07Ma6aSol6 - Math 6a Solution Set 6 Problem 1 Write a...

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Math 6a - Solution Set 6 Problem # 1 Write a paragraph on dominance in graphs, fire fighting, and the spread of computer viruses, with reference to the work of Hartnell. Write a paragraph on what binary search trees have to do with port security, with reference to the work of anyone at Rutgers University. Solution. Note that this is not a ”question,” but rather, a writing assignment, and thus it does not have a ”solution.” Problem # 2 If you have 25 droids arranged on a 5 × 5 grid, show how each droid can move exactly one space forward, back, left or right (no diagonal moves allowed and no droid can remain stationary), or show it cannot be done. Solution. Label the rows and columns of the grid with elements from the set { 1 , 2 , 3 , 4 , 5 } and then label the square in row i and column j with ( i, j ) and color the square ( i, j ) black if i + j is even and color it white if i + j is odd. Then, note that a droid on square ( i, j ) can move to square ( i 0 , j 0 ) if and only if either the move is horizontal, so i = i 0 and | j - j 0 | = 1 , in which case | i - i 0 + j - j 0 | = 1 , or the move is vertical, j = j 0 and | i - i 0 | = 1 in which case we also have | i - i 0 + j - j 0 | = 1 . Thus, if a given droid moves from ( i, j ) to ( j, j 0 ) then 1 = | i - i 0 + j - j 0 | = | ( i + j ) - ( i 0 + j 0 ) | and it follows that the sums i + j and i 0 + j 0 differ by 1 which means that i + j is even and ( i, j ) is black if and only if i 0 + j 0 is odd and ( i, j ) is white. Thus, droids on black squares can only move to white squares and droids from white squares can only move to black squares. But note that in rows 1 , 3 , and 5 there are 3 black squares and in rows 2 and 4 there are 2 black squares, and thus there are a total 9 + 4 = 13 black squares, and thus 25 - 13 = 12 white squares. So, if each of the droids moves, that means the 13 droids on black squares much each move to one of 12 whites squares, which implies, by the pigeonhole principle that two such droids must move to the same square, an impossibility, since each droid must move to a unique square. Thus, this cannot be done. Problem # 3 1
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2 Fix m 1 According to Liu, page 138, every regular m -ary tree of length h has a number t of leaves, where m + ( m - 1)( h - 1) t m h .
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