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{"payload":{"allShortcutsEnabled":false,"path":"Count Occurrences","repo":{"id":283265754,"defaultBranch":"master","name":"swift-algorithm-club","ownerLogin":"orchca","currentUserCanPush":false,"isFork":true,"isEmpty":false,"createdAt":"2020-07-28T16:18:29.000Z","ownerAvatar":"https://avatars.githubusercontent.com/u/3979323?v=4","public":true,"private":false,"isOrgOwned":false},"currentUser":null,"refInfo":{"name":"master","listCacheKey":"v0:1616582547.416779","canEdit":false,"refType":"branch","currentOid":"bcd9eb3e6b8bb7c8f67eb962b065b16ef5c9f25b"},"tree":{"items":[{"name":"CountOccurrences.playground","path":"Count Occurrences/CountOccurrences.playground","contentType":"directory"},{"name":"CountOccurrences.swift","path":"Count Occurrences/CountOccurrences.swift","contentType":"file"},{"name":"README.markdown","path":"Count Occurrences/README.markdown","contentType":"file"}],"templateDirectorySuggestionUrl":null,"readme":{"displayName":"README.markdown","richText":"\u003carticle class=\"markdown-body entry-content container-lg\" itemprop=\"text\"\u003e\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCount Occurrences\u003c/h1\u003e\u003ca id=\"user-content-count-occurrences\" class=\"anchor\" aria-label=\"Permalink: Count Occurrences\" href=\"#count-occurrences\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eGoal: Count how often a certain value appears in an array.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe obvious way to do this is with a \u003ca href=\"/orchca/swift-algorithm-club/blob/master/Linear%20Search\"\u003elinear search\u003c/a\u003e from the beginning of the array until the end, keeping count of how often you come across the value. That is an \u003cstrong\u003eO(n)\u003c/strong\u003e algorithm.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eHowever, if the array is sorted you can do it much faster, in \u003cstrong\u003eO(log n)\u003c/strong\u003e time, by using a modification of \u003ca href=\"/orchca/swift-algorithm-club/blob/master/Binary%20Search\"\u003ebinary search\u003c/a\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eLet's say we have the following array:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eIf we want to know how often the value \u003ccode\u003e3\u003c/code\u003e occurs, we can do a regular binary search for \u003ccode\u003e3\u003c/code\u003e. That could give us any of these four indices:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n * * * *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n * * * *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eBut that still doesn't tell you how many other \u003ccode\u003e3\u003c/code\u003es there are. To find those other \u003ccode\u003e3\u003c/code\u003es, you'd still have to do a linear search to the left and a linear search to the right. That will be fast enough in most cases, but in the worst case -- when the array consists of nothing but \u003ccode\u003e3\u003c/code\u003es -- it still takes \u003cstrong\u003eO(n)\u003c/strong\u003e time.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe trick is to use two binary searches, one to find where the \u003ccode\u003e3\u003c/code\u003es start (the left boundary), and one to find where they end (the right boundary).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn code this looks as follows:\u003c/p\u003e\n\u003cdiv class=\"highlight highlight-source-swift notranslate position-relative overflow-auto\" dir=\"auto\" data-snippet-clipboard-copy-content=\"func countOccurrences\u0026lt;T: Comparable\u0026gt;(of key: T, in array: [T]) -\u0026gt; Int {\n var leftBoundary: Int {\n var low = 0\n var high = array.count\n while low \u0026lt; high {\n let midIndex = low + (high - low)/2\n if a[midIndex] \u0026lt; key {\n low = midIndex + 1\n } else {\n high = midIndex\n }\n }\n return low\n }\n\n var rightBoundary: Int {\n var low = 0\n var high = array.count\n while low \u0026lt; high {\n let midIndex = low + (high - low)/2\n if a[midIndex] \u0026gt; key {\n high = midIndex\n } else {\n low = midIndex + 1\n }\n }\n return low\n }\n\n return rightBoundary - leftBoundary\n}\"\u003e\u003cpre\u003e\u003cspan class=\"pl-en\"\u003efunc\u003c/span\u003e countOccurrences\u003cspan class=\"pl-c1\"\u003e\u0026lt;\u003c/span\u003eT\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003eComparable\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e(\u003c/span\u003eof key\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003eT\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e in array\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e[\u003c/span\u003e\u003cspan class=\"pl-smi\"\u003eT\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e]\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e)\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e-\u0026gt;\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003eInt\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003eleftBoundary\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003eInt\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003elow\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e0\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ehigh\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e array\u003cspan class=\"pl-kos\"\u003e.