(a) The SCC algorithm returns the SCC's one by one in a reverse topological order (sink to source). How would you make a small modification to the algorithm to return the SCC's in topological order (source to sink) instead? (b) Consider two algorithms for the same problem: • Algorithm A, which runs in O(n) and produces a correct answer with probability 0.7, and a wrong answer with probability 0.3. • Algorithm B, which runs in O(n log n) and produces a correct answer with probability 0.99, and a wrong answer with probability 0.01. Which of the two algorithms should you use to build a more asymptotically (in n) efficient algorithm with probability 0.99 of producing a correct answer? Justify. Assume that you can always check if an answer is correct in constant time. (c) Argue (very briefly) that any algorithm to compute the minimum spanning tree of a graph G = (V,E) must use at least 0(|E|) time. %3D (d) Show that it is possible to achieve negative regret in the expert's problem if the losses are chosen ahead of time. Give the losses and your choices of experts. (e) Show that if there is no unique minimum weight edge in a graph G, but exactly two minimum-weight edges ej and e2, then both ej and ez must be included in any MST of G.
(a) The SCC algorithm returns the SCC's one by one in a reverse topological order (sink to source). How would you make a small modification to the algorithm to return the SCC's in topological order (source to sink) instead? (b) Consider two algorithms for the same problem: • Algorithm A, which runs in O(n) and produces a correct answer with probability 0.7, and a wrong answer with probability 0.3. • Algorithm B, which runs in O(n log n) and produces a correct answer with probability 0.99, and a wrong answer with probability 0.01. Which of the two algorithms should you use to build a more asymptotically (in n) efficient algorithm with probability 0.99 of producing a correct answer? Justify. Assume that you can always check if an answer is correct in constant time. (c) Argue (very briefly) that any algorithm to compute the minimum spanning tree of a graph G = (V,E) must use at least 0(|E|) time. %3D (d) Show that it is possible to achieve negative regret in the expert's problem if the losses are chosen ahead of time. Give the losses and your choices of experts. (e) Show that if there is no unique minimum weight edge in a graph G, but exactly two minimum-weight edges ej and e2, then both ej and ez must be included in any MST of G.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Transcribed Image Text:(a) The SCC algorithm returns the SCC's one by one in a reverse topological order (sink to source). How
would you make a small modification to the algorithm to return the SCC's in topological order (source
to sink) instead?
(b) Consider two algorithms for the same problem:
Algorithm A, which runs in O(n) and produces a correct answer with probability 0.7, and a
wrong answer with probability 0.3.
Algorithm B, which runs in O(n log n) and produces a correct answer with probability 0.99, and
a wrong answer with probability 0.01.
Which of the two algorithms should you use to build a more asymptotically (in n) efficient algorithm
with probability 0.99 of producing a correct answer? Justify. Assume that you can always check if an
answer is correct in constant time.
(c) Argue (very briefly) that any algorithm to compute the minimum spanning tree of a graph G = (V,E)
must use at least 0(E) time.
(d) Show that it is possible to achieve negative regret in the expert's problem if the losses are chosen
ahead of time. Give the losses and your choices of experts.
(e) Show that if there is no unique minimum weight edge in a graph G, but exactly two minimum-weight
edges ej and e2, then both ej and ez must be included in any MST of G.
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