Problem sets with comments/solutions

• 1. Asymptotic complexity (in preparation)
• 2. Minimum spanning trees (in preparation)
• 3. Directed graphs with comments/solutions (Consultations 2)
• 4. Heaps with comments/solutions (Consultations 2)
• 5. Isomorphism with comments/solutions (Consultations 1)
• 6. Combinatorial generation with comments/solutions (Consultations 1)
• 7. Finite automata and text search I with comments/solutions
• 8. Finite automata and text search II with comments/solutions
• 9. Finite automata and text search III with comments/solutions

Extra online consultations will be organized during semester weeks 10. - 14. to discuss presented solutions in problem set 1.-8.

Consultation 1

Consultation date: Tue 24.11., 16:15-17:45

The discussed topics of the consultation were topics 5. and 6. - Isomorphism and Combinatorial generation (see the documents above), those correspond to lectures 5. and 6.

Consultation 2

The discussed topics of the consultation will be topics 3. and 4. - Directed graphs and Heaps (see the documents above), those correspond to lectures 3. and 4.

Your course of action:

1. Note the consultation date: Mon 30.11., 11:00-12:30

2. Check the solutions/comments of the given topics 3. and 4. above. Mark the solutions which, in your opinion, need more explanation or corrections and which you would like to understand better.

3. Go to the poll Advanced Algorithms Consultation 2 and tick the boxes at the corresponding problems from topics 3. and 4. above, where you demand more explanation.
Complete this step before 10:00 PM Sunday 29.11.
The consultations will discuss primarily the problems in the highest demand.

Introduction and repetitions

• Asymptotic complexity recap. Training problems are here. Another set in Czech.
• Graphs and their representations, BFS, DFS. Solve example exam problems. Additional training problems can be found here.
• For more thorough or necessary basic acquaintance with the related topics consult the book Problems on Algorithms. Student are expected to be able to solve any problem marked by one cap in chapters 2. and 3. both during the semester and at the exam. We recommend problems 64. - 66., 144. - 148., 163. - 170. as an easy start.

Upload system

• An introduction to Upload system, 0-th programming optional (not graded) task, its specification can be found here.
  Write a program that solves the task. Upload it and check how the system asseses it. Follow strictly the rules specified on page Upload System.
  

Training homework problem

• The solution and submission is not mandatory and it is not graded if submitted. The problem itself is extremely simple, you can utilize it is to get acquainted with the. Upload System BRUTE.
  

Training problem: Problem statement and public data

• Spanning trees and minimum spanning trees of graphs: example problems. More problems for training here.

First homework problem

Laser Highways

Exam topics

• Asymptotic notations, asymptotic complexity, graph representations.
• Boruvka's, Kruskal's, Prim's algorithms. Their implementation and data structures, ie. priority queues and Union-Find.
• Prim's algorithm without a queue.
• Union-Find implementations.

Note:
Supplementary problems for missing online practices are available: in pdf .
Solve problems 1-5 and one of problems 6-10 according to your choice. Send a photo of your solutions by 6.10., to berezovs@fel.cvut.cz with cc to varhaiho@fel.cvut.cz.

Additional:
Problems for personal training: Directed graphs, strongly connected components, Eulerian graphs:
set 1, set 2.

Exam topics

• Eulerian graph (directed and undirected), Eulerian path/cycle, method of finding them.
• Strong and weak connectivity, strong and weak components, graph condensation. Algorithms which find strong components or produce a condensation.
• Topological ordering of a graph using DFS or using gradual roots/leaves removal.

• Binary, d-ary, binomial heaps. Fibonacci heaps. Example problems.
• Additional problems for training here

Exam topics

• The heap rule. Standard operations on heaps. Binary, d-ary and binomial heap.
• Binary and d-ary heap implemented in an array.
• Fibonacci heap, amortized complexity of its operations.

• Graph and tree isomorphism. Example problems.
• Additional problems for training here
  

Exam topics

• Graph isomorphism, definition, invariants, algorithm for finding all isomorphisms.
• Isomorphism and certificates of undirected trees, computing the certificate for a tree and reconstructing a tree from a certificate.

• Combinatorial generation of subsets, permutations and their ranks. Example problems.
• Additional problems for training here
  

Kreher, Stinson: Combinatorial Algorithms, notes:

• page 30, 31
• page 32, 33
• page 34, 35
• page 36, 37
• page 38, 39
• page 40, 41
• page 42, 43
• page 44, 45
• page 52, 53
• page 54, 55
• page 56, 67

Exam topics

• Combinations, variations, permutations with or without repetition, definitions, formulas. Binomial coefficients, their basic relations, Pascal triangle.
• Operations(functions) Rank and UnRank for ordered permutations and k-element subsets of given set.
• Gray code - definition, properties, operations Rank and UnRank.

• Finite automata and regular expressions. Example problems.
• Additional problems for training here
  

Exam topics

• Deterministic and non-deterministic finite automaton (DFA and NFA), definition. Languages accepted by finite automata.
• Algorithm for transforming NFA to DFA, simulation of NFA work without transformation to DFA. Epsilon-transitions and their removal from an automaton.
• Regular expressions. Definition, algorithm for transformation of regular expression to NFA.

• Approximate pattern matching and language operations. Example problems
• Text Searching. Additional problems for training here
  

Exam topics

• Construction of NFA for pattern matching. Impelementation of NFA/DFA by a table or by code.
• Hamming and Levenshtein distance. NFA/DFA for search of substrings which have nonzero distance from a given pattern.

• Approximate text searching example problems.
• Additional problems for training here.
  

Exam topics

• Finding dictionary entries in a text with the help of dictionary automaton.
• Number of states of dictionary NFA, DFA.
• Approximate text search based on previous methods.

Primes and pseudorandom numbers – Example problems .

Exam topics

• Generation of random numbers and prime numbers. Their properties. Prime factorization, exact and randomized primality tests.

• Skip List, B and B+ trees example problems.

Exam topics

• Skip list
• Search trees B and B+. Asymptotic complexity of particular search tree operations.

• 2-3-4 trees and B+ trees, splay trees. Additional problems for training here
  

Exam topics
2-3-4 trees and B+ trees. Asymptotic complexity of particular search tree operations.

• KD trees, Nearest neighbours problem. Example problems incl. previous week.

Exam topics
KD trees, search for Nearest Neighbour in 2D.

In preparation

• Trie, suffix trie, binary trie

Exam topics

• Trie, suffix trie, binary trie

In preparation
* Radix trie, Patricia trie, segment tree.

Exam topics
* Radix trie, Patricia trie, segment tree.

• Binary search tree
• AVL tree
• B tree
• B+ tree
• Red-black tree
• Splay tree
• 2-3-4 tree

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