Homework #6 -- due Monday Wk 7

#	suggested problems	topic
4	CLR Exercise 15.4-6 on page 356	longest increasing subsequence
5	CLR Problem 15-1 on page 364	bitonic euclidean tsp
6	Baase Exercise 10.9 on page 477	optimal bst
7	Find an optimal binary search tree for the probabilities p1 = p2 = p3 = p4 = p5 = 0.05 q0=0.0, q1=0.05, q2=0.10, q3=0.15, q4=0.20, q5=0.25	optimal bst
8	Find the optimal binary search tree consisting of n elements having probabilities p1 = p2 = ... = pn = 0.0 qi < 0.5 qi +1, for 0 < i < n Give an explanation of why it is optimal.	optimal bst
9	Baase Exercise 10.16 on page 478	dynamic pgm - binomial coefficients
10	Baase Exercise 10.21 on page 479	dynamic pgm - partition problem
11	The Strahler number of a binary tree is defined as follows: An empty tree has Strahler number 0.  If the binary tree T has subtrees TL and TR, the Strahler number S(T) of T is defined by max{ S(TL),S(TR) },         if S(TL) ≠ S(TR) S(TL) + 1,                 otherwise Give an algorithm (possibly recursive) to find the Strahler number of a binary tree.	dynamic pgm
12	One version of the game of Nim is played with counters that are arranged in three rows.  A position (the pattern of counters) can be conveniently represented by an ordered triple of integers describing the number of counters in each row. The game is played by two persons who move in turns.  Each move consists of removing one or more counters from one of the rows.  A player wins if he/she removes all remaining counters, i.e., the position <0,0,0> results from his/her move. Any position in which the next player to move, if he/she moves correctly, is guaranteed to eventually win (regardless of which moves his/her opponent may make) is called a winning position.  Let WIN(i, j, k) be true iff <i, j, k> is a winning position. Give a recurrence for the values of WIN, including boundary conditions. Using your recurrence from part a, produce a polynomial-time algorithm that determines whether any given position <a, b, c> is a winning position. Analyze the time and space complexity of your algorithm as a function of a, b, c. What is the complexity as a function of n when a=b=c=n? Hint: A good solution would be θ(n4). A better solution would be θ(n3).	Nim
13	Let T be a complete binary tree with n = 2k-1 nodes. The nodes are numbered in level order with the root labelled 1, its two children labelled 2 and 3, etc. Let Ti be the subtree whose root is node i, and so T1 = T. Every node i has a value VAL[i] which can be positive or negative or zero. We define WT(i) to be the weight of subtree i, given by the sum of all values of the nodes in Ti. The problem is to find the maximum weight of all subtrees of T. Give a recurrence for WT(i). Give a divide-and-conquer algorithm that finds the maximum WT(i) given the array VAL. What are the space and time complexities of your D&C algorithm? Give a dynamic programming algorithm that finds the maximum WT(i) given the array VAL. What are the space and time complexities of your dynamic programming algorithm?	DC+dyn pgm
14	DPV exercise 6.4 on p.178 You are given a string of n characters s[1...n], which you believe to be a corrupted text document in which all punctuation has vanished (so that it looks something like "itwasthebestoftimes...").  You wish to reconstruct the document using a dictionary, which is available in the form of a Boolean function dict(*) such that, for any string w, dict(w) has value 1 if w is a valid word, and has value 0 otherwise. Give a dynamic programming algorithm that determines whether the string s[*] can be reconstituted as a sequence of valid words.  The running time should be at most O(n2), assuming that each call to dict takes unit time. In the event that the string is valid, make your algorithm output the corresponding sequence of words.	dynamic pgm
15	DPV exercise 6.5 on p.178 We are given a checkerboard which has 4 rows and n columns, and has an integer written in each square.  We are also given a set of 2n pebbles, and we wish to place some or all of these on the checkerboard (each pebble can be placed on exacyly one square) so as to maximize the sum of integers in the squares that are covered by pebbles.  There is one constraint: for a placement of pebbles to be legal, no two of them can be on horizontally or vertically adjacent squares (diagonal adjacency is fine). Determine the number of legal patterns that can occur in any column (in isolation, ignoring the pebbles in adjacent columns) and describe these patterns. Call two patterns compatible if they can be placed on adjacent columns to form a legal placement.  Let us consider subproblems consisting of the first k columns 1≤k≤n.  Each subproblem can be assigned a type, which is the pattern occurring in the last column. Using the notions of compatibility and type, give an O(n)-time dynamic programming algorithm for computing an optimal placement.	dynamic pgm