\u003c/span\u003ecount\n \u003cspan class=\"pl-k\"\u003ewhile\u003c/span\u003e low \u003cspan class=\"pl-c1\"\u003e\u0026lt;\u003c/span\u003e high \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003elet\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003emidIndex\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e low \u003cspan class=\"pl-c1\"\u003e+\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e(\u003c/span\u003ehigh \u003cspan class=\"pl-c1\"\u003e-\u003c/span\u003e low\u003cspan class=\"pl-kos\"\u003e)\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e/\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e2\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003eif\u003c/span\u003e \u003cspan class=\"pl-en\"\u003ea\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e[\u003c/span\u003emidIndex\u003cspan class=\"pl-kos\"\u003e]\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e\u0026lt;\u003c/span\u003e key \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n low \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e midIndex \u003cspan class=\"pl-c1\"\u003e+\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e1\u003c/span\u003e\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e \u003cspan class=\"pl-k\"\u003eelse\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n high \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e midIndex\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e low\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003erightBoundary\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-smi\"\u003eInt\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003elow\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e0\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003evar\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ehigh\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e array\u003cspan class=\"pl-kos\"\u003e.\u003c/span\u003ecount\n \u003cspan class=\"pl-k\"\u003ewhile\u003c/span\u003e low \u003cspan class=\"pl-c1\"\u003e\u0026lt;\u003c/span\u003e high \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003elet\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003emidIndex\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e low \u003cspan class=\"pl-c1\"\u003e+\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e(\u003c/span\u003ehigh \u003cspan class=\"pl-c1\"\u003e-\u003c/span\u003e low\u003cspan class=\"pl-kos\"\u003e)\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e/\u003c/span\u003e\u003cspan class=\"pl-c1\"\u003e2\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003eif\u003c/span\u003e \u003cspan class=\"pl-en\"\u003ea\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e[\u003c/span\u003emidIndex\u003cspan class=\"pl-kos\"\u003e]\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e\u0026gt;\u003c/span\u003e key \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n high \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e midIndex\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e \u003cspan class=\"pl-k\"\u003eelse\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e{\u003c/span\u003e\n low \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e midIndex \u003cspan class=\"pl-c1\"\u003e+\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e1\u003c/span\u003e\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e low\n \u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\n\n \u003cspan class=\"pl-k\"\u003ereturn\u003c/span\u003e rightBoundary \u003cspan class=\"pl-c1\"\u003e-\u003c/span\u003e leftBoundary\n\u003cspan class=\"pl-kos\"\u003e}\u003c/span\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eNotice that the variables \u003ccode\u003eleftBoundary\u003c/code\u003e and \u003ccode\u003erightBoundary\u003c/code\u003e are very similar to the \u003ca href=\"/orchca/swift-algorithm-club/blob/master/Binary%20Search\"\u003ebinary search\u003c/a\u003e algorithm. The big difference is that they don't stop when they find the search key, but keep going. Also, notice that we constrain the type \u003ccode\u003eT\u003c/code\u003e to be Comparable so that the algorithm can be applied to an array of Strings, Ints or other types that conform to the Swift Comparable protocol.