Algorithm

Choosing Oranges

8th JLTi Code Jam – Oct 2017

Orange is one of my favourite fruits that I buy for our Friday Fun Session participants. How would you choose the good ones from hundreds of them; especially, on the way to office, when you stop by the supermarket, in the morning rush hour?

To speed up the selection while at the same time choosing the good – firm, smooth and heavier compare to its size – I have devised a selection process. I would go from left to right, scoring each of the oranges, in a scale from 0 to 9, 9 being the best; and once a row is done, I go to the next row and so on. As I go and score, I would also choose the best one among each consecutive, say 5 oranges.

How that would look like?

Input:

5

1 3 5 7 3 5 9 1 2 5

Output: 7, 9

Explanation:

The first line says: choose the best among consecutive 5. The second line shows the score for each of the 10 oranges. The first 5 are: 1, 3, 5, 7, and 3; best among them is 7. We choose 7. The next 5 are:  3, 5, 7, 3, and 5; best among them 7 – already chosen. Move on to the next 5: 5, 7, 3, 5, and 9; best among them 9, pick that. Move to the next 5: 7, 3, 5, 9, and 1; best among them is 9, already chosen. Next 5 are: 3, 5, 9, 1, and 2; once again the best among them 9 is already chosen. Final 5 are: 5, 9, 1, 2, and 5; same as before. We cannot move further as we don’t have 5 oranges after this point.

We end up with two oranges: 7 and 9. I am not doing a bad job of selecting the best oranges for you, am I?

Input:

4

1, 3, 5

Output: None

The first line says: choose the best among 4. However, the second line shows only 3 oranges. Obviously we cannot choose any.

Input:

3

1 2 4 9

Output: 4, 9

Choose 4 from 1, 2 and 4. And then choose 9 from the next consecutive 3: 2, 4 and 9. And we are done!

Task: If we have a total of n oranges and we got to choose the best from each consecutive m, I am looking for a solution having better than O(mn) time complexity.

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Pseudo-polynomial Complexity

33rd Friday Fun Session (Part 2) – 15th Sep 2017

The complexity for FaaS solution is O(n), where n is the largest day number. It looks like polynomial. However, it is actually pseudo-polynomial.

Size of input

Complexity is measured in terms of the size of input, say, in bits. Suppose, there are b bits in n. Then O(n) = O(2b) and hence, it is exponential.

Let’s assume n increases from 10 to 1125899906842624. More specifically, lunch schedule, as used in the previous example, changes from 1, 3, 4, 5, 6, 7, 10 to 1, 3, 4, 5, 6, 7, 1125899906842624. We still have the same 7 days to go for lunch. Yet, we are running 1,125,899,906,842,624 loops. In our layman understanding, the problem is still the same and should have taken the same amount of time to execute, and yet, for the latter, the algorithm takes way too long!

Spot a pseudo-polynomial

This is how we spot a pseudo-polynomial time algorithm. Ideally, we would like to express the complexity using the number of inputs; here, it should have been 7. But the above algorithm works in a way, where the complexity has been expressed in one of the numeric values of the input, the maximum value of the input – 1125899906842624, to be precise. This is where we are tricked into believing it to be a polynomial time algorithm, linear (polynomial) in the (max) numeric value of the input. But if we apply the definition of complexity that takes into consideration the size/length of the input, then it is actually exponential.

To be more specific, if we look at the input size, 4 bits are required to represent 10, while 50 bits are required to represent 1,125,899,906,842,624. Complexity has gone from O(24) = 10 loops to O(250) = 1,125,899,906,842,624 loops.

That is essentially exponential in the number of bits, meaning exponential in the size of the input but polynomial in the numeric value of the input. Algorithm with this kind of running time is called pseudo-polynomial.

Truly polynomial

At this point, you might wonder what is a truly polynomial time algorithm. For example, when we add n numbers using a loop running n times, we say, the complexity of it to be O(n). But here this n can also be written as 2b. So, shall we also say, adding n numbers is a pseudo-polynomial time algorithm?

Well, when we say, adding n numbers, we implicitly say, we want to find the sum of n 32 bit numbers/integers. Then the size of n numbers is 32 * n. Once again, the formal definition of complexity is defined in terms of input size, in bits. What is the input size here? The size here is 32n. The complexity is O(32n) and removing the constant terms it is O(n), a truly polynomial time algorithm.

