{"id":12581,"date":"2022-10-01T05:06:41","date_gmt":"2022-09-30T23:36:41","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-in-a-square-matrix-where-each-cell-is-black-or-white-design-an-algorithm-to-find-the-max-white-sub-square-closed\/"},"modified":"2022-10-01T05:06:41","modified_gmt":"2022-09-30T23:36:41","slug":"solved-in-a-square-matrix-where-each-cell-is-black-or-white-design-an-algorithm-to-find-the-max-white-sub-square-closed","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-in-a-square-matrix-where-each-cell-is-black-or-white-design-an-algorithm-to-find-the-max-white-sub-square-closed\/","title":{"rendered":"[Solved] In a square matrix, where each cell is black or white. Design an algorithm to find the max white sub-square [closed]"},"content":{"rendered":"<p> [ad_1]<br \/>\n<\/p>\n<div id=\"answer-29468546\" class=\"answer js-answer accepted-answer js-accepted-answer\" data-answerid=\"29468546\" data-parentid=\"29468151\" data-score=\"0\" data-position-on-page=\"1\" data-highest-scored=\"1\" data-question-has-accepted-highest-score=\"1\" itemprop=\"acceptedAnswer\" itemscope itemtype=\"https:\/\/schema.org\/Answer\">\n<div class=\"post-layout\">\n<div class=\"votecell post-layout--left\"><\/div>\n<div class=\"answercell post-layout--right\">\n<div class=\"s-prose js-post-body\" itemprop=\"text\">\n<p>First note that your solution is NOT <code>O(n^2)<\/code>, it is more like <code>O(n^4)<\/code>, because for each cell, you look for the largest matrix that can be of size up to <code>O(n^2)<\/code> itself, so it is totalling to <code>O(n^4)<\/code>.<\/p>\n<p>It can be done however in <code>O(n^2)<\/code>:<\/p>\n<p>First, define 2 auxillary functions (implemented as matrices):<\/p>\n<pre><code>whitesLeft(x,y) = number of adjacent white cells on the same row to the left from x,y - exclusive of x,y\nwhitesUp(x,y) = number of adjacent white cells on the same column up from x,y -  exclusive of x,y\n<\/code><\/pre>\n<p>Now, define the recursive function:<\/p>\n<pre><code>D(0,y) = 0     if (0,y) is black\n         1     otherwise\nD(x,0) = 0     if (x,0) is black\n         1     otherwise\nD(x,y) = 0                                                         if  x,y is black\n         min{ D(x-1,y-1), whitesLeft(x,y), whitesUp(x,y) } + 1     otherwise\n<\/code><\/pre>\n<p>The idea is to calculate with <code>D(x,y)<\/code> the biggest square that ends (bottom right) in <code>(x,y)<\/code>. This square is calculated by finding the minimal cells to the left, cells to the up and submatrix that ends at <code>(x-1,y-1)<\/code> that has only white cells.<\/p>\n<p>Note that <code>D(x,y)<\/code> can be calculated efficiently with D<a rel=\"nofollow noopener\" target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Dynamic_programming\">ynamic Programming<\/a> in <code>O(n^2)<\/code>, and the pre-processing to calculate the auxillary functions (matrices) is also <code>O(n^2)<\/code>.<\/p>\n<p>When you are done, post prcess <code>D<\/code> to find the largest value of <code>D(x,y)<\/code> for any <code>(x,y)<\/code> to get the result &#8211; this is also <code>O(n^2)<\/code>.<\/p>\n<\/p><\/div>\n<div class=\"mt24\"><\/div>\n<\/div>\n<p>            <span class=\"d-none\" itemprop=\"commentCount\"><\/span> <\/p><\/div>\n<\/div>\n<p>[ad_2]<\/p>\n<p>solved In a square matrix, where each cell is black or white. Design an algorithm to find the max white sub-square [closed] <\/p>\n","protected":false},"excerpt":{"rendered":"<p>[ad_1] First note that your solution is NOT O(n^2), it is more like O(n^4), because for each cell, you look for the largest matrix that can be of size up to O(n^2) itself, so it is totalling to O(n^4). It can be done however in O(n^2): First, define 2 auxillary functions (implemented as matrices): whitesLeft(x,y) &#8230; <a title=\"[Solved] In a square matrix, where each cell is black or white. Design an algorithm to find the max white sub-square [closed]\" class=\"read-more\" href=\"https:\/\/jassweb.com\/solved\/solved-in-a-square-matrix-where-each-cell-is-black-or-white-design-an-algorithm-to-find-the-max-white-sub-square-closed\/\" aria-label=\"More on [Solved] In a square matrix, where each cell is black or white. Design an algorithm to find the max white sub-square [closed]\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[457,324,859],"class_list":["post-12581","post","type-post","status-publish","format-standard","hentry","category-solved","tag-algorithm","tag-c","tag-data-structures"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>[Solved] In a square matrix, where each cell is black or white. 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