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Questions and Answers
What does Hall’s marriage theorem require in a bipartite graph?
What do greedy algorithms assume about obtaining the global optimum?
What is combinatorial optimization aimed at determining?
In the context of Hall’s Marriage Theorem, what happens if both sets in a bipartite graph are not covered by the matching?
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In combinatorial optimization, what does selecting the local optimum at the current time lead to?
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What is the main characteristic of a greedy algorithm approach to problem-solving?
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What technique is used for finding a solution to a problem with constraints defined by a system of inequalities?
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In the context of the text, what do vertices represent in a graph?
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What does Hall’s theorem state about the possibility of a matching between men and women?
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Why did the matching fail in the given example involving Maria, Sophia, and Kenneth?
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What is the reason for generating the same hash for all the given keys in the poorly designed hash function?
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What term is used to describe a matching where all sets are covered by the matching?
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Why does the text recommend rounding up to the nearest prime number when choosing the size of the hash table?
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What happens if m ≥ f for every set of women based on Hall’s theorem?
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In the given example, what is the size of the hash table used for computing hashes?
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Which factor could help in avoiding generating similar hashes for keys with the same characters?
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What could be a consequence of having multiple keys with different permutations but generating the same hash?
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How does considering the position of characters improve a hash function?
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What is the purpose of a disjoint-set data structure?
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Which data structure employs the find operation to determine if two elements belong to the same subset?
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What type of lists do Fibonacci heaps use for its operations?
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In a disjoint-set data structure, how are elements organized?
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What is the time complexity for key operations in Fibonacci heaps?
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Which of the following is NOT a resource needed for implementing union-find and Fibonacci heap data structures?
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Which statement is true about the hash function used in the given scenario?
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What is the correct order in which the keys were inserted into the hash table in the first scenario?
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Which keys hash to the same value in the second scenario?
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What would be the resultant hash table after inserting keys 14, 18, 12, 6, 9, and 8 using linear probing?
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In the first scenario, how many elements were inserted into the hash table?
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What type of probing method was used in the scenarios described?
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Study Notes
Hash Table and Hash Function
- A good hash function should generate unique hashes for different keys.
- A poorly designed hash function may generate the same hash for different keys, leading to collisions.
- To improve the hash function, consider the position of the characters as part of the computation.
Hash Table Operations
- A hash table of length 10 uses the hash function h(k) = k mod 10 and linear probing.
- After inserting 6 values into an empty hash table, the table is as shown below:
- 0 1 2 3 4 5 6 7 8 9
- 42 23 34 52 46 33
Hashing and Collisions
- Given the input 35, 70, 61, 30, and 45, and the hash function h(k) = k mod 8, the following statements are true:
- 61 and 45 hash to the same value
- 35 and 30 hash to the same value
- The keys 14, 18, 12, 6, 9, and 8 are inserted into an initially empty hash table of length 7 using the hash function h(k) = k mod 7 and linear probing.
- The resultant hash table is:
- 0 1 2 3 4 5 6
- 12
Combinatorial Optimization
- Combinatorial optimization aims to determine the optimal answer from a list of possible solutions to a problem.
- Hall's Marriage Theorem is used for finding a matching that covers at least one side of the bipartite graph.
- The theorem states that for a set of f women, denote m to represent the number of men that at least 1 of these women prefer. If m ≥ f for every set of women, then a matching is possible.
- Dijkstra's algorithm uses a greedy approach to find the shortest-path between nodes in a graph.
- Linear programming is an optimization technique used for finding a solution to a problem with constraints defined by a system of inequalities.
Data Structures for Combinatorial Optimization
- Data structures for combinatorial optimization include union-find and Fibonacci heap data structures.
- Fibonacci heaps use doubly linked lists to enable operations like cutting off a portion of a list, merging two lists, and determining the minimum (or maximum) value to be performed in O(N) time.
- A disjoint-set data structure keeps track of a set of elements that have been divided into several overlapping subsets.
- To determine whether two elements are in the same subset, the find operation on a disjoint-set data structure is employed.
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Description
Test your knowledge on bipartite graph matching, Hall's marriage theorem, decision variables, and greedy algorithms. Explore concepts related to data structures and algorithms in practical applications.