Solution – FaaS

33rd Friday Fun Session – 15th Sep 2017

Given a lunch schedule – a sequence of days when lunch is planned, and three price plans – daily, weekly and monthly, we want to get the cheapest lunch price.

This is the solution to JLTi Code Jam – Aug 2017 problem.

Let us walk through an example

Let us take an example as mentioned here: 1, 2, 4, 5, 17, 18. Since first day is 1 and last day is 18, it can be put under a month that consists of 20 consecutive days (not calendar month). We can use a monthly plan. But it would be too expensive (S$ 99.99) for just 6 days.

The days: 1, 2, 4 and 5 fall within a week that requires consecutive 5 days (not a calendar week). We have an option to buy a weekly plan for these 4 days that would cost S$ 27.99. However, that would be higher than had we bought day-wise for 4 days at a price of S$24.

Dynamic Programming

In general, at any given day, we have three options:

  1. We buy lunch for this day alone, using daily price S$ 6. Add that to the best price found for the previous day.
  2. We treat this as the last day of a week, if applicable, and buy a weekly plan at a cost of S$ 27.99. Add that to the best price for the day immediately prior to the first day of this week.
  3. We treat this as the last day of a month, if applicable, and buy a monthly plan at a cost of S$ 99.99. Add that to the best price for the day immediately prior to the first day of this month.

This is an optimization problem that can be solved with dynamic programming where we use the result of already solved sub-problems.

Bottom-up

We have two options: top-down and bottom-up. We realize that, at the end, all the sub-problems (for each of the days) have to be solved. We also find that it is easy to visualize the problem bottom-up. And if we do use bottom-up then the required space would be limited by the last day number.

Hence, we will solve it using bottom-up dynamic programming.

Blue colored days are when lunch is scheduled.

DP table1.png

On day 1:

Cost S$ 6.

On day 2:

Daily basis: S$ 6 + price at day 1 = S$ 12

Weekly basis: S$ 27.99

Monthly basis: S$ 99.99

Best price: S$ 12

On day 3:

No lunch schedule, cost of previous day S$ 12 is its cost.

On day 4:

Daily basis: S$ 6 + price at day 3 = S$ 18

Weekly basis: S$ 27.99

Monthly basis: S$ 99.99

Best price: S$ 18

On day 5:

Daily basis: S$ 6 + price at day 4 = S$ 24

Weekly basis: S$ 27.99

Monthly basis: S$ 99.99

Best price: S$ 24

From day 6 to day 16:

No lunch schedule, cost of previous day will be carried forward: S$ 24.

On day 17:

Daily basis: S$ 6 + price at day 16 = S$ 30

Weekly basis: S$ 27.99 + price at day 12 = S$ 51.99

Monthly basis: S$ 99.99

Best price: S$ 30

On day 18:

Daily basis: S$ 6 + price at day 17 = S$ 36

Weekly basis: S$ 27.99 + price at day 13 = S$ 51.99

Monthly basis: S$ 99.99

Best price: S$ 36

Finally, the best price is S$ 36.

Another example

Let us work with another example: 1, 3, 4, 5, 6, 7, 10.

DP table2

On day 7:

Daily basis: S$ 6 + price at day 6 = S$ 36

Weekly basis: S$ 27.99 + price at day 2 = S$ 33.99

Monthly basis: S$ 99.99

Best price: S$ 33.99

Finally, the best price at the end is S$ 39.99.

Complexity

The complexity is O(n), where n is the largest day number. It is a pseudo-polynomial time algorithm.

GitHub: FaaS

Team Lunch

7th JLTi Code Jam – Sep 2017

Since our JLTi Mumbai colleagues started vising our Singapore office, we are having more team/project lunches. Usually, a number of them come together, and after a short while they also leave together. It is only few days before they leave that we start organizing team lunches. Suppose, there are three colleagues belonging to three different teams, then there would be three team lunches, one for each team.

However, not all members work for an exclusive team. For example, I create an impression as if I work for more than one team, and due to the good grace of those teams, I also get invited in their team lunches.

However, due to the rush of deliverables, that is the norm here, the team lunches are squeezed in the last few days, and at times, multiple team lunches fall on the same day, typically on the last day.

That is all fine and good for most. However, I have a big problem. If two team lunches fall on the same day, and I belong to both, I miss one for obvious reason. I skip lunch does not necessarily mean I skip free lunches.

Hence, I decided to write a small program that will take the team composition in certain way and output the minimum days required to schedule the lunches so that people working on multiple teams don’t miss out any.

