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Chapter 7: Q15P (page 323)
Show that is closed under the star operation. (Hint: Use dynamic programming. On input , build a table indicating for each whether the substring
The solution is,
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In a directed graph, the indegreeof a node is the number of incoming edges and the outdegreeis the number of outgoing edges. Show that the following problem is NP-complete. Given an undirected graph G and a designated subset C of G鈥檚 nodes, is it possible to convert G to a directed graph by assigning directions to each of its edges so that every node in C has in-degree 0 or outdegree 0, and every other node in G has indegree at least 1?
The difference hierarchyis defined recursively as
(Here .) For example, a language in D2P is the difference of two NP languages. Sometimes is called DP (and may be written DP). Let
.Show that Z is complete for DP. In other words, show that Z is in DP and every language in DP is polynomial time reducible to Z.
Show that is closed under union and concatenation.
Modify the algorithm for context-free language recognition in the proof of Theorem 7.16 to give a polynomial time algorithm that produces a parse tree for a string, given the string and a CFG, if that grammar generates the string.
Let Analyse the algorithm given on page 185 to show that this language is in .
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