{"id":7250,"date":"2020-08-15T21:59:32","date_gmt":"2020-08-16T04:59:32","guid":{"rendered":"https:\/\/zxi.mytechroad.com\/blog\/?p=7250"},"modified":"2020-08-15T23:46:21","modified_gmt":"2020-08-16T06:46:21","slug":"leetcode-1551-minimum-operations-to-make-array-equal","status":"publish","type":"post","link":"https:\/\/zxi.mytechroad.com\/blog\/math\/leetcode-1551-minimum-operations-to-make-array-equal\/","title":{"rendered":"\u82b1\u82b1\u9171 LeetCode 1551. Minimum Operations to Make Array Equal"},"content":{"rendered":"\n

You have an array arr<\/code> of length n<\/code> where arr[i] = (2 * i) + 1<\/code> for all valid values of i<\/code> (i.e. 0 <= i < n<\/code>).<\/p>\n\n\n\n

In one operation, you can select two indices x<\/code> and y<\/code> where 0 <= x, y < n<\/code> and subtract 1<\/code> from arr[x]<\/code> and add 1<\/code> to arr[y]<\/code> (i.e. perform arr[x] -=1 <\/code>and arr[y] += 1<\/code>). The goal is to make all the elements of the array equal<\/strong>. It is guaranteed<\/strong> that all the elements of the array can be made equal using some operations.<\/p>\n\n\n\n

Given an integer n<\/code>, the length of the array. Return the minimum number of operations<\/em> needed to make all the elements of arr equal.<\/p>\n\n\n\n

Example 1:<\/strong><\/p>\n\n\n\n

Input:<\/strong> n = 3\nOutput:<\/strong> 2\nExplanation:<\/strong> arr = [1, 3, 5]\nFirst operation choose x = 2 and y = 0, this leads arr to be [2, 3, 4]\nIn the second operation choose x = 2 and y = 0 again, thus arr = [3, 3, 3].\n<\/pre>\n\n\n\n
Example 2:<\/strong><\/p>\n\n\n\n
Input:<\/strong> n = 6\nOutput:<\/strong> 9\n<\/pre>\n\n\n\n
Constraints:<\/strong><\/p>\n\n\n\n
  • 1 <= n <= 10^4<\/code><\/li><\/ul>\n\n\n\n
    Solution: Math<\/strong><\/h2>\n\n\n\n
    1: Find the mean (final value) of the array, assuming x, easy to show x == n
    2: Compute the sum of an arithmetic progression of (x – 1) + (x – 3) + … for n \/\/ 2 pairs

    e.g. n = 6
    arr = [1, 3, 5, 7, 9, 11]
    x = (1 + 2 * n – 1) \/ 2 = 6 = n
    steps = (6 – 1) + (6 – 3) + (6 – 5) = (n \/\/ 2) * (n – (1 + n – 1) \/ 2) = (n \/\/ 2) * (n – n \/\/ 2) = 3 * 3 = 9

    e.g. n = 5
    arr = [1,3,5,7,9]
    x = (1 + 2 * n  – 1) \/ 2 = 5 = n
    steps = (5 – 1) + (5 – 3)= (n\/\/2) * (n – n \/\/ 2) = (n \/\/ 2) * ((n + 1) \/\/ 2) = 2 * 3 = 6<\/p>\n\n\n\n
    Time complexity: O(1)
    Space complexity: O(1)<\/p>\n\n\n\n
    \n
    C++<\/h2>\n
    \n\n
    class Solution {\npublic:\n  int minOperations(int n) {    \n    return (n \/ 2) * ((n + 1) \/ 2);\n  }\n};\n<\/pre>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"
    You have an array arr of length n where arr[i] = (2 * i) + 1 for all valid values of i (i.e. 0 <= i < n). In one operation, you can select…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[49],"tags":[],"class_list":["post-7250","post","type-post","status-publish","format-standard","hentry","category-math","entry","simple"],"_links":{"self":[{"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/posts\/7250","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/comments?post=7250"}],"version-history":[{"count":3,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/posts\/7250\/revisions"}],"predecessor-version":[{"id":7259,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/posts\/7250\/revisions\/7259"}],"wp:attachment":[{"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/media?parent=7250"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/categories?post=7250"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/zxi.mytechroad.com\/blog\/wp-json\/wp\/v2\/tags?post=7250"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}