Yes, I am not the only person but there are some other colleagues who also work across more than one team. Let us also assume that, for our 7 or 8 teams, it might be easy to calculate it manually. But when the number of team exceeds, say 100, then a program is a must.

Input:

1 2

Output: 2

Explanation:

Input 1 2 (1 and 2 separated by a space) means there are one or more members who belong to both team 1 and team 2.

Output 2 means, at least 2 days are required to arrange lunches for the teams. On day 1, one of the two teams can go for lunch. On the second day, the other can go.

How many teams are there? Well, there are at least 2. There can be more, but that is irrelevant. Suppose there are 4 more teams – team 3, team 4, team 5 and team 6, they can go either on first day or on second day. This is because, no member working in those 4 teams work for a second team. After all, the input says, only team 1 and team 2 have some common members.

Input:

1 2

2 3

Output: 2

We have some members common to team 1 and team 2. And there are some members, who are common to team 2 and team 3, as shown in the second line.

Each line in the input would indicate the presence of common members between two teams, where the two team numbers are separated by a space. There would be at least one line of input, meaning somebody would run this program only if there exist at least one member working for more than one team.

For the above input, we would still require at least two days to avoid any conflict. On one day team 1 and team 3 can go. Team 2 must go on a separate day.

Input:

1 2

2 3

1 3

Output: 3

Now we need 3 separate days. Team 1 cannot go on the same day as team 2 or team 3. This is because team 1 has members working for both team 2 and team 3. Similarly, team 2 cannot go for lunch on the same day as team 3 as they have common members. Hence, team 1, team 2 and team 3 – all need exclusive lunch days.

Task: Given a list of team pairs (like 1 2 is a team pair, as shown in input) sharing common members, we need to write a program, that would output the minimum number of days required to set aside for team lunches, so that nobody who work across multiple teams misses his/her share of team lunches.

Solution – Scoring Weight Loss

29th Friday Fun Session – 4th Aug 2017

Given a sequence of weights (decimal numbers), we want to find the longest decreasing subsequence. And the length of that subsequence is what we are calling weight loss score. This is essentially the standard longest increasing subsequence (LIS) problem, just the other way.

This is the solution to JLTi Code Jam – Jul 2017 problem.

Let us walk through an example

Let us take the example as mentioned here: 95, 94, 97, 89, 99, 100, 101, 102, 103, 104, 105, 100, 95, 90. The subsequence can start at any value, and a value in a subsequence must be strictly lower than the previous value. Any value in the input can be skipped. The soul goal is to find the longest subsequence of decreasing values. Here one of the longest decreasing subsequences could be:  105, 100, 95, 90 and the length would be 4.

Even though, in our weight loss example, we have to find the length of longest decreasing subsequence, the standard problem is called longest increasing subsequence. Essentially the problems are the same. We can have a LIS solution and can pass it the negative of the input values. Alternatively, in the algorithm, we can alter the small to large, greater than to smaller than etc. We chose the former.

We will use two approaches to solve this problem: one is a dynamic programming based solution having O(n2) complexity, another is, let’s call it Skyline solution having O(n log n) complexity.

Dynamic Programming Solution

Let’s work with this example: 95, 96, 93, 101, 91, 90, 95, 100 – to see how LIS would work.

When the first value, 95 comes, we know it alone can make a subsequence of length 1. Well, each value can make a subsequence on its own of length 1.

When the second value 96 comes, we know it is greater than 95. Since 95 already made a subsequence of length 1, 96 can sit next to it and make a subsequence of length 2. And it would be longer than a subsequence of its own of length 1.

When the value 93 comes, it sees it cannot sit next to any value that appeared prior to it (95 and 96). Hence, it has to make a subsequence of its own.

When the value 101 comes, it knows that it can sit next to any prior values (95, 96 and 93). After all, it is bigger than each of them. It then computes the score it would make if it sits next to each of them, separately. The scores would be 2, 3, and 2, if it sits next to 95, 96 and 93 respectively. Of course, it would choose 96. The subsequence is 95, 96, 101 and the score is 3.

So we see, we can go from left to right of the input, and then for each of the previous values, it sees whether it can be placed after it. If yes, it computes the possible score. Finally, it chooses the one that gives it the highest score as its predecessor.

So we are using the solutions already found for existing overlapping sub-problems (the scores already computed for its preceding input values, that we can reuse) and can easily compute its own best score from them. Hence, it is called a dynamic programming solution.

The following table summarizes it.

