Tuesday, May 12, 2015

List of Topics for programming Competitions

1.  Basic Geometry/Euclidean Geometry/Coordinate Geometry/ [3-D variants of everything].
2.  Computational Geometry.
a.  Graham Scan algorithm for Convex Hull O(n * log(n)).
b.  Online construction of 3-D convex hull in O(n^2).
c.  Bentley Ottmann algorithm to list all intersection points of n line segments in O((n + I) * logn).
    Suggested Reading -
d.  Rotating Calipers Technique.
    Suggested Reading - http://cgm.cs.mcgill.ca/~orm/rotcal.html
    Problems - Refer the article for a list of problems which can be solved using Rotating Calipers technique.
e.  Line Sweep/Plane Sweep algorithms -
    Area/Perimeter of Union of Rectangles.
    Closest pair of points.
    Suggested Reading -
    Problems - Follow the tutorial for list of problems.
f.  Area of Union of Circles.
g.  Delayunay Triangulation of n points in O(n * logn).
h.  Voronoi Diagrams of n points in O(n * logn) using Fortunes algorithm.
i.  Point in a polygon problem -
    O(n) solution without preprocessing.
    O(logn) algorithm with O(n * logn) preprocessing for convex polygons.
j.  Problems on computational geometry -
    CultureGrowth, PolygonCover on Topcoder.
k.  Suggested Reading -
    Computational Geometry: Algorithms and applications. Mark De Burg.
3.  String Algorithm.
a.  KnuthMorrisPratt algorithm.
    Problems - NHAY, PERIOD on SPOJ.
    Suggested Reading -
1.  Cormen chapter on Strings.
b.  Aho Corasick algorithm.
    Problems - WPUZZLES on SPOJ.
c.  Suffix Arrays
    O(n^2 * logn) Naive method of suffix array construction
    O(n * logn^2) method of suffix array construction
    O(n * logn) method of suffix array construction.
    O(n) method of suffix array construction
    O(n) LCA preprocess on Suffix Arrays to solve a variety of string problems.
d.  Suffix Trees
    O(n) construction of Suffix trees using Ukkenon’s algorithm.
    O(n) construction of Suffix Trees if provided with Suffix Arrays using Farach’s algorithm.
e.  Suffix Automata
    O(n) Suffix Automaton construction.
f.  Dictionary Of Basic Factors
    O(n * logn) method of DBF construction using Radix Sort.
g.  Manachar’s algorithm to find Lengh of palindromic substring of a string centered at a position for each position in the string. Runtime -> O(n).
h.  Searching and preprocessing Regular Expressions consisting of ‘?’, ‘*’.
i.  Multi-dimensional pattern matching.
j.  Problems on Strings [can be solved with a variety of techniques] -
4.  Basic Graphs [beginner].
a.  Representation of graphs as adjacency list, adjacency matrix, incidence matrix and edge list and uses of different representations in different scenarios.
b.  Breadth First Search.
    problems -
1.  PPATH, ONEZERO, WATER on SPOJ
c.  Depth First Search.
d.  Strongly Connected Components.
    problems -
1.  TOUR and BOTTOM on SPOJ.
e.  Biconnected Components, Finding articulation points and bridges].
    problems -
1.  RELINETS, PT07A on SPOJ.
f.  Dijkstra algorithm -
    problems -
1.  SHPATH on SPOJ.
g.  Floyd Warshall algorithm -
    problems -
1.  COURIER on SPOJ.
h.  Minimum Spanning Tree
    problems -
1.  BLINNET on SPOJ.
i.  Flood-fill algorithm
j.  Topological sort
k.  Bellman-Ford algorithm.
l.  Euler Tour/Path.
    problems - WORDS1 on SPOJ.
m.  Suggested reading for most of the topics in Graph algorithms -
    Also refer to the tutorial for problems concerning these techniques.
    Cormen chapter 22 to 24.
5.  Flow networks/ matching etc etc. [Interdiate/Advanced].
a.  Maximum flow using Ford Fulkerson Method.
    Suggested Reading -
    problems - TAXI, POTHOLE, IM, QUEST4, MUDDY, EN, CABLETV, STEAD, NETADMIN, COCONUTS, OPTM on SPOJ.
b.  Maximum flow using Dinics Algorithm.
    Problems - PROFIT on spoj.
c.  Minimum Cost Maximum Flow.
    Successive Shortest path algorithm.
    Cycle Cancelling algorithm.
    Suggested Reading -
d.  Maximum weighted Bipartite Matching (Kuhn Munkras algorithm/Hungarian Method)
e.  Stoer Wagner min-cut algorithm.
f.  Hopcroft Karp bipartite matching algorithm.
    problems - ANGELS on SPOJ.
g.  Maximum matching in general graph (blossom shrinking)
h.  Gomory-Hu Trees.
    i) Problems - MCQUERY on Spoj.