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eTo test this algorithm, copy the code to a playground and then do:\u003c/p\u003e\n\u003cdiv class=\"highlight highlight-source-swift notranslate position-relative overflow-auto\" dir=\"auto\" data-snippet-clipboard-copy-content=\"let a = [ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n\ncountOccurrences(of: 3, in: a) // returns 4\"\u003e\u003cpre\u003e\u003cspan class=\"pl-k\"\u003elet\u003c/span\u003e \u003cspan class=\"pl-s1\"\u003ea\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e=\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e[\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e0\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e1\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e1\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e6\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e8\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e10\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e11\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e11\u003c/span\u003e \u003cspan class=\"pl-kos\"\u003e]\u003c/span\u003e\n\n\u003cspan class=\"pl-en\"\u003ecountOccurrences\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e(\u003c/span\u003eof\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e \u003cspan class=\"pl-c1\"\u003e3\u003c/span\u003e\u003cspan class=\"pl-kos\"\u003e,\u003c/span\u003e in\u003cspan class=\"pl-kos\"\u003e:\u003c/span\u003e a\u003cspan class=\"pl-kos\"\u003e)\u003c/span\u003e // returns 4\u003c/pre\u003e\u003c/div\u003e\n\u003cblockquote\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eRemember:\u003c/strong\u003e If you use your own array, make sure it is sorted first!\u003c/p\u003e\n\u003c/blockquote\u003e\n\u003cp dir=\"auto\"\u003eLet's walk through the example. The array is:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eTo find the left boundary, we start with \u003ccode\u003elow = 0\u003c/code\u003e and \u003ccode\u003ehigh = 12\u003c/code\u003e. The first mid index is \u003ccode\u003e6\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWith a regular binary search you'd be done now, but here we're not just looking whether the value \u003ccode\u003e3\u003c/code\u003e occurs or not -- instead, we want to find where it occurs \u003cem\u003efirst\u003c/em\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSince this algorithm follows the same principle as binary search, we now ignore the right half of the array and calculate the new mid index:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3 | x, x, x, x, x, x ]\n *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3 | x, x, x, x, x, x ]\n *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAgain, we've landed on a \u003ccode\u003e3\u003c/code\u003e, and it's the very first one. But the algorithm doesn't know that, so we split the array again:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1 | x, x, x | x, x, x, x, x, x ]\n *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1 | x, x, x | x, x, x, x, x, x ]\n *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eStill not done. Split again, but this time use the right half:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ x, x | 1 | x, x, x | x, x, x, x, x, x ]\n *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ x, x | 1 | x, x, x | x, x, x, x, x, x ]\n *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe array cannot be split up any further, which means we've found the left boundary, at index 3.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow let's start over and try to find the right boundary. This is very similar, so I'll just show you the different steps:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n *\n\n[ x, x, x, x, x, x, x | 6, 8, 10, 11, 11 ]\n *\n\n[ x, x, x, x, x, x, x | 6, 8, | x, x, x ]\n *\n\n[ x, x, x, x, x, x, x | 6 | x | x, x, x ]\n *\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode\u003e[ 0, 1, 1, 3, 3, 3, 3, 6, 8, 10, 11, 11 ]\n *\n\n[ x, x, x, x, x, x, x | 6, 8, 10, 11, 11 ]\n *\n\n[ x, x, x, x, x, x, x | 6, 8, | x, x, x ]\n *\n\n[ x, x, x, x, x, x, x | 6 | x | x, x, x ]\n *\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe right boundary is at index 7. The difference between the two boundaries is 7 - 3 = 4, so the number \u003ccode\u003e3\u003c/code\u003e occurs four times in this array.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eEach binary search took 4 steps, so in total this algorithm took 8 steps. Not a big gain on an array of only 12 items, but the bigger the array, the more efficient this algorithm becomes. For a sorted array with 1,000,000 items, it only takes 2 x 20 = 40 steps to count the number of occurrences for any particular value.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eBy the way, if the value you're looking for is not in the array, then \u003ccode\u003erightBoundary\u003c/code\u003e and \u003ccode\u003eleftBoundary\u003c/code\u003e return the same value and so the difference between them is 0.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThis is an example of how you can modify the basic binary search to solve other algorithmic problems as well. 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