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In the Floyd-Warshall all shortest path algorithm, what is the purpose of introducing an intermediate vertex k during iteration?

None of these

To evaluate whether a path from i to j can be shortened via k

To reduce the number of edges adjacent to vertex k in the graph

To prioritize vertices with higher degrees during path updates for each intermediate vertex k

To remove all self-loops from the graph going through vertex k

To detect cycles in the graph including vertex k

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Which of the following scenarios would cause the greedy approach (rational knapsack algorithm, choosing items by highest value-to-weight ratio) to fail in solving the 0/1 (or binary) knapsack optimally?

Items have fractional weights

None of these

All items have the same value and weight

All item values are zero

Knapsack capacity is very large

The item with the best value/weight ratio cannot be combined with others due to its weight

100%

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Based on the discussion from our lectures, what is the running time of the preprocessing step of the string matching algorithm with finite automata (i.e., of the step where we compute the transition function)? Recall that n, and m are the sizes of the text and pattern, respectively, and Σ is the alphabet. (Choose the tightest bound.)

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Consider inserting the keys 21, 22, 31, 4,15, 28, 17, 88, 59 (in this order) into a hash table of length m=11 using linear probing with the auxiliary hash function h'(k)=k. Based on our lecture discussion, which of the choices below represents the resulting hash table?