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go through all the later prices for later_time in xrange earlier_time 1 len stock_prices later_price stock_prices later_time see what our profit would be if we bought at the earlier price and sold at the later price potential_profit later_price earlier_price update max_profit if we can do better max_profit max max_profit potential_profit return max_profit what s our runtime now well our outer for loop goes through all the times and prices but our inner for loop goes through one fewer price each time so our total number of steps is the sum n n 1 n 2 2 1 which is still time we can do better if we re going to do better than we re probably going to do it in either or comes up in sorting and searching algorithms where we re recursively cutting the list in half it s not obvious that we can save time by cutting the list in half here let s first see how well we can do by looping through the list only once since we re trying to loop through the list once let s use a greedy approach where we keep a running max_profit until we reach the end we ll start our max_profit at 0 as we re iterating how do we know if we ve found a new max_profit at each iteration our max_profit is either the same as the max_profit at the last time step or the max profit we can get by selling at the current_price how do we know when we have case 2 the max profit we can get by selling at the current_price is simply the difference between the current_price and the min_price from earlier in the day if this difference is greater than the current max_profit we have a new max_profit so for every price we ll need to keep track of the lowest price we ve seen so far see if we can get a better profit here s one possible solution def get_max_profit stock_prices min_price stock_prices 0 max_profit 0 for current_price in stock_prices ensure min_price is the lowest price we ve seen so far min_price min min_price current_price see what our profit would be if we bought at the min price and sold at the current price potential_profit current_price min_price update max_profit if we can do better max_profit max max_profit potential_profit return max_profit we re finding the max profit with one pass and constant space are we done let s think about some edge cases what if the price stays the same what if the price goes down all day if the price doesn t change the max possible profit is 0 our function will correctly return that so we re good but if the value goes down all day we re in trouble our function would return 0 but there s no way we could break even if the price always goes down how can we handle this well what are our options leaving our function as it is and just returning zero is not a reasonable option we wouldn t know if our best profit was negative or actually zero so we d be losing information two reasonable options could be return a negative profit what s the least badly we could have done raise an exception we should not have purchased stocks yesterday in this case it s probably best to go with option 1 the advantages of returning a negative profit are we more accurately answer the challenge if profit is revenue minus expenses we re returning the best we could have done it s less opinionated we ll leave decisions up to our function s users it would be easy to wrap our function in a helper function to decide if it s worth making a purchase we allow ourselves to collect better data it matters if we would have lost money and it matters how much we would have lost if we re trying to get rich we ll probably care about those numbers how can we adjust our function to return a negative profit if we can only lose money initializing max_profit to 0 won t work well we started our min_price at the first price so let s start our max_profit at the first profit we could get if we buy at the first time and sell at the second time min_price stock_prices 0 max_profit stock_prices 1 stock_prices 0 but we have the potential for an index out of bounds error here if stock_prices has fewer than 2 prices we do want to raise an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it if len stock_prices 2 raise valueerror getting a profit requires at least 2 prices min_price stock_prices 0 max_profit stock_prices 1 stock_prices 0 ok does that work no max_profit is still always 0 what s happening if the price always goes down min_price is always set to the current_price so current_price min_price comes out to 0 which of course will always be greater than a negative profit when we re calculating the max_profit we need to make sure we never have a case where we try both buying and selling stocks at the current_price to make sure we re always buying at an earlier price never the current_price let s switch the order around so we calculate max_profit before we update min_price we ll also need to pay special attention to time 0 make sure we don t try to buy and sell at time 0 we ll greedily walk through the list to track the max profit and lowest price so far for every price we check if we can get a better profit by buying at min_price and selling at the current_price we have a new min_price to start we initialize min_price as the first price of the day max_profit as the first profit we could get we decided to return a negative profit if the price decreases all day and we can t make any money we could have raised an exception instead but returning the negative profit is cleaner makes our function less opinionated and ensures we don t lose information def get_max_profit stock_prices if len stock_prices 2 raise valueerror getting a profit requires at least 2 prices we ll greedily update min_price and max_profit so we initialize them to the first price and the first possible profit min_price stock_prices 0 max_profit stock_prices 1 stock_prices 0 start at the second index 1 time we can t sell at the first time since we must buy first and we can t buy and sell at the same time if we started at index 0 we d try to buy and sell at time 0 this would give a profit of 0 which is a problem if our max_profit is supposed to be negative we d return 0 for current_time in xrange 1 len stock_prices current_price stock_prices current_time see what our profit would be if we bought at the min price and sold at the current price potential_profit current_price min_price update max_profit if we can do better max_profit max max_profit