mid2008 - CS300 Algorithms 2008 Fall semester M id-term...

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CS300 Algorithms 2008 Fall semester Mid-term Exam. Name : ID No. : Dept. : question score 1 2 3 4 5 6 7 8 Total **Caution: Do not use Pseudo Code while describing algorithms.
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(20pts) 1. Let T be a linear algebraic decision tree (LADT) for solving a decision problem π . Let W and D be defined as follows: W = { ( x 1 , x 2 , …, x n ) | ( x 1 , x 2 , …, x n ) eventually leads to a “halt and yes” node}, D = { i D | i D is the domain of leaf node i l for all d i 1 }, where ( x 1 , x 2 , …, x n ) represents an instance of input data to and are the leaf nodes of T . Let be the disjoint connected components of W . (5pts) 1.1 Show that i D , d i 1 are convex. (5pts) 1.2 Suppose that a mapping Y is defined as follows: Y : {w 1 , w 2 , …, w p ) {1, 2, …, d } such that Show that if and only if ) ( i w Y , p i 1 are distinct, where and are the number of leaf nodes of and that of disjoint connected components of , respectively, i.e., . (5pts) 1.3 Show that ) ( i w Y , p i 1 are distinct. (5pts) 1.4 A lower bound in time complexity for solving is p i w i 1 , }. | { min ) ( φ = j i j i D w j w Y L W ). (log 2 W L d j l j 1 , p W d L = = and W W T
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(20pts) 2. An element uniqueness problem (EUP hereafter) is stated as follows: EUP : Given N real numbers i x , N i 1 , decide that any two are equal. Answer the following questions: (5pts) 2.1 Let W be defined as follows: W = } 1 | ) ,..., , {( 2 1 N j i all for x x x x x j i N < Show that W is an -dimensional subset representing the set of all inputs that eventually leads to “halt and yes,” and then count the number of disjoint connected components of W . (5pts) 2.2 Based on the answer of problem 2.1, derive a lower bound in time complexity for solving EUP in the worst case. Is this bound tight? Why or why not? (10pts)
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mid2008 - CS300 Algorithms 2008 Fall semester M id-term...

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