373: CS Combinatorial Algorithms, Fall 2000
Homework 2 (due September 28, 2000 at midnight)
Name: Net ID: Name: Net ID: Name: Net ID:
Alias:
U 3/4 1
Alias:
U 3/4 1
Alias:
U 3/4 1
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team turns in just one solution, and every member of a team gets the same grade. Since 1-unit graduate students are required to solve problems that are worth extra credit for other students, 1-unit grad students may not be on the same team as 3/4-unit grad students or undergraduates. Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3/4-unit grad student, or 1-unit grad student by circling U, 3/4 , or 1, respectively. Staple this sheet to the top of your homework.
Required Problems
1. Faster Longest Increasing Subsequence (15 pts) Give an O(n log n) algorithm to nd the longest increasing subsequence of a sequence of numbers. [Hint: In the dynamic programming solution, you dont really have to look back at all previous items. There was a practice problem on HW 1 that asked for an O(n2 ) algorithm for this. If you are having difculty, look at the HW 1 solutions.] 2. S ELECT(A, k) (10 pts) Say that a binary search tree is augmented if every node v also stores |v |, the size of its subtree. (a) Show that a rotation in an augmented binary tree can be performed in constant time. (b) Describe an algorithm S CAPEGOAT S ELECT(k ) that selects the k th smallest item in an augmented scapegoat tree in O(log n) worst-case time. (c) Describe an algorithm S PLAY S ELECT(k ) that selects the k th smallest item in an augmented splay tree in O(log n) amortized time.
CS 373
Homework 2 (due 9/28/00)
Fall 2000
(d) Describe an algorithm T REAP S ELECT(k ) that selects the k th smallest item in an augmented treap in O(log n) expected time. [Hint: The answers for (b), (c), and (d) are almost exactly the same!]
3. Scapegoat trees (15 pts) (a) Prove that only one subtree gets rebalanced in a scapegoat tree insertion. (b) Prove that I (v ) = 0 in every node of a perfectly balanced tree. (Recall that I (v ) = max{0, |T | |s| 1}, where T is the child of greater height and s the child of lesser height, and |v | is the number of nodes in subtree v . A perfectly balanced tree has two perfectly balanced subtrees, each with as close to half the nodes as possible.) (c) Show that you can rebuild a fully balanced binary tree from an unbalanced tree in O(n) time using only O(log n) additional memory. For 5 extra credit points, use only O(1) additional memory.
4. Memory Management (10 pts) Suppose we can insert or delete an element into a hash table in constant time. In order to ensure that our hash table is always big enough, without wasting a lot of memory, we will use the following global rebuilding rules: After an insertion, if the table is more than 3/4 full, we allocate a new table twice as big as our current table, insert everything into the new table, and then free the old table. After a deletion, if the table is less than 1/4 full, we we allocate a new table half as big as our current table, insert everything into the new table, and then free the old table. Show that for any sequence of insertions and deletions, the amortized time per operation is still a constant. Do not use the potential methodit makes the problem much too hard!
5. Fibonacci Heaps: S ECOND M IN (10 pts) (a) Implement S ECOND M IN by using a Fibonacci heap as a black box. Remember to justify its correctness and running (b) time. Modify the Fibonacci Heap data structure to implement the S ECOND M IN operation in constant time, without degrading the performance of any other Fibonacci heap operation. 2
CS 373
Homework 2 (due 9/28/00)
Fall 2000
Practice Problems
1. Amortization (a) Modify the binary double-counter (see class notes Sept 12) to support a new operation S IGN, which determines whether the number being stored is positive, negative, or zero, in constant time. The amortized time to increment or decrement the counter should still be a constant. [Hint: Suppose p is the number of signicant bits in P , and n is the number of signicant bits in N . For example, if P = 17 = 100012 and N = 0, then p = 5 and n = 0. Then p n always has the same sign as P N . Assume you can update p and n in O(1) time.] (b) Do the same but now you cant assume that p and n can be updated in O(1) time.
2. Amortization Suppose instead of powers of two, we represent integers as the sum of Fibonacci numbers. In other words, instead of an array of bits, we keep an array of ts, where the ith least signicant t indicates whether the sum includes the ith Fibonacci number Fi . For example, the t string 101110 represents the number F6 + F4 + F3 + F2 = 8 + 3 + 2 + 1 = 14. Describe algorithms to increment and decrement a t string in constant amortized time. [Hint: Most numbers can be represented by more than one t string. This is not the same representation as on Homework 0!]
3. Rotations (a) Show that it is possible to transform any n-node binary search tree into any other n-node binary search tree using at most 2n 2 rotations. (b) Use fewer than 2n 2 rotations. Nobody knows how few rotations are required in the worst case. There is an algorithm that can transform any tree to any other in at most 2n 6 rotations, and there are pairs of trees that are 2n 10 rotations apart. These are the best bounds known.
4. Give an efcient implementation of the operation C HANGE K EY(x, k ), which changes the key of a node x in a Fibonacci heap to the value k . The changes you make to Fibonacci heap data structure to support your implementation should not affect the amortized running time of any other Fibonacci heap operations. Analyze the amortized running time of your implementation for cases in which k is greater than, less than, or equal to key [x].
5. Detecting overlap (a) You are given a list of ranges represented by min and max (e.g., [1,3], [4,5], [4,9], [6,8], [7,10]). Give an O(n log n)-time algorithm that decides whether or not a set of ranges contains a pair that overlaps. You need not report all intersections. If a range completely covers another, they are overlapping, even if the boundaries do not intersect. 3
CS 373
Homework 2 (due 9/28/00)
Fall 2000
(b) You are given a list of rectangles represented by min and max x- and y -coordinates. Give an O(n log n)-time algorithm that decides whet her or not a set of rectangles contains a pair that overlaps (with the same qualications as above). [Hint: sweep a vertical line from left to right, performing some processing whenever an end-point is encountered. Use a balanced search tree to maintain any extra info you might need.] 6. Comparison of Amortized Analysis Methods A sequence of n operations is performed on a data structure. The ith operation costs i if i is an exact power of 2, and 1 otherwise. That is operation i costs f (i), where: f (i) = i, i = 2k , 1, otherwise
Determine the amortized cost per operation using the following methods of analysis: (a) Aggregate method (b) Accounting method (c) Potential method
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