Exam Preparation Algorithms Course: Sample Exam SoftUni Team Technical Trainers Software University

2  The practical exam checks your skills to design and implement algorithms efficiently  4 problems for 6 hours  Graphs, dynamic programming, recursion, combinatorics, greedy, …  Automated judge system with real-time feedback: Algorithms: Practical Exam

3  Group-Permutations: we are given n capital Latin letters. Generate all their permutations, so that the same letters stay together. Example:  Example: n = 7, letters = { B, C, A, B, A, C, B }. Result:  AABBBCC  AACCBBB  BBBAACC  BBBCCAA  CCAABBB  CCBBBAA Sample Problem #1 – Combinatorics

4  Remove duplicate letters: { B, C, A, B, A, C, B }  { A, B, C }  Permute the letters and replace each letter with its original number of occurrences:  ABC  AABBBCC  ACB  AACCBBB  BAC  BBBAACC  BCA  BBBCCAA  CAB  CCAABBB  CBA  CCBBBAA Group-Permutations – Solution

5  Bridges: we have two sequences of points (natural numbers): north[0 … n-1] and south[0 … m-1]  Connect with bridges as many as possible equal numbers from both sequences so that there are no crossing bridges  Print the max number of bridges Sample Problem #2 – Dynamic Programming

6 Bridges – Solution Assume f(x, y) = max bridges using north[0…x] and south[0…y] f(x, y) = 0 when x < 0 or y < 0 f(x, y) = max ( f(x-1, y), f(x, y-1) ; 1 + f(x-1, y), 1 + f(x, y-1) when north[x] == south[y] )

7  You are given a directed graph. Find all cycles of length 3. Example:  Cycles (in lexicographical order): {0  1  6}, {0  13  2}, {0  13  9}, {4  6  10}, {6  11  8} Sample Problem #3 – Graphs

8  For each 3 vertices {u, v, w} check if they form a loop  To skip the duplicates, require that u < v and u < w  To improve performance, use sorted adjacency list for each vertex Cycles of Length 3 – Solution for u = 0 … n-1 for v = children-in-increasing-order(u) where u < v for w = children-in-increasing-order(v) where u < w if edge-exists(w  u) print { u  v  w } u v w

9  Line Inverter:  We are given a board of size n x n  Each cell holds black ( 0 ) or white ( 1 ) cell  At each step we can invert a row or invert a column  Find the minimal number of inversions to make the board black  Or print -1 if this is not possible Sample Problem #4 – Other

10  Line-Inverter problem has a simple greedy solution: 1. Invert all rows that have a white cell in the first column 2. Invert all columns that have a white cell in the first row 3. If the board is black, we have solution, otherwise  no solution  Another solution is found by:  First inverting the first column, then running the previous algorithm  Take the better of these two algorithms Line Inverter: Solution

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License  This course (slides, examples, labs, videos, homework, etc.) is licensed under the "Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International" licenseCreative Commons Attribution- NonCommercial-ShareAlike 4.0 International 12  Attribution: this work may contain portions from  "Fundamentals of Computer Programming with C#" book by Svetlin Nakov & Co. under CC-BY-SA licenseFundamentals of Computer Programming with C#CC-BY-SA  "Data Structures and Algorithms" course by Telerik Academy under CC-BY-NC-SA licenseData Structures and AlgorithmsCC-BY-NC-SA

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