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Chapter 2: Q14E (page 85)
You are given an array of elements, and you notice that some of the elements are duplicates; that is, they appear more than once in the array. Show how to remove all duplicates from the array in time .
All duplicates are removed from the array in time:
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In this problem we will develop a divide-and-conquer algorithm for the following geometric task.
CLOSEST PAIRInput: A set of points in the plane,
Output: The closest pair of points: that is, the pair for which the distance between and , that is,
,
is minimized.
For simplicity, assume that n is a power of two, and that all the x-coordinates role="math" localid="1659237354869" are distinct, as are the y-coordinates.
Here鈥檚 a high-level overview of the algorithm:
.Find a value for which exactly half the points have , and half have . On this basis, split the points into two groups, L and R.
鈥� Recursively find the closest pair in L and in R. Say these pairs are and with distances and respectively. Let d be the smaller of these two distances.
鈥� It remains to be seen whether there is a point in Land a point in R that are less than distance dapart from each other. To this end, discard all points with or and sort the remaining points by y-coordinate.
鈥� Now, go through this sorted list, and for each point, compute its distance to the seven subsequent points in the list. Let be the closest pair found in this way.
鈥� The answer is one of the three pairs role="math" localid="1659237951608" , whichever is closest.
(a) In order to prove the correctness of this algorithm, start by showing the following property: any square of size in the plane contains at most four points of L.
(b) Now show that the algorithm is correct. The only case which needs careful consideration is when the closest pair is split between L and R.
(c) Write down the pseudocode for the algorithm, and show that its running time is given by the recurrence:
Show that the solution to this recurrence is .
(d) Can you bring the running time down to ?
Question: You are given an infinite array in which the first n cells contain integers in sorted order and the rest of the cells are filled with . You are not given the value of n. Describe an algorithm that takes an integer x as input and finds a position in the array containing x, if such a position exists, in O(log n) time. (If you are disturbed by the fact that the array A has infinite length, assume instead that it is of length n, but that you don鈥檛 know this length, and that the implementation of the array data type in your programming language returns the error message whenever elements are accessed.)
In justifying our matrix multiplication algorithm (Section 2.5), we claimed the following block wise property: if X and Y are n matrices, and
,
where A,B,C,D,E,F,G, and H are sub-matrices, then the product XY can be expressed in terms of these blocks:
Prove this property.
Suppose we want to evaluate the polynomial P(x) = a0 + a1x + a2x2 + ... + anxn at point x.
A way merge operation. Suppose you have sorted arrays, each with elements, and you want to combine them into a single sorted array of elements.
(a)Here鈥檚 one strategy: Using the merge procedure from Section 2.3, merge the first two arrays, then merge in the third, then merge in the fourth, and so on. What is the time complexity of this algorithm, in terms of and ?
(b) Give a more efficient solution to this problem, using divide-and-conquer.
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