i.  Chinese Postman Problem.
j.  Suggested Reading for the full category ->
    Network flow - Algorithms and Applications by Ahuja
    Cormen book chapter 25.
6.  Dynamic Programming.
a.  Suggested Reading - Dynamic Programming(DP) as a tabulation method
    Cormen chapter on DP
b.  Standard problems (you should really feel comfortable with these types)
c.  State space reduction
d.  Solving in the reverse - easier characterizations looking from the end
e.  Counting/optimizing arrangements satisfying some specified properties
f.  Strategies and expected values
g.  DP on probability spaces
h.  DP on trees
i.  DP with datastructures
j.  Symmetric characterization of DP state
k.  A good collection of problems
7.  Greedy.
a.  Suggested Reading -
    Chapter on Greedy algorithms in Cormen.
    http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=greedyAlg
b.  problems - refer to the topcoder tutorial.
8.  Number Theory.
a.  Modulus arithmetic - basic postulates [Including modular linear equations ,  Continued fraction and Pell's equation]
    Suggested Reading -
1.  Chapter 1 from Number Theory for Computing by SY Yan [ Recommended ]
2.  31.1, 31.3 and 31.4 from Cormen
    Problems
b.  Fermat's theorem, Euler Totient theorem ( totient function, order , primitive roots )
    Suggested Reading
1.  1.6, 2.2 from Number Theory by SY Yan
2.  31.6 , 31.7 from Cormen
    Problems
c.  Chinese remainder theorem
    Suggested Reading
1.  31.5 from Cormen
2.  1.6 from Number Theory by SY Yan
    Problems
1.  Project Euler 271
d.  Primality tests -
    Deterministic O(sqrt(n) ) approach
    Probabilistic primality tests - Fermat primality test, Miller-Rabin Primality test
1.  Suggested Reading -
b.  Cormen 31.8
c.  2.2 from Number Theory by SY Yan
2.  Problems -
a.  PON, PRIC, SOLSTRAS on SPOJ
e.  Prime generation techniques - Sieve of Erastothenes
    Suggested Problems - PRIME1 on SPOJ
f.  GCD using euclidean method
    Suggested Reading
1.  31.2 Cormen
    Problems -
1.  GCD on SPOJ
g.  Logarithmic Exponentiation
    Suggested Reading -
h.  Integer Factorization
    Naive O(sqrt(n)) method
    Pollard Rho factorization
    Suggested Reading
1.  2.3 from Number Theory SY Yan
2.  31.9 Cormen
    Problems -
i.  Stirling numbers
j.  Wilson theorem
    nCr % p  in O(p) preprocess and O(log n ) query 
k.  Lucas Theorem
l.  Suggested Reading for Number Theory -
    Number theory for computing by Song Y Yan  [ Simple book describing concepts in details ]
    Concepts are also superficially covered in Chapter 31 of Introduction to Algorithms by Cormen
m.  Problems on Number Theory -
9.  Math (Probability, Counting, Game Theory, Group Theory, Generating functions, Permutation Cycles, Linear Algebra)
a.  Probability.
Syllabus
    Basic probability and Conditional probability
1.  Suggested problems
    Random variables, probability generating functions
    Mathematical expectation + Linearity of expectation
1.  Suggested problems
    Special discrete and continuous probability distributions
1.  Bernoulli, Binomial, Poisson, normal distribution
2.  Suggested Problem
    Suggested Readings
1.  Cormen appendix C (very basic)
5.  William Feller, An introduction to probability theory and its applications
b.  Counting
Syllabus
    Basic principles - Pigeon hole principle, addition, multiplication rules
1.  Suggested problems
a.  http://acm.timus.ru/problem.aspx?space=1&num=1690
b.  http://www.topcoder.com/stat?c=problem_statement&pm=10805
3.  Suggested readings
a.  http://en.wikipedia.org/wiki/Combinatorial_principles
b.  http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=combinatorics
c.  http://www.maa.org/editorial/knot/pigeonhole.html
    Inclusion-exclusion
1.  Suggested readings
a.  http://en.wikipedia.org/wiki/Inclusion–exclusion_principle
2.  Suggested problems
a.  http://www.topcoder.com/stat?c=problem_statement&pm=4463&rd=6536
b.  http://www.topcoder.com/stat?c=problem_statement&pm=10238
    Special numbers
1.  Suggested reading - Stirling, eurlerian, harmonic, bernoulli, fibonnacci numbers
a.  http://en.wikipedia.org/wiki/Stirling_number
b.  http://en.wikipedia.org/wiki/Eulerian_numbers
c.  http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
