[Solved] CS721: Advanced Algorithms and Analysis Programming Assignment: Maxima

Let x, y, and z denote the three coordinate axes in the three dimensional space R³. A point p R³ is specified by its coordinates along the three axes: p = (x(p),y(p),z(p)).

For p,q R³, we say that p is dominated by q, if x(p) x(q), y(p) y(q) and z(p) z(q).

Let S = {p₀,p₁,,p_n1} be a set of n points in R³. p_i S is called a maximal element of S, if p_i is not dominated by any other element of S. The set of all maximal elements of S is denoted by maxima(S). The maxima problem is: Given S, find maxima(S).

You will write an efficient program to solve this problem, using the C++ or Java Standard Library functions.

One brute-force algorithm to solve this problem is as follows: Compare each point p_i S against all the other points in S to determine if p_i is dominated by any of those points; if p_i is not dominated by any of them, add it to the output set maxima(S). This algorithm takes (n) time for each point p_i, for a total of (n²) time.

You will implement an efficient algorithm that is expected to run in (n log n) time. The program must be in C++ or Java only. This handout discusses the implementation in C++, using the Standard Template Library (STL). The algorithm consists of three steps.

1. Input: The point set S is in an input file. The first line contains the value of n (the number of points). Following that, there will be n lines, each line containing the x, y and z coordinates of one point.

The points must be read and stored in an array POINTS[0..(n 1)] of POINT objects. The POINTS[i] object corresponds to point p_i, and has four fields: the x, y and z-coordinates (all double); the boolean field maximal indicating whether or not the point is maximal.

1. Sorting: Sort the points (i.e., the array POINTS) according to their z-coordinates, and reindex them such that z(p₀) z(p₁) z(p_n1).

The sorting must be done using the sort library function: sort(POINTS, POINTS+n); this requires that you implement the operator<.

p < q if: z(p) < z(q), or z(p) = z(q) and y(p) < y(q), or z(p) = z(q) and y(p) = y(q) and x(p) < x(q).

III. Finding the Maxima: Process the points, one-by-one, in decreasing order of z-coordinates.

Suppose that, at some instant, you have processed the points p_n1,p_n2,,p_i+1. You must maintain the 2-dimensional maxima of these points in the (x,y)-plane; this will be called maxima₂[(i + 1)..(n 1)]. It forms a staircase in the (x,y)-plane.

CONSIDER maintaining maxima₂[(i+1)..(n1)] in a Binary Search Tree (BST) keyed on the y-coordinate; each node of BST corresponds to a point in maxima₂[i+1..n]. As we traverse these nodes in decreasing order of y-coordinate, their x-coordinates increase (thus forming a staircase).

Now we want to process p_i. p_i has a smaller z-coordinate than the points processed so far (namely, p_n1,p_n2,,p_i+1). So, p_i maxima(S) iff it is not dominated in the (x,y)-plane by any of the points in maxima₂[i + 1..n]; i.e., p_i is not under the staircase.

The test of whether p_i is dominated by any of these points (in the (x,y)-plane) is performed as follows: Find the point q in BST such that y(q) is just above (or equal to) y(p_i); then p_i maxima(S) iff x(p_i) > x(q).

If x(p_i) > x(q), add p_i to maxima(S); also, we need to update the BST so that it represents maxima₂[i..n]. The update of the BST is done as follows: First, consider the nodes in the BST whose y-coordinates are less than y(p_i), in decreasing order of their y-coordinates; delete them one-by-one, until you come across a point p with x(p) > x(p_i). Finally, insert p_i into the BST.

Each insertion or deletion in a BST takes time (height of the tree). To achieve the (nlogn) runtime, the above BST must be a height-balanced binary search tree: At any istant, the height of the tree is (logof number nodes in the tree). One example of such a tree is Red-Black Tree (see Chapter 13 in the text book). You will use the STL data structure map. It is implemented as a Red-Black Tree. In the class, I will explain how to use map.

Variable MaxNum has the number of elements in Maxima(S). Print out MaxNum and Maxima(S). For each point in Maxima(S), also print out its array index (i.e., the index of the point in POINTS array).

Your program should be modular, and contain appropriate procedures/functions. No comments or other documentation is needed. Use meaningful names for all variables.

You will run your program on 10 different sets of points; your program should have a loop for this. All the point sets are in the input file infile.txt.

The name of your program file must be maxima.[cpp or java] (corresponding to C++ or Java programs, respectively); the name of your output file must be outfile.txt.