Connect and share knowledge within a single location that is structured and easy to search. 9}, {4, {4, 6}, 12}, {5, {4, 6}, 15}}. Please don't take this as being condescending; I did not come up this answer on the spot. Implementation of the stack-based algorithm in plain Javascript (with linear time complexity): function maxRectangle(mask) { Determine the longest valid line to the EAST that begins at each point and save its length in the corresponding cell of LINES. WebMax rectangle. There is a function in Mathematica for finding a maximum matching: FindIndependentEdgeSet. To learn more, see our tips on writing great answers. I'm not going to post pseudo-code for this because the idea is completely experimental, and I have no idea if the perf will be any better for a loose pixel grid ;). For each label, I want to find 2 largest rectangles. 1 & 0 & 0 & 1 & 1 \\ If you do it correctly you will consider each pixel only once. I'm just not sure how to implement such an algorithm. 1 & 0 & 0 & 0 & 0 \\ Find centralized, trusted content and collaborate around the technologies you use most. And you can get the subsets to search over (corresponding to the coordinates of the non-zero elements in the matrix above). My own solution is to find the biggest rectangle, using the same algorithm as in @j_random_hacker answer, then split the remaining area into 4 regions, and The idea of using an increasing stack to find the largest rectangle in a histogram and implementation is from this answer by Pei. Definitions: A clique is a complete subgraph. \right) \{2,4,5,6\} & \{1,2,4,5,6\} & 20 \\ This solution So it's pretty impressive to see someone working through the problem! \end{array} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. I believe you want the 1st interpretation, but this does not seem to have fast solutions. \text{rows} & \text{columns} & \text{area} \\ \hline We can interpret your matrix $A$ as a bipartite incidence matrix: $a_{ij}=1$ means that vertex $i$ of the first partition is connected to vertex $j$ of the second partition of a bipartite graph. Okkes's links to Tushar Roy's YouTube videos has been extremely useful; especially, Maximum Rectangular Area in Histogram and Maximum Size Rectangle of All 1's Dynamic Programming. I would like to recover the coordinates of where the sub-matrix is residing. A straight forward approach would be to do a loop through all the potential rectangles in the grid, figure out their area, and if it is greater than the current highest area, select it as the highest: Then you simply need to find the potential rectangles. public class Test { According to Knig's theorem, in bipartite graphs, the minimum vertex cover can be formed from a maximum matching. Thanks Jerry, there seem to be two issues: 1 the largest sub-matrix in the example is 5x3, your answer results in (after transpose) a 6x3 sub-matrix; Also, I am concerned the Subset will explode beyond memory capacity once we work with large matrices? Is there an accessibility standard for using icons vs text in menus? Hope this helps. \end{array} \right)$. \hline Asking for help, clarification, or responding to other answers. As you were working through your ideas though you were getting close. \hline There are multiple interpretations, e.g. \hline \end{array} There's an obvious dynamic programming approach, but this at least gets the ball rolling. Plus, it is often useful to code up an obviously correct solution before you worry about bug-finding in more clever solutions. Once, S(x,y) are know for all 1<=x <= m, and for all 1 <= y <= n, we simply need to find the x, and y for which S(x,y) is the largest; this step also takes O(mn) time. Its's fairly easy to bound the area of the rectangular submatrix, in order to restrict the search quite a bit. \{2,3,4,5,6\} & \{4,5,6\} & 15 \\ \right),\left( for (int i=0; i < M; i++) { How do you determine purchase date when there are multiple stock buys? \hline This finds ONLY SQUARE sub matrix. $. Plotting Incidence function of the SIR Model. They examine the data one row (or scan line) at a time building up the answer you are looking for, in your case candidate rectangles. 1 & 1 & 1 & 1 & 1 \\ Largest rectangle in Matrix 3 = 3 units. 1 & 1 & 1 & 1 & 1 \\ \end{array} ADDENDUM:Several edits later, I think this answer deserves a group upvote. What happens if you connect the same phase AC (from a generator) to both sides of an electrical panel? Thank you for your valuable feedback! 0 & 0 & 0 & 0 & 0 \\ It seems that you need a more stringent condition; if your current condition is satisfied for a small isolated submatrix of 1s, you miss a larger block in other parts of the input matrix. The pattern is either very sparse, or there are very few large submatrices. For the lower bound, it's a little quirky to calculate. \left( I wanted to know if this particular algorithm had a name or something. A question remains: what does "largest" submatrix, or equivalently "largest" biclique mean? By clicking Post Your Answer, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct. Windows system "regions" and "dirty rectangles", as well as general "temporal caching" might be good inspiration here for more efficiency. Apologies, there is a larger selection possible in my answer that I omitted. Help us improve. Use MathJax to format equations. Maximum vertex biclique finding is equivalent to finding a maximum independent vertex set in the bipartite complement graph. The best answers are voted up and rise to the top, Not the answer you're looking for? \hline Look at the list of max possible values, and the max of that. Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. // for topleft corner. In the simplified example below, it would be elements {2, 1} towards {2,6} and {5,1} towards {6,6}. I was wondering, that's all. This problem can be solved in polynomial time by reducing it to bipartite matching. The lack of evidence to reject the H0 is OK in the case of my research - how to 'defend' this in the discussion of a scientific paper? However, the blocks are likely to overlap. We can use Hashing technique I'm the author of that Dr. Dobb's article and get occasionally asked about an implementation. There are also a lot of z-buffer tricks if this is for a graphics algorithm Use dynamic programming approach. Examples: The naive solution for this problem is to check every possible rectangle in given 2D array by counting the total number of 1s and 0s in that rectangle. 