d.  http://en.wikipedia.org/wiki/Bernoulli_number
e.  http://en.wikipedia.org/wiki/Fibonnaci_numbers
f.  Concrete mathematics by Knuth
2.  Suggested problems
a.  http://www.topcoder.com/stat?c=problem_statement&pm=1643
b.  http://www.topcoder.com/stat?c=problem_statement&pm=8202&rd=11125
c.  http://www.topcoder.com/stat?c=problem_statement&pm=8725
d.  http://www.topcoder.com/stat?c=problem_statement&pm=2292&rd=10709
    Advanced counting techniques - Polya counting, burnsides lemma
1.  Suggested reading
a.  http://en.wikipedia.org/wiki/Burnside's_lemma
b.  http://petr-mitrichev.blogspot.com/2008/11/burnsides-lemma.html
2.  Suggested Problems
a.  http://www.topcoder.com/stat?c=problem_statement&pm=9975
b.  http://www.spoj.pl/problems/TRANSP/
 c. Game theory
Syllabus     
    Basic principles and Nim game
1.  Sprague grundy theorem, grundy numbers
2.  Suggested readings
a.  http://en.wikipedia.org/wiki/Sprague%E2%80%93Grundy_theorem
b.  http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=algorithmGames
c.  http://www.ams.org/samplings/feature-column/fcarc-games1
d.  http://www.codechef.com/wiki/tutorial-game-theory
3.  Suggested problems
a.  http://www.topcoder.com/stat?c=problem_statement&pm=3491&rd=6517
b.  http://www.topcoder.com/stat?c=problem_statement&pm=3491&rd=6517
    Hackenbush
1.  Suggested readings
a.  http://en.wikipedia.org/wiki/Hackenbush
b.  http://www.ams.org/samplings/feature-column/fcarc-partizan1
2.  Suggested problems
a.  http://www.cs.caltech.edu/ipsc/problems/g.html
b.  http://www.spoj.pl/problems/PT07A/
  d. Linear Algebra
       Syllabus     
    Matrix Operations
1.  Addition and subtraction of matrices
a.  Suggested Reading
i.  Cormen 28.1
2.  Multiplication ( Strassen's algorithm ), logarithmic exponentiation
a.  Suggested reading
i.  Cormen 28.2
ii. Linear Algebra by Kenneth Hoffman Section 1.6
b.  Problems
i.  http://uva.onlinejudge.org/external/111/11149.html
3.  Matrix transformations [ Transpose, Rotation of Matrix, Representing Linear transformations using matrix ]
a.  Suggested Reading
i.  Linear Algebra By Kenneth Hoffman Section 3.1,3.2,3.4,3.7
b.  Problems
i.  http://www.topcoder.com/stat?c=problem_statement&pm=6877
ii. JPIX on Spoj
4.  Determinant , Rank and Inverse of Matrix [ Gaussean Elimination , Gauss Jordan Elimination]
a.  Suggested Reading
i.  28.4 Cormen
ii. Linear Algebra by Kenneth Chapter 1
b.  Problems
i.  http://www.topcoder.com/stat?c=problem_statement&pm=8174
ii. http://www.topcoder.com/stat?c=problem_statement&pm=6407&rd=9986
iii.         http://www.topcoder.com/stat?c=problem_statement&pm=8587
iv. HIGH on Spoj
5.  Solving system of linear equations
a.  Suggested Reading
i.  28.3 Cormen
ii. Linear Algebra by Kenneth Chapter 1
b.  Problems -
i.  http://www.topcoder.com/stat?c=problem_statement&pm=3942&rd=6520
6.  Using matrix exponentiation to solve recurrences
a.  Suggested Reading
b.  Problems
i.  REC, RABBIT1 , PLHOP on spoj
ii. http://www.topcoder.com/stat?c=problem_statement&pm=6386 , http://www.topcoder.com/stat?c=problem_statement&pm=7262, http://www.topcoder.com/stat?c=problem_statement&pm=6877
7.  Eigen values and Eigen vectors
a.  Problems
i.  http://www.topcoder.com/stat?c=problem_statement&pm=2423&rd=4780
    Polynomials
1.  Roots of a polynomial [  Prime factorization of a polynomial, Integer roots of a polynomial, All real roots of a polynomial ]
a.  Problems
i.  http://www.topcoder.com/stat?c=problem_statement&pm=8273&rd=10798
ii. POLYEQ , ROOTCIPH on Spoj
2.  Lagrange Interpolation
a.  Problems
i.  http://www.topcoder.com/stat?c=problem_statement&pm=10239
ii. http://www.topcoder.com/stat?c=problem_statement&pm=8725
   e. Permutation cycles
    Suggested Reading
1.  Art of Computer Programming by Knuth Vol. 3
    Problems
1.  ShuffleMethod, Permutation and WordGame on topcoder.
   f. Group Theory
    Bernside Lemma, Polias theorem
1.  Suggested Reading
a.   Hernstein's topics in algebra
2.  Problems
a.  TRANSP on spoj
b.  http://www.topcoder.com/stat?c=problem_statement&pm=9975
b.  Generating functions
    Suggested Reading
1.  Herbert Wilf's generating functionology
2.  Robert Sedgewick and Flajoulet's Combinatorial analysis

