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Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 403 — #411<br />
10.11. Summary of Relational Properties 403<br />
Problem 10.16.<br />
Describe a sequence consisting of the integers from 1 to 10,000 in some order so<br />
that there is no increasing or decreasing subsequence of size 101.<br />
Problem 10.17.<br />
What is the smallest number of partially ordered tasks <strong>for</strong> which there can be more<br />
than one minimum time schedule, if there are unlimited number of processors?<br />
Explain your answer.<br />
Problem 10.18.<br />
The following DAG describes the prerequisites among tasks f1; : : : ; 9g.<br />
9<br />
8<br />
3<br />
5<br />
7<br />
2<br />
4<br />
6<br />
1
“mcs” — 2017/3/3 — 11:21 — page 402 — #410 402 Chapter 10 Directed graphs & Partial Orders king a b king king d c not a king Figure 10.11 A 4-chicken tournament in which chickens a, b and d are kings. . The King Chicken Theorem means that if the player with the most victories is defeated by another player x, then at least he/she defeats some third player that defeats x. In this sense, the player with the most victories has some sort of bragging rights over every other player. Un<strong>for</strong>tunately, as Figure 10.11 illustrates, there can be many other players with such bragging rights, even some with fewer victories. Problems <strong>for</strong> Section 10.5 Practice Problems Problem 10.14. What is the size of the longest chain that is guaranteed to exist in any partially ordered set of n elements? What about the largest antichain? Problem 10.15. Let fA; :::; H g be a set of tasks that we must complete. The following DAG describes which tasks must be done be<strong>for</strong>e others, where there is an arrow from S to T iff S must be done be<strong>for</strong>e T . (a) Write the longest chain. (b) Write the longest antichain. (c) If we allow parallel scheduling, and each task takes 1 minute to complete, what is the minimum amount of time needed to complete all tasks?
“mcs” — 2017/3/3 — 11:21 — page 403 — #411 10.11. Summary of Relational Properties 403 Problem 10.16. Describe a sequence consisting of the integers from 1 to 10,000 in some order so that there is no increasing or decreasing subsequence of size 101. Problem 10.17. What is the smallest number of partially ordered tasks <strong>for</strong> which there can be more than one minimum time schedule, if there are unlimited number of processors? Explain your answer. Problem 10.18. The following DAG describes the prerequisites among tasks f1; : : : ; 9g. 9 8 3 5 7 2 4 6 1
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