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Chapter 5: Q32E (page 167)

Question:Show how to implement the stingy algorithm for Horn formula satisfiability (Section 5.3) 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.)

Short Answer

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Answer:This challenge of determining whether such a combination of conceptual Horn sentences seems to be satisfiable or not is known as Horn-satisfiability. The indicated literals are the simplest representation of something like the Horn formula.

Step by step solution

01

Build direct bipartiate(graph):

The entire process runs in linear time since each literal is only marked true once (with respect to the number of literals, because of the graph building).

Build a directed(bipartiate) graph, connecting the Horn clauses to their respective positive literal(if they have one), and connecting the literals to all chauses where they appear negated.

02

 Calculation of Propagate for programme functionality:

[UPDATE] Do the same for each clause c that does not contain a negating literal. {

assuming c doesn't have a significant literal STOPunsatisfactory,

disseminate the otherwise (c)

}

eventually, this unit programme functionality

propagate(c) :-

suppose c(positive )'s literal hasn't been certified true yet {

mark p true

considering all(p,c') in the graph's Edge set {

remove p from c'

Disseminate if c' contains nothing more negated literals (c')
}

}

Percussion seems to be the task of deciding whether a given mixture of theoretical Horn sentences appears to really be satisfiable or not. The literals shown are the most basic representations of things like with the Horn formula.

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Most popular questions from this chapter

The basic intuition behind Huffman鈥檚 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 file composed of m different words, with frequencies f1,...,fm. Suppose also that for the ithword, the cost per bit of encoding is ci. Thus, if we find a prefix-free code where the ithword has a codeword of length Ii, then the total cost of the encoding will be localid="1659078764835" 鈭�fi路ci路li.

Show how to modify Huffman鈥檚 algorithm to find the prefix-free encoding of minimum total cost.

We use Huffman's algorithm to obtain an encoding of alphabet {a,b,c}with frequencies fa,fb,fc. In each of the following cases, either give an example of frequencies (fa,fb,fc)that would yield the specified code, or explain why the code cannot possibly be obtained (no matter what the frequencies are).

(a) Code:{0,10,11}

(b) Code:{0,1,00}

(c) Code:{10,01,00}

Ternary Huffman. Trimedia Disks Inc. has developed 鈥渢ernary鈥� 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 modified Huffman algorithm for compressing sequences of characters from an alphabet of size n, where the characters occur with known frequencies f1, 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 prefix of another codeword and so as to obtain the maximum possible compression. Prove that your algorithm is correct

Show that for any integer n that is a power of 2 , there is an instance of the set cover problem (Section 5.4) with the following properties:

  1. There are n elements in the base set.
  2. The optimal cover uses just two sets.
  3. The greedy algorithm picks at least log n sets.

Thus the approximation ratio we derived in the chapter is tight.

The following statements may or may not be correct, In each case, either prove it (if it is correct) or give a counter-example (if it isn鈥檛 correct). Always assume that the graph G=(V,E)is undirected. Do not assume that edge weights are distinct unless this is specifically stated.

  1. If a graph G has more than |V|-1edges, and there is a unique heaviest edge, then this edge cannot be part of a minimum spanning tree.
  2. If G has a cycle with a unique heaviest edge e, then e cannot be part of any MST.
  3. Let e be any edge of minimum weight in G. Then e must be part of some MST.
  4. If the lightest edge in a graph is unique, then it must be part of every MST.
  5. If e is part of some MST of G, then it must be a lightest edge across some cut of .
  6. If G has a cycle with a unique lightest edge e must be part of every MST.
  7. The shortest-path tree computed by Dijkstra鈥檚 algorithm is necessarily an MST.
  8. The shortest path between two nodes is necessarily part of some MST.
  9. Prim鈥檚 algorithm works correctly when there are negative edges.
  10. (For any r>0, define an r-path to be a path whose edges all have weight <r). If G contains an r-path from node s to t , then every MST of G must also contain an r-path from node s to node t.
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