10. Data Structures.
i.  Basic
a.  Arrays/Stacks/Queues :
    Problems
    Reading:
1.  CLRS: section 10.1

           b. Singly/Doubly Linked List :
    Problems
1.  https://www.spoj.pl/problems/POSTERS/
    Reading: CLRS: section 10.2, Mark Allen Weies Chapter 3
            c. Hash Tables :
    Problems
    Reading: CLRS: Chapter 11, Mark Allen Weies Chapter 5
            d. Circular linked list / queue
    Problems
            e. Binary/nary  Trees
    Reading
1.  CLRS: section 10.4
2.  CLRS: Chapter 12
3.  Mark Allen Weies Chapter 4
            f. Heaps
    Problems
    Reading : Mark Allen Weies Chapter 6
    ii. Advanced
a.  Trie (Keyword tree)
    Problems
    Reading
b.  Interval trees / Segment Trees
    Problems
    Reading
c.  Fenwick(Binary Indexed) trees
    Problems
d.  Disjoint data structures
    Problems
    Reading:
2.  Mark Allen Weies Chapter 8
e.  Range minimum Query(RMQ)
    Problems
f.  Customized interval/segment trees (Augmented DS)
    Problems
    Reading: CLRS: Chapter 14 (augmented DS)
          g.  AVL Trees
    Problems
    Reading

    iii. Miscellaneous (Not to be covered)  
a.  Splay Trees
b.  B/B+ Trees
c.  k-d Trees
d.  Red-black Trees
e.  Skip List
f.  Binomial/ Fibonacci heaps

    iv. Exercices
2.  https://www.spoj.pl/problems/HELPR2D2/ (Hint: Interval Trees)
5.  https://www.spoj.pl/problems/HEAPULM/ (Hint: Interval Trees)
6.  https://www.spoj.pl/problems/CORNET/ (Hint: Disjoint )


11. Search Techniques/Bruteforce writing techniques/Randomized algorithms.
a.  Backtracking - [Beginner].
    problems ->
1.  N queens problems
2.  Knights Tour
3.  Sudoku Problem
4.  Tiling Problem.
5.  15 puzzle.
b.  Dancing Links and Algorithm X given by Knuth - [Advanced]
    problems - PRLGAME, SUDOKU, NQUEEN on SPOJ
    Suggested reading -
c.  Binary Search - [Beginner].
    poblems - AGGRCOW on SPOJ. Refer the tutorial for more problems.
    finding all real roots of a polynomial using binary search. [intermediate].
    Suggested Reading -
d.  Ternary Search - [Intermediate].
    problems -
e.  Meet in the middle [Intermediate].
    problems -
f.  Hill Climbing [Advanced].
g.  Regular Iteration to reach a fixed point [Advanced].
    Newton-Raphson method to find root of a mathematical function.
    Iterations to solve linear non-homogeneous system of equations.
h.  Randomized Algorithms [Intermediate]-
    Quick-Sort.
12. General programming issues in contests ->
a.  Arithmetic Precision - [Beginner].
    Suggested Reading -
b.  Representing sets with bitmasks and manipulating bitmasks - [Beginner].
    Suggested Reading -
    problems - refer to the tutorial link in Suggested reading section.


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