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \right) So temp[i] indicates sum of elements from left to right in row i. Here's a rough outline of how it would work. 8 & 0 & 0 & 0 & 0 \\ I added the size (thousands long, less than a hundred wide and that I am looking for non-contiguous solutions. I'll refer to rectangles using (x, y, width, height). You are now done with scan line 0, so you increment your scan line. Was there a supernatural reason Dracula required a ship to reach England in Stoker? Explanation 2. My bad, I meant to say rectangle. Consider a function S(x,y) such that S(x,y) holds the area of the largest rectangle where (x,y) are the lowest-right-most corner cell of the rectangle; x is the row co-ordinate and y is the column co-ordinate of the rectangle. \begin{array}{ccccc} import java.util.Stack; Assume that a rectangle can be formed by swapping any number of columns with each Maximum size of rectangle in a binary matrix Author Pranav Gautam 0 upvotes Table of contents 1. 1 & 1 & 1 & 1 & 1 & 1 \\ f There might be optimization opportunities here using quadtrees, and in ensuring that you keep split rectangles "in order", top to bottom, left to right, in case you need to re-combine rectangles farther down in the algorithm. Fortunately, this is only for a hobby project I am working on. Why do "'inclusive' access" textbooks normally self-destruct after a year or so? It is a little tricky to describe. I like a region growing approach for this. Edit The pattern is either very sparse, or there are very few I have a function to find 1 largest rectangle for a particular label called def find_largest_rect(). The point of finding an independent edge set (matching) is that this can be solved in polynomial time, and is equivalent (through a series of transformations) to out original problem, @Sander But you are right that the implementation would have been a bit shorter with. Try, for example, Finding the largest rectangular submatrix, The maximum edge biclique problem is NP-complete, http://tryalgo.org/en/matching/2016/08/05/konig/, Maximum Size Rectangle of All 1's Dynamic Programming, Semantic search without the napalm grandma exploit (Ep. \right)\right\} This is the only suggested answer with a correct algorithm. would be a thorough, but maybe not-the-most-efficient way to go about it. Thanks a bunch guys. Enhance the article with your expertise. Thanks, unfortunately . its mostly rectangular squares I am dealing with here. $\qquad mat=\left( \begin{array}{ccc} \begin{array}{ccccc} 2. MathJax reference. A-143, 9th Floor, Sovereign Corporate Tower, Sector-136, Noida, Uttar Pradesh - 201305, We use cookies to ensure you have the best browsing experience on our website. c++ - Finding largest rectangle in 2D array - Stack Overflow 600), Medical research made understandable with AI (ep. Potential shapes can be seen $ What is the meaning of the blue icon at the right-top corner in Far Cry: New Dawn? Is it rude to tell an editor that a paper I received to review is out of scope of their journal? At each point we are looping through the width of the matrix to find the largest rectangle at that point, this slows the complexity to w*h*w. So the complexity is: O(w*h) Blurry resolution when uploading DEM 5ft data onto QGIS. 0 & 0 & 0 & 0 & 0 \\ 600), Medical research made understandable with AI (ep. \right)$, $ 0&0&0&1&1&1\\ Examining row 0 you have the beginnings of two rectangles (I am assuming you are only interested in the black square). \left( Update: Putting together pieces from several sources (links below) to identify the largest contiguous rectangle in a binary matrix: {{1, 1, 1, 1, 0, 0}, {2, 2, 2, 2, 1, 1}, {0, 0, 0, 3, 2, 2}, {1, 1, 0, I implemented the solution of Dobbs in Java. No warranty for anything. package com.test; I've edited my answer to include what I think may be a more efficient "second step. We can think of the matrix as the bipartite incidence matrix of a bipartite graph. @lassevk // 4. Outer double-for-loop to consider all possible positions I am sure there are more efficient ways of calculating the potential rectangles. I was just trying to give you a solution that would work, since you didn't stipulate any requirements on performance or what you've already done. How can overproduction of electric power be a problem to the grid? obstacles which split existing rectangles, obstacles which require you to remove a rectangle from the active list, Active List: (0,1,8,1), (1,0,2,2), (4,0,4,2), (9, 0, 1, 2), grow NORTH as far as possible by adding rows, grow SOUTH as far as possible by adding rows, save the resulting area for the seed pixel used, After looping through each point in ARRAY, pick the seed pixel with the largest area result, Create a second array of integers called LINES that has the same dimensions as ARRAY. Input 1.2. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Number all the rows in your image from 0..6, I'll work from the bottom up. I could not use, @Sander That would be basically the same as the method I already provided in the previous section (find cliques). e.g., east and south, or north and west. Is there a way to smoothly increase the density of points in a volume using the 'Distribute points in volume' node? It is the largest we have seen to far. I am the author of the Maximal Rectangle Solution on LeetCode, which is what this answer is based on. Since the stack-based solution has already How do I identify the largest non-contiguous rectangular sub-matrix consisting only of 1-entries? Will try to understand this do you think the, @Sanders, although we are considering only 2-subsets of the non-zero positions. It's just there are a lot of students on right now posting homework problems with no work, so basically asking people to do their homework for them :) I figured a high level generic answer was enough of a hint, if this happened to be the case.
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