1. (12 points) Give a table-driven parse table for the following grammar: 1) X ax 2) X P 3) P DT 4) T P 5) T ε 7) D 1

Transcription

1 1. (12 points) Give a table-driven parse table for the following grammar: 1) X ax 2) X P 3) P DT 4) T P 5) T ε 6) D 0 7) D 1 4. Recall that our class database has the following schemas: SNAP(StudentID, Name, Address, Phone) CP(Course, Prerequisite) CDH(Course, Day, Hour) CR(Course, Room) CSG(Course, StudentID, Grade) (a) (5 points) Give a datalog query and rule or rules for the query: Find all prerequisites (including prerequisites of prerequisites) for CS 936. (b) (5 points) Give a relational algebra expression for the query: List the name and phone numbers of all those who live at the same address as those who have an A grade in CS236. (c) (5 points) Give a relational algebra expression for the query : List all students who took CS236 at 1:00 pm in room 3718 HBLL who do not have a D or an E grade.

2 5. Consider the set A, and the relation R on A. A = {a, b, c, d, x, y, z} R = {(a, b), (a, z), (z, x), (x, c), (b, x), (d, c), (b, d)}. (a) (2 points) How many elements are in 2 A? (b) (6 points) Give the matrix for R + (the transitive closure of R). (c) (6 points) Determine the properties of R. Circle all that apply. reflexive irreflexive symmetric anti-symmetric asymmetric transitive (d) (2 points) Which properties does R* (the reflexive transitive closure) add to R +? Page 2/8

3 5. (continued) (e) (5 points) Draw the Hasse diagram for R* with the element a at the top. (f) (2 points) List the maximal element(s) of R*. (g) (5 points) Give a topological sort of R +. (h) (5 points) Using a matrix representation for R*, compute the symmetric closure, R (s), for R*, making the result a partition, P. (i) (2 points) Give the blocks of P. Page 3/8

4 6. (a). (3 points) For the given domain space and range space, show an injection that is not a bijection. a) D (b). (3 points) For the given domain space and range space, show a surjection that is not a bijection. b) (c). (3 points) For the given domain space and range space, show a bijection. c) Page 4/8

5 8. Consider the following graph G. (a). (5 points) Give the adjacency matrix, M, for this graph. (b). (5 points) Compute M 2. (c) (5 points) Assume the first pivot chosen is d. Give the matrix after one iteration of Warshall s algorithm. Page 5/8

6 11. Consider the following graph G. b e h a d g c f (a). (8 points) Create a depth first search tree starting from node a. If there is a choice to be made between which node to consider next, choose the node with the label closest to the beginning of the alphabet. (b). (6 points) Add all other edges from the graph G not yet found in the DFS tree (make them dashed lines). Label them as to whether they are forward, backward, or cross arcs. (c). (4 points) Add the postfix numbering to the DFS tree. (d). (3 points) What is it about this DFS tree that shows there is a cycle in the original graph? Page 6/8

7 12. Consider the following graph G. 5 a 4 6 h b c 7 14 f 10 d 2 e g 11 (a). (8 points) Assume you are using Dijkstra s algorithm to find the shortest path from a to all other nodes. List the nodes in the order in which they are settled. Page 7/8

8 13. Consider the following graph G. 5 a 4 6 h b c 7 14 f 10 d 2 e g 11 (a). (7 points) Using Kruskal s algorithm, list the edges (e.g. {x, y}) of the minimal cost spanning tree in the order in which they are added to the result. (c). (7 points) Using Prim s algorithm, and assuming the first node chosen is h, list nodes of the minimal cost spanning tree in the order in which they are added to the result. Page 8/8