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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\"\u003eMinimum Coin Change\u003c/h1\u003e\u003ca id=\"user-content-minimum-coin-change\" class=\"anchor\" aria-label=\"Permalink: Minimum Coin Change\" href=\"#minimum-coin-change\"\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\"\u003eMinimum Coin Change problem algorithm implemented in Swift comparing dynamic programming algorithm design to traditional greedy approach.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWritten for Swift Algorithm Club by Jacopo Mangiavacchi\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/shiablue/swift-algorithm-club/blob/master/MinimumCoinChange/eurocoins.gif\"\u003e\u003cimg src=\"/shiablue/swift-algorithm-club/raw/master/MinimumCoinChange/eurocoins.gif\" alt=\"Coins\" data-animated-image=\"\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eIntroduction\u003c/h1\u003e\u003ca id=\"user-content-introduction\" class=\"anchor\" aria-label=\"Permalink: Introduction\" href=\"#introduction\"\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\"\u003eIn the traditional coin change problem you have to find all the different ways to change some given money in a particular amount of coins using a given amount of set of coins (i.e. 1 cent, 2 cents, 5 cents, 10 cents etc.).\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor example using Euro cents the total of 4 cents value of money can be changed in these possible ways:\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003eFour 1 cent coins\u003c/li\u003e\n\u003cli\u003eTwo 2 cent coins\u003c/li\u003e\n\u003cli\u003eOne 2 cent coin and two 1 cent coins\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp dir=\"auto\"\u003eThe minimum coin change problem is a variation of the generic coin change problem where you need to find the best option for changing the money returning the less number of coins.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor example using Euro cents the best possible change for 4 cents are two 2 cent coins with a total of two coins.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eGreedy Solution\u003c/h1\u003e\u003ca id=\"user-content-greedy-solution\" class=\"anchor\" aria-label=\"Permalink: Greedy Solution\" href=\"#greedy-solution\"\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\"\u003eA simple approach for implementing the Minimum Coin Change algorithm in a very efficient way is to start subtracting from the input value the greater possible coin value from the given amount of set of coins available and iterate subtracting the next greater possible coin value on the resulting difference.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor example from the total of 4 Euro cents of the example above you can subtract initially 2 cents as the other biggest coins value (from 5 cents to above) are to bigger for the current 4 Euro cent value. Once used the first 2 cents coin you iterate again with the same logic for the rest of 2 cents and select another 2 cents coin and finally return the two 2 cents coins as the best change.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eMost of the time the result for this greedy approach is optimal but for some set of coins the result will not be the optimal.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIndeed, if we use the a set of these three different coins set with values 1, 3 and 4 and execute this greedy algorithm for asking the best change for the value 6 we will get one coin of 4 and two coins of 1 instead of two coins of 3.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch1 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eDynamic Programming Solution\u003c/h1\u003e\u003ca id=\"user-content-dynamic-programming-solution\" class=\"anchor\" aria-label=\"Permalink: Dynamic Programming Solution\" href=\"#dynamic-programming-solution\"\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\"\u003eA classic dynamic programming strategy will iterate selecting in order a possible coin from the given amount of set of coins and finding using recursive calls the minimum coin change on the difference from the passed value and the selected coin. For any interaction the algorithm select from all possible combinations the one with the less number of coins used.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe dynamic programming approach will always select the optimal change but it will require a number of steps that is at least quadratic in the goal amount to change.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eIn this Swift implementation in order to optimize the overall performance we use an internal data structure for caching the result for best minimum coin change for previous values.\u003c/p\u003e\n\u003c/article\u003e","errorMessage":null,"headerInfo":{"toc":[{"level":1,"text":"Minimum Coin Change","anchor":"minimum-coin-change","htmlText":"Minimum Coin Change"},{"level":1,"text":"Introduction","anchor":"introduction","htmlText":"Introduction"},{"level":1,"text":"Greedy Solution","anchor":"greedy-solution","htmlText":"Greedy Solution"},{"level":1,"text":"Dynamic Programming 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