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Apparently one of the questions was: Q. Given X-Y coordinates of two rectangles, determine if they intersect or not. I'll try for an answer -- I looked up not

Apparently one of the questions was: Q. Given X-Y coordinates of two rectangles, determine if they intersect or not. I'll try for an answer -- I looked up nothing, and it took longer to type in the answer than it took to think of it! A. Given positive integers m and n and m points A(i) = (a_i1, a_i2), i = 1, 2, ..., m and B(j) = (b_j1, b_j2), j = 1, 2, ..., n, determine if the convex hull of the points A(i) intersects the convex hull of the points B(j). So, look for a line that has all the points A(i) on one side and all the points B(j) on the other side. Suppose we have numbers u, v, w, and consider the set of all (x,y) such that ux + vy = w Then those points (x,y) form a line, and every line can be written in this form. Then we seek u, v, w so that all the A(i) are on one side of the line and all the B(j) are on the other side. So, without loss of generality, we seek u, v, w so that for all i ua_i1 + va_i2 >= w and for all j ub_i1 + vb_i2 <= w Or, we seek u, v, w to solve the linear program maximize z = u + v + w subject to ua_i1 + va_i2 >= w ub_i1 + vb_i2 <= w for all i, j. The simplex algorithm will determine if this linear program is feasible, that is, if u, v, w exist to satisfy the m + n linear constraints, or not feasible. If the program is feasible, then the two convex hulls are separated by the line the set of all (x,y) such that ux + vy = w Else the linear program is not feasible, no line exists separating the convex hulls, and the two convex hulls overlap. Yes, we could tweak this simple formulation to something a little more involved and, then, determine if the two convex hulls just touch but otherwise don't overlap. This solution solves the problem about rectangles in the OP as a special case. Let's see: Given a closed convex set C with points A(i), is the convex hull of the points A(i) also in C? Well, the intersection of any collection of closed sets is closed (the union of any collection of open sets are open). The intersection of any collection convex sets is convex. Then, the intersection of any collection of closed, convex sets is closed and convex. The convex hull of a set of points is the smallest closed convex set that contains all the points, that is, the intersection of all closed convex sets that contain all the points and, thus, is closed and convex. Here convex hull is well defined since the intersection is unique. A line in X-Y and one side of that line is closed and convex. So the convex hull of a set of points on the line and some one side of it is closed, convex, and a subset of the line and that side of it. Our algorithm needs this result. Is that a sufficiently good answer?

MRD85 8y ago
If I was an interviewer I wouldn't like this response for most jobs, although it would be fine for a small subset of jobs. The overlap of two rectangles is an easy question with an easy answer and a long, more complicated answer would indicate that the interviewee missed social queues about what is expected in the answer and may be a difficult person to work with.
- graycat 8y ago
  You have an easier answer that is correct? Looking at the "solution" in the OP, there was a HUGE assumption NOT stated, implied, etc. in the statement of the question: The assumption was that the sides of the rectangles were parallel to the orthogonal X-Y axes. So, the question was badly stated. For how the question was STATED, my solution may be about the easiest. > The overlap of two rectangles is an easy question with an easy answer You have an "easy" answer to how the question was stated? Until I hear or think of a much easier answer, I have to conclude that your statement of an "easy question" is false. The OP certainly did NOT have a solution to the question as stated. > If I was an interviewer I wouldn't like this response for most jobs, although it would be fine for a small subset of jobs. The interview questions were to be for "most jobs". Then "most jobs" should prefer a correct answer, that is, correct for the question as stated. Or, the question could be important for many jobs working with graphics, rectangles, triangles, etc., and there commonly we won't have specific orientations, say, rectangle sides parallel to the X-Y axes. > social queues about what is expected The interview questions are supposed to be challenging and to let the candidate show their capabilities. In that context, "what is expected" should not limit the power of an answer. If the company wants only people who know not much more and not much less than the people already there, then the company, should not be looking for new people, is on the way downhill, also because of that should not be hiring, and no good candidate should want to work there. The OP questions were to be about "data structures and algorithms". Well, I used the simplex algorithm, seriously ranked as one of the best pieces of engineering in the 20th century. A special case of that algorithm is min cost capacitated flows on networks, and there is a really cute algorithm there and a cute use of a spanning tree on a network. So, my answer was good for "algorithms and data structures".

MRD85 8y ago

If I was an interviewer I wouldn't like this response for most jobs, although it would be fine for a small subset of jobs. The overlap of two rectangles is an easy question with an easy answer and a long, more complicated answer would indicate that the interviewee missed social queues about what is expected in the answer and may be a difficult person to work with.

graycat 8y ago
You have an easier answer that is correct? Looking at the "solution" in the OP, there was a HUGE assumption NOT stated, implied, etc. in the statement of the question: The assumption was that the sides of the rectangles were parallel to the orthogonal X-Y axes. So, the question was badly stated. For how the question was STATED, my solution may be about the easiest. > The overlap of two rectangles is an easy question with an easy answer You have an "easy" answer to how the question was stated? Until I hear or think of a much easier answer, I have to conclude that your statement of an "easy question" is false. The OP certainly did NOT have a solution to the question as stated. > If I was an interviewer I wouldn't like this response for most jobs, although it would be fine for a small subset of jobs. The interview questions were to be for "most jobs". Then "most jobs" should prefer a correct answer, that is, correct for the question as stated. Or, the question could be important for many jobs working with graphics, rectangles, triangles, etc., and there commonly we won't have specific orientations, say, rectangle sides parallel to the X-Y axes. > social queues about what is expected The interview questions are supposed to be challenging and to let the candidate show their capabilities. In that context, "what is expected" should not limit the power of an answer. If the company wants only people who know not much more and not much less than the people already there, then the company, should not be looking for new people, is on the way downhill, also because of that should not be hiring, and no good candidate should want to work there. The OP questions were to be about "data structures and algorithms". Well, I used the simplex algorithm, seriously ranked as one of the best pieces of engineering in the 20th century. A special case of that algorithm is min cost capacitated flows on networks, and there is a really cute algorithm there and a cute use of a spanning tree on a network. So, my answer was good for "algorithms and data structures".