potential_profit update min_price so it s always the lowest price we ve seen so far min_price min min_price current_price return max_profit time and space we only loop through the list once this one s a good example of the greedy approach in action greedy approaches are great because they re fast usually just one pass through the input but they don t work for every problem how do you know if a problem will lend itself to a greedy approach best bet is to try it out and see if it works trying out a greedy approach should be one of the first ways you try to break down a new question to try it on a new problem start by asking yourself suppose we could come up with the answer in one pass through the input by simply updating the best answer so far as we went what additional values would we need to keep updated as we looked at each item in our input in order to be able to update the best answer so far in constant time in this case the best answer so far is of course the max profit that we can get based on the prices we ve seen so far the additional value is the minimum price we ve seen so far if we keep that updated we can use it to calculate the new max profit so far in constant time the max profit is the larger of the previous max profit the max profit we can get by selling now the current price minus the minimum price seen so far try applying this greedy methodology to future questions actual_language c actual_language_short_name c default content n python content python3 content ruby content java content javascript content cpp content csharp content php content swift content objectivec content question__stock price__throw an error default content throw an exception python content raise an exception actual_language python actual_language_short_name python python3 content raise an exception actual_language python actual_language_short_name python3 ruby content raise an exception actual_language ruby actual_language_short_name ruby c content abort actual_language c actual_language_short_name c swift content throw an error actual_language swift actual_language_short_name swift java content throw an exception javascript content throw an exception cpp content throw an exception csharp content throw an exception php content throw an exception objectivec content throw an exception question__stock price__thrown an error default content thrown an exception python content raised an exception actual_language python actual_language_short_name python python3 content raised an exception actual_language python actual_language_short_name python3 ruby content raised an exception actual_language ruby actual_language_short_name ruby c content aborted actual_language c actual_language_short_name c java content thrown an exception javascript content thrown an exception cpp content thrown an exception csharp content thrown an exception php content thrown an exception swift content thrown an exception objectivec content thrown an exception question__stock price__out of bounds result default content an index out of bounds error javascript content reading undefined values actual_language javascript actual_language_short_name javascript ruby content reading nil values actual_language ruby actual_language_short_name ruby java content an arrayindexoutofboundsexception actual_language java actual_language_short_name java cpp content undefined behavior actual_language cpp actual_language_short_name cpp c content undefined behavior actual_language c actual_language_short_name c csharp content an indexoutofrangeexception actual_language csharp actual_language_short_name csharp swift content an index out of range error actual_language swift actual_language_short_name swift python content an index out of bounds error python3 content an index out of bounds error php content an index out of bounds error objectivec content an index out of bounds error question__stock price__error requires two prices text default content we do want to in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it swift content we could throw an error in that case or we use swift s optional return values to just return nil the second way is a bit more common so let s do that actual_language swift actual_language_short_name swift python content we do want to raise an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it python3 content we do want to raise an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it ruby content we do want to raise an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it java content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it javascript content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it c content we do want to abort in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it cpp content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it csharp content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it php content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it objectivec content we do want to throw an exception in that case since profit requires buying and selling which we can t do with less than 2 prices so let s explicitly check for this case and handle it question__stock price__standard list default content list python content list python3 content list csharp content array ruby content array java content array javascript content array c content array swift content array php content array objectivec content array cpp content vector question__stock price__a standard list default content a list python content a list python3 content a list csharp content an array ruby content an array java content an array javascript content an array c content an array swift content an array php content an array objectivec content an array cpp content a vector question__stock price__function default content function csharp content method ruby content method java content method python content function python3 content function javascript content function c content function cpp content function php content function swift content function objectivec content function var codetypes words code blocks vars funcs code inlines var mergeobjects function objecta objectb objecta objecta objectb objectb for var key in objectb objecta key objectb key return objecta if window contenttranslations window contenttranslations eltranslations else for var i 0 i share tweet share share 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