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Maximum number of rectangles which includes a certain point

By Usu, history, 7 years ago, In English

Hey! Does anybody have a solution in n log n for this? Given n rectangles, determinate a point which is included by a maximum number of rectangles (including the boundaries).

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7 years ago, hide # |
 
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It can be solved with sweep line algo and segment tree in $$$O(N \lg N)$$$.

  1. For each rectangle $$$[x_1, x_2] \times [y_1, y_2]$$$, produce two events $$$(x_1, (y_1, y_2), +)$$$ and $$$(x_2, (y_1, y_2), -)$$$.
  2. Sort all the events (there are $$$2N$$$ such many) by $$$x$$$.
  3. From minimum $$$x$$$ to maximum $$$x$$$, if it's from left side of some rectangle ($$$(x_1, (y_1, y_2), +)$$$), add 1 to all the points in the closed interval $$$[y_1, y_2]$$$, otherwise, subtract 1 to those in the closed interval $$$[y_1, y_2]$$$.
  4. After each addition or subtraction, find the maximum value among all possible $$$y$$$ which is a candidate to the answer(update the answer).
  5. Addition/Subtraction on an interval and querying maximum value can be done with segment tree in $$$O(\lg C)$$$ per operation, where $$$C$$$ is the range of values. If we first normalize all the value $$$y$$$, it can be reduced to $$$O(\lg N)$$$.
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    7 years ago, hide # ^ |
     
    Vote: I like it +24 Vote: I do not like it

    I think that when sweeplining, for a certain x, you should perform all addition events for that x, then check for a new max, then all subtraction events for that x. Otherwise you might miss the correct answer if you perform a subtraction, then a query, then an addition.

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      7 years ago, hide # ^ |
       
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      Yes, I just give a rough idea. Those parts should be taken care when implementing.

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        7 years ago, hide # ^ |
         
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        It’s a very nice sketch. I just wanted to add the detail since I had also been thinking of the same solution for a little bit. I hope it helps the asker.

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          7 years ago, hide # ^ |
           
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          Yes, it helped me. Is this algorithm similar to that one: "find the number of intersections of segments, all segments parallel either Ox either Oy"? I think they are 90% the same, am I wrong?

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            7 years ago, hide # ^ |
             
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            You can think of each segment of length L as an 1xL or an Lx1 rectangle, so the rectangle problem is a generalization of the line segment problem.

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