}
|
|
Let's think about the bipartite graph, with vertex set A and B, satisfying follow property:
|
|
|
|
\begin{itemize}
|
|
\item If there's edge from node $i$ to node $j$ on DAG, then there's edge connecting $i^{th}$ node of A and $j^{th}$ node of B, and vice versa.
|
|
\end{itemize}
|
|
|
|
Then following holds:
|
|
|
|
$|\texttt{Minimum Path Cover of DAG}| = |\texttt{Maximum Matching on Bipartite Graph}|$
|
|
|
|
|
|
\subsubsection{Maximum Antichain on DAG(Dilworth's Theorem)}
|
|
On DAG, $$|\texttt{Minimum Path Cover}| = |\texttt{Maximum Antichain}|$$
|
|
|
|
|
|
\Algorithm{Dominator Tree}
|
|
{For directed graph $G = (V, E)$ and some vertex $s \in V$ which can reach every vertex in $V$, $u$ is domonator of $v$ iff $u \neq v$ and every path from $s$ to $v$ must go through $u$. Dominator tree is the tree that connects $\textrm{idom}(v)$(Immediate Dominator) and $v$.}
|
|
{}
|
|
{cpp}
|
|
{source/Graph/DominatorTree.cpp}
|
|
|
|
\Algorithm{Network Flow - Dinic}
|
|
{Construct graph with \texttt{connect(from, to, capacity, isDirected);}. Find the flow from $S$ to $T$ with \texttt{flow(S, T);}. }
|
|
{$\mathcal O(V^2E)$, but it works like magic.}
|
|
{cpp}
|
|
{source/Graph/Dinic.cpp}
|
|
|
|
\Algorithm{MCMF - with SPFA}
|
|
{Construct graph with \texttt{connect(from, to, capacity, cost);}. Find the maximum flow and corresponding minimum cost from $S$ to $T$ with \texttt{flow(S, T);}.}
|
|
{$\mathcal O(VEf)$, but it works like magic.}
|
|
{cpp}
|
|
{source/Graph/MCMF.cpp}
|
|
|
|
|
|
\Algorithm{General Matching}
|
|
{We can use Tutte matrix. First, select arbitrary prime $p$. Second, construct $T_{i,j} = \begin{cases}
|
|
r_{i,j} & \text{if } (i,j) \in E \text{ and } i < j \\
|
|
-r_{j,i} & \text{if } (i,j) \in E \text{ and } i > j \\
|
|
0 & \text{otherwise}
|
|
\end{cases}$, with $r_{i, j}$ be the random integer in $[1, p-1]$. After construct $T$, the maximum size of $G$ is $\textrm{rank}(T)/2$ with high probability.
|
|
|
|
If we want the result of matching, use:}
|
|
{$O(n^3)$}
|
|
{cpp}
|
|
{source/Graph/Matching.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Tree}
|
|
|
|
\Algorithm{HLD(Heavy Light Decomposition)}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Tree/HLD.cpp}
|
|
|
|
\Algorithm{Centroid Tree}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Tree/Centroid.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Data Structure}
|
|
|
|
\Algorithm{PBDS - Policy-Based Data Structure}
|
|
{}
|
|
{Equivalent to std::set}
|
|
{cpp}
|
|
{source/DS/PBDS.cpp}
|
|
|
|
% \Algorithm{rope}{}{}{cpp}{}
|
|
|
|
\Algorithm{Union and Find - Queue Undoing}
|
|
{}
|
|
{$\mathcal O(\log^2N)$}
|
|
{cpp}
|
|
{source/DS/UF_QUndo.cpp}
|
|
|
|
\Algorithm{Fenwick Tree}{}{}{cpp}
|
|
{source/DS/Fenwick.cpp}
|
|
|
|
\Algorithm{Segment Tree}
|
|
{}
|
|
{$\mathcal O(\log N)$}
|
|
{cpp}
|
|
{source/DS/SegmentTree.cpp}
|
|
|
|
% \Algorithm{Segment Tree Generalization}
|
|
% {}
|
|
% {$\mathcal O(\log N)$}
|
|
% {cpp}
|
|
% {source/DS/SegmentTree.cpp}
|
|
|
|
\Algorithm{Segment Tree Beats}
|
|
{}
|
|
{$\mathcal O(\log N)$ on updating and querying}
|
|
{cpp}
|
|
{source/DS/STBeats.cpp}
|
|
|
|
\Algorithm{Li-Chao Tree}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/DS/LiChaoTree.cpp}
|
|
|
|
\Algorithm{Splay Tree}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/DS/SplayTree.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Numerical Analysis}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\section{Technic}
|
|
\Algorithm{Cartesian Tree}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Technic/CartesianTree.cpp}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Misc}
|
|
\Algorithm{Negative Division on C/C++}
|
|
{Procedure : (1) make both dividend and divisor positive, and calc the quotient (2) decide the sign of quotient, and calc $r$ as $a-b \times q$. Following shows the floored division, not the truncated division.}
|
|
{}
|
|
{cpp}
|
|
{source/Misc/NegDiv.cpp}
|
|
|
|
|
|
|
|
\Algorithm{Fast Input}
|
|
{Fast Input with fread. Do not use with scanf, cin, or other input function. Use \texttt{forr(i, n) read(arr[i]);} instead of \texttt{forr(i, n) scanf("\%d", arr+i);}. Use \texttt{read(s+1)} instead of \texttt{scanf("\%s", s+1);}.}
|
|
{}
|
|
{cpp}
|
|
{source/Misc/FastI.cpp}
|
|
|
|
\Algorithm{MT19937 Random Number}
|
|
{}
|
|
{}
|
|
{cpp}
|
|
{source/Misc/mt19937.cpp}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\begin{center}
|
|
\bigskip
|
|
--- Document end ---
|
|
\end{center}
|
|
|
|
\end{multicols}
|
|
|
|
\end{document}
|