MIT6_851S10_assn01_sol

MIT6_851S10_assn01_sol - 6.851 Advanced Data Structures...

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Unformatted text preview: 6.851 Advanced Data Structures (Spring’10) Prof. Erik Demaine Dr. Andr´ Schulz e Problem 1 TA: Aleksandar Zlateski Sample Solutions Transposing a matrix. Consider a point set {(xi , i)} of k 2 points on a k 2 × k 2 lattice representing �� � � the access � equence. For each point (xi , i) we introduce three new points at (xi − 1, i), k xi , i s� k �� and k xi , i . k The newly formed set is Aborally Satisfied, hence it represents a valid BST execution. The set contains O(k 2 ), giving amortized cost of O(1) per access. Logarithmic redux. Consider the access sequence, the point set X = {(xi , i)} of m points on a n × m lattice. Let x be the median of all x ∈ X . Inserting m points (x, i) will ensure that each ˆ ˆ rectangle connecting a point left of or at x and a point right of or at x contains a point. ˆ ˆ Now consider the two subsets of X , Xx≤x and Xx≥x , each with at most m points, and at most ˆ ˆ n 2 distinct x values. We recursively apply the same technique, to obtained point set that is Aborally Satisfied. We get the number of newly inserted points by solving the recursion N (m, n ) = 2N (m, n , n) + m. 2 2 The total number of accesses is then O(N (m, n) + m) = O(m log n + m) = O(m log n). 1 MIT OpenCourseWare http://ocw.mit.edu 6.851 Advanced Data Structures Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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This note was uploaded on 03/31/2011 for the course EECS 6.851 taught by Professor Erikdemaine during the Spring '10 term at MIT.

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MIT6_851S10_assn01_sol - 6.851 Advanced Data Structures...

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