CMPSCI 250: Introduction to Computation

Second Midterm Exam

David Mix Barrington

16 November 2010

Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points plus 10 extra credit. Actual scale is A=84, C=54.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 15 points
  Q2: 30+10 points
  Q3: 35 points
  Q4: 15 points
  Q5: 20 points
 Total: 100+10 points

• Question 1 deals with a sequence of integers S(n), defined for all naturals n as follows. S(0) is defined to be 0. If n+1 is even, S(n+1) is defined to be S(n) + 2n + 1. If n+1 is odd, S(n+1) is defined to be S(n) - (2n+1). S(n) can equivalently be defined as the sum from 1 to n of (-1)ⁱ(2i-1). You may use either definition.

  For example, S(1) = -1, S(2) = -1 + 3 = 2, and S(3) = -1 + 3 - 5 = -3.

• Question 1 (15): Find a rule that specifies S(n) for any possible arbitrary natural n. (This might, for example, be one rule for even n and another for odd n.) Prove your rule to be correct for all n, by induction.

• The next question involves tilings, like the tilings with L-shaped tiles in Section 4.11 of the book. We are again tiling square blocks, with the lower right corner square missing. But this time our tiles are "I-shaped" rather than L-shaped -- they are 1 by 3 rectangles. Each rectangle can be placed either horizontally or vertically.

• Question 2 (30+10): Let P(n), where n is a positive natural, be the statement "an n by n square block, with the lower right square missing, can be tiled with 1 by 3 rectangles". (The statement P(0) makes no sense and we will not worry about it.) We are interested in the set of positive naturals n for which P(n) is true. In particular we are interested in the values of P(300), P(301), and P(302).

  • (a,10) State and prove a theorem that tells, for any positive natural n, whether the number of squares in an n by n block, with one square missing, is divisible by three. (Note: This can be proved either with or without induction.) Does your theorem determine the values of any of P(300), P(301), or P(302)?

  • (b,10) Prove the statement "P(n) → P(n+3)" where n is an arbitrary natural.

  • (c,5) Prove that P(1) is true. What, if anything, do this fact and part (b) allow you to conclude about values of P(n), in particular P(300), P(301), or P(302)?

  • (d,5) Prove that P(2) is false. What, if anything, do this fact and part (b) allow you to conclude about values of P(n), in particular P(300), P(301), or P(302)?

  • (e, 10XC) Either prove that P(302) is true or prove that it is false. (Hint: We can color the squares of our figure with three colors -- we color every square (i,j) red if (i+j)%3 = 0, blue if (i+j)%3 = 1, and green if (i+j)%3 = 2. There is a useful fact about the coloring of the figures involved in the statements P(n) when n%3 = 2. If you can prove this fact by induction, you can use it to determine P(302).)

• The next question deals with the following real-Java class. An IntTree object is a tree where the leaves each store int values, so that the entire tree stores a set of int values, in no particular order.

  
        public class IntTree {
           boolean isLeaf;
           int leafValue; // only defined if isLeaf is true
           IntTree left;  // left subtree if isLeaf is false
           IntTree right; // right subtree if isLeaf is false
        }
   

  A valid IntTree object is either (1) a leaf, where isLeaf is true and leafValue is an int, or (2) a composite tree, where isLeaf is false and both left and right are valid IntTree objects. Nothing else is an IntTree.

• Question 3 (35): Answer the following questions:

  • (a,5) Prove, by induction on the definition of valid IntTree objects, that every IntTree has at least one leaf.

  • (b,10) Write a Java instance method for this class, such that if t is an IntTree, t.max() returns the largest leaf value in t.

  • (c,10) Prove, by induction on the definition of valid IntTree objects, that your max method terminates and returns the maximum leaf value in the tree, when called from any valid IntTree.

  • (d,10) Write a Java instance method maxUpTo for the IntTree class that takes an int argument cap and returns an int. If there is any leaf value in the tree t that is less than or equal to cap, t.maxUpTo(cap) should return the largest such value. If there is no such value, t.maxUpTo(cap) should return cap + 1.

• The last question deals with the following labeled (undirected) graph G. The node set of G is {a,b,c,d,e} and it has six labeled edges: (a,b) with label 6, (a,c) with label 1, (a,d) with label 2, (b.d) with label 3, (b,e) with label 3, and (c,d) with label 1.

• Question 4 (20): Answer the following questions:

  • (a,10) Carry out both a depth-first and a breadth-first search of G, with start node a and goal node e. When there are multiple edges out of a node, we explore the lowest-cost edges first. For part (a), assume that the two searches do mark nodes when they are explored, and do not explore them again.

  • (b,5) How do the same two searches behave if we do not mark explored nodes, and thus may visit them again? (Be sure to answer for both the BFS and the DFS.)

  • (c,5) Carry out a uniform-cost search (not an A^* search) of G, again starting from a with e as the goal node. (Note that a uniform-cost search always recognizes previously explored nodes.)

Last modified 21 December 2010