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Selection and Adversary arguments Sung Yong Shin CS Dept., KAIST

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Avg. 66.1 Std. 30.1 100 ↑ A+ 91~99 A0 88~90 A- 86~87 B+ 80~85 B0 78~79 B- 76~77 C+ 70~75 C0 60~69 C- 50~59 D0 40~49 D- 0~39 F Midterm Result
Contents 1. Introduction 2. Finding max. and min. 3. Finding the 2th largest key 4. The Selection Problem 5. A lower bound for finding the median

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1. Introduction SP : (Selection Problem) Given a set of n real numbers, find the k th smallest one, 1 k n . How can you solve it? well, … (1) Sort the numbers. (2) Pick the k th smallest one. O( n log n ) Any better way?
What is a trivial lower bound in time complexity for solving SP ? T L ( n ) = ( n ) Why? What if only considering comparisons? well, ...

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P : Given a set S of n real numbers, find the largest one. | W | | L | W = {(?, ?, x 1 ), (?, ?, x 2 ), ···, (?, ?, x n )} | W | = n T L ( n ) = log 2 | W | = log 2 n However, this is not tight !!! Why? n = 3 S = { x 1 , x 2 , x 3 } 1 : 2 L = {1 1 , 1 2 , 1 3 , 1 4 } < > 2 : 3 1 : 3 < > < > 1 1 1 2 1 3 1 4 x 3 x 2 x 3 x 1 ( x 1 , x 2 , x 3 ) (?, ?, x 2 ) ( x 2 , x 1 , x 3 ) (?, ?, x 1 )
Adversary Arguments Z 1000 = {0, 1, …… , 999} Guess the number in Z 1000 that I have in mind? A Guessing Game !!! You can change your mind as long as your answers(responses) are consistent !!! Maximize the number of leaves in a decision tree.

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2. Finding Max. and Min. MM : Given a set of n real numbers, find max and min. max = the largest number min = the smallest number How can you solve MM ? x 1 x 2 x 3 x 4 …… x 2 n -1 x 2 m n = 2 m W { x 1 1 , x 2 1 , x 3 1 ,……, x m 1 } max L { x 1 2 , x 2 2 , x 3 2 ,……, x m 2 } min How many comparisons? m …… dividing m -1 …… finding max m -1 …… finding min Any better way? 2 2 3 2 3 - = - n m
What information is needed for finding max and min? Finding max : All numbers except max itself must lose at least once in some comparisons. (n-1 losses) Finding min : All numbers except min itself must win at least once in some comparisons. (n-1 wins) 1 unit of information (1 win) or (1 loss) (2n - 2) units of information are needed !!! max min x 1 x 2 x 3 x 4 x 5 x 6 L L L L L W W W W W

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Status of a number ( x i , s i ) Status W at least one win, no lost L at least one lost, no win WL wins and losts N no comparisons Status of keys x and y Units of new compared by an algorithm Adversary response New Status information N,N
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