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(see the box in page 145) Would you expect it to be larger or smaller than your answer above Explain (d) Do you think that this is the limit of how much English text can be compressed What features of the English language, besides letters and their frequencies, should a better compression scheme take into account 519 Entropy Consider a distribution over n possible outcomes, with probabilities p 1 , p2 , , pn (a) Just for this part of the problem, assume that each pi is a power of 2 (that is, of the form 1/2k ) Suppose a long sequence of m samples is drawn from the distribution and that for all 1 i n, the ith outcome occurs exactly mpi times in the sequence Show that if Huffman encoding is applied to this sequence, the resulting encoding will have length
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(b) Now consider arbitrary distributions that is, the probabilities pi are not restricted to powers of 2 The most commonly used measure of the amount of randomness in the distribution is the entropy n 1 pi log pi i=1 For what distribution (over n outcomes) is the entropy the largest possible The smallest possible 520 Give a linear-time algorithm that takes as input a tree and determines whether it has a perfect matching: a set of edges that touches each node exactly once A feedback edge set of an undirected graph G = (V, E) is a subset of edges E E that intersects every cycle of the graph Thus, removing the edges E will render the graph acyclic Give an ef cient algorithm for the following problem: Input: Undirected graph G = (V, E) with positive edge weights we Output: A feedback edge set E E of minimum total weight e E we 521 In this problem, we will develop a new algorithm for nding minimum spanning trees It is based upon the following property: Pick any cycle in the graph, and let e be the heaviest edge in that cycle Then there is a minimum spanning tree that does not contain e (a) Prove this property carefully (b) Here is the new MST algorithm The input is some undirected graph G = (V, E) (in adjacency list format) with edge weights {we } sort the edges according to their weights for each edge e E, in decreasing order of we : if e is part of a cycle of G: G = G e (that is, remove e from G) return G Prove that this algorithm is correct (c) On each iteration, the algorithm must check whether there is a cycle containing a speci c edge e Give a linear-time algorithm for this task, and justify its correctness (d) What is the overall time taken by this algorithm, in terms of |E| Explain your answer 522 You are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E ) with respect to these weights; you may assume G and T are given as adjacency lists Now suppose the weight of a particular edge e E is modi ed from w(e) to a new value w(e) You wish to quickly update the minimum spanning tree T to re ect this change, without recomputing the entire tree from scratch There are four cases In each case give a linear-time algorithm for updating the tree (b) e E and w(e) < w(e) (d) e E and w(e) > w(e) (c) e E and w(e) < w(e) (a) e E and w(e) > w(e)
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523 Sometimes we want light spanning trees with certain special properties Here s an example
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Input: Undirected graph G = (V, E); edge weights we ; subset of vertices U V Output: The lightest spanning tree in which the nodes of U are leaves (there might be other leaves in this tree as well) (The answer isn t necessarily a minimum spanning tree) Give an algorithm for this problem which runs in O(|E| log |V |) time (Hint: When you remove nodes U from the optimal solution, what is left ) 524 A binary counter of unspeci ed length supports two operations: increment (which increases its value by one) and reset (which sets its value back to zero) Show that, starting from an initially zero counter, any sequence of n increment and reset operations takes time O(n); that is, the amortized time per operation is O(1) 525 Here s a problem that occurs in automatic program analysis For a set of variables x1 , , xn , you are given some equality constraints, of the form xi = xj and some disequality constraints, of the form xi = xj Is it possible to satisfy all of them For instance, the constraints x1 = x2 , x2 = x3 , x3 = x4 , x1 = x4 cannot be satis ed Give an ef cient algorithm that takes as input m constraints over n variables and decides whether the constraints can be satis ed 526 Graphs with prescribed degree sequences Given a list of n positive integers d 1 , d2 , , dn , we want to ef ciently determine whether there exists an undirected graph G = (V, E) whose nodes have degrees precisely d1 , d2 , , dn That is, if V = {v1 , , vn }, then the degree of vi should be exactly di We call (d1 , , dn ) the degree sequence of G This graph G should not contain self-loops (edges with both endpoints equal to the same node) or multiple edges between the same pair of nodes (a) Give an example of d1 , d2 , d3 , d4 where all the di 3 and d1 + d2 + d3 + d4 is even, but for which no graph with degree sequence (d1 , d2 , d3 , d4 ) exists (b) Suppose that d1 d2 dn and that there exists a graph G = (V, E) with degree sequence (d1 , , dn ) We want to show that there must exist a graph that has this degree sequence and where in addition the neighbors of v1 are v2 , v3 , , vd1 +1 The idea is to gradually transform G into a graph with the desired additional property i Suppose the neighbors of v1 in G are not v2 , v3 , , vd1 +1 Show that there exists i < j n and u V such that {v1 , vi }, {u, vj } E and {v1 , vj }, {u, vi } E / ii Specify the changes you would make to G to obtain a new graph G = (V, E ) with the same degree sequence as G and where (v1 , vi ) E iii Now show that there must be a graph with the given degree sequence but in which v 1 has neighbors v2 , v3 , , vd1 +1 (c) Using the result from part (b), describe an algorithm that on input d1 , , dn (not necessarily sorted) decides whether there exists a graph with this degree sequence Your algorithm n should run in time polynomial in n and in m = i=1 di 527 Alice wants to throw a party and is deciding whom to call She has n people to choose from, and she has made up a list of which pairs of these people know each other She wants to pick as many people as possible, subject to two constraints: at the party, each person should have at least ve other people whom they know and ve other people whom they don t know Give an ef cient algorithm that takes as input the list of n people and the list of pairs who know each other and outputs the best choice of party invitees Give the running time in terms of n
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528 A pre x-free encoding of a nite alphabet assigns each symbol in a binary codeword, such that no codeword is a pre x of another codeword Show that such an encoding can be represented by a full binary tree in which each leaf corresponds to a unique element of , whose codeword is generated by the path from the root to that leaf (interpreting a left branch as 0 and a right branch as 1) 529 Ternary Huffman Trimedia Disks Inc has developed ternary hard disks Each cell on a disk can now store values 0, 1, or 2 (instead of just 0 or 1) To take advantage of this new technology, provide a modi ed Huffman algorithm for compressing sequences of characters from an alphabet of size n, where the characters occur with known frequencies f 1 , f2 , , fn Your algorithm should encode each character with a variable-length codeword over the values 0, 1, 2 such that no codeword is a pre x of another codeword and so as to obtain the maximum possible compression Prove that your algorithm is correct 530 The basic intuition behind Huffman s algorithm, that frequent blocks should have short encodings and infrequent blocks should have long encodings, is also at work in English, where typical words like I, you, is, and, to, from, and so on are short, and rarely used words like velociraptor are longer However, words like fire!, help!, and run! are short not because they are frequent, but perhaps because time is precious in situations where they are used To make things theoretical, suppose we have a le composed of m different words, with frequencies f1 , , fm Suppose also that for the ith word, the cost per bit of encoding is ci Thus, if we nd a pre x-free code where the ith word has a codeword of length li , then the total cost of the encoding will be i fi ci li Show how to modify Huffman s algorithm to nd the pre x-free encoding of minimum total cost 531 A server has n customers waiting to be served The service time required by each customer is known in advance: it is ti minutes for customer i So if, for example, the customers are served in i order of increasing i, then the ith customer has to wait j=1 tj minutes We wish to minimize the total waiting time
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Give an ef cient algorithm for computing the optimal order in which to process the customers 532 Show how to implement the stingy algorithm for Horn formula satis ability (Section 53) in time that is linear in the length of the formula (the number of occurrences of literals in it) (Hint: Use a directed graph, with one node per variable, to represent the implications) 533 Show that for any integer n that is a power of 2, there is an instance of the set cover problem (Section 54) with the following properties: i There are n elements in the base set ii The optimal cover uses just two sets iii The greedy algorithm picks at least log n sets Thus the approximation ratio we derived in the chapter is tight