DP table.png

There are two longest subsequences each with length 3. For a certain value, if we need to know the preceding value, we can backtrace and find from which earlier value its score is computed. That way, we can complete the full subsequence ending with this value.

Since for each of the input values we are looping all the preceding values, the complexity is O(n2).

Skyline Solution

In this approach, we would retain all incompatible and hence promising subsequences since any of them could lead to the construction of one of the final longest subsequences. Only at the end of the input we would know which one is the longest. Since we are retaining all incompatible subsequences I am calling it Skyline, inspired by Skyline operator.

It is obvious but let me state here, all these solutions are standard, already found and used. However, Skyline is a name I am using as I find it an appropriate term to describe this method.

If there are two apples: one big and another small, and if you are asked to choose the better one, you would choose the big one. However, if you are given an apple and an orange, you cannot, as they are incomparable. Hence you need to retain both.

When a value comes it can be one of the below three types:

Smallest value (case 1)

  1. It won’t fit at the end of any existing subsequences. Because the value is smaller than all the end values for all existing subsequences.
  2. There is no other way but to create a new subsequence with this value.
  3. We can safely discard all single value subsequences existed so far. After all, the new subsequence with the smallest value can be compared with each of them and it is clearly superior to them (score for each such subsequence is 1 and the end (and only) value for the new one is the smallest – hence it can accept more future input values than the rests).
  4. In the list of subsequences we can retain the single value subsequence at first. Meaning, every time the new smallest value comes, we simply replace the existing smallest value listed as the first subsequence.

Biggest value (case 2)

  1. The opposite of the previous case is: the new value is bigger than the end values of each of the existing subsequences.
  2. So it can fit at the end of all existing subsequences. So which one to choose?
  3. Suppose, it is the end of the input. In that case, we would like it to go at the end of the longest subsequence found so far and make it longer by one more.
  4. However, if it is not the end of the input and suppose there are some future input values coming that are bigger than the end value of the present longest subsequence and smaller than the present input value. By placing the present input value at the end of the present longest subsequence we will jeopardize a more promising possibility in future.
  5. So we should rather copy the longest subsequence found so far and add this new value at the end of it, making it the new longest.
  6. At the same time, we retain the previous longest subsequence as it is, that by now is the second longest subsequence.
  7. We will add this new and longest subsequence at the end of the list.

Middle value (case 3)

  1. We have a third case where the input value can fit the end of some subsequences and cannot fit at the end of the rest subsequences.
  2. This is because this new value is bigger than the end values of some sun-sequences and smaller than the same for the rests.
  3. So which one to choose? Of course, we have to choose one where it can fit, meaning from those whose end values are smaller than the input value.
  4. And we would like to choose one with the largest end element (yet it is smaller than the input value).
  5. However, we cannot just over-write it for the same reason as stated earlier (case 2, promising reasoning). Rather we copy it, add the new value at the end of it and add it to the list.
  6. Where – at the end of the list?
  7. No, we would insert in next to the subsequence from which we copied and extended it.
  8. And we can safely discard all other subsequences with the same length as this newly created  subsequence. After all, the length is the same and it’s end element is smaller than the end elements of the rests having equal length of it.
  9. Shall we run a loop over the list to find those to be deleted? No, we just need to find the next subsequence and if its length is the same as the newly created subsequence we delete it. No more checking is required.
  10. Why so? Please read the second point as stated below.

So we have handled all possible input values. The list of subsequences that we have created would have some nice properties:

  1. As we go from the first subsequence to the last in the list of subsequences, the length will gradually increase.
  2. There would be a maximum of one subsequence with a certain length.
  3. To find whether the input value is a case 1 or case 2 or case 3 type, we can easily run a binary search with O(log n) complexity over the end elements of the subsequences in the list. Since we would like to do so for each of the n input values, the complexity of this approach would be O(n log n).
  4. For doing the above we can use the list, just that we need to look at the end elements. Then why are we retaining the complete list?
  5. The answer is: to output the longest subsequence as well.
  6. Could we do it without saving the complete subsequence?
  7. We leave it for another day.

Walking through an example

Let’s go through the same example as used earlier: 95, 96, 93, 101, 91, 90, 95, 100.

95 (case 1)

95

96 (case 2)

95

95, 96

93 (case 1)

93

95, 96

101 (case 2)

93

95, 96

95, 96, 101

91 (case 1)

91

95, 96

95, 96, 101

90 (case 1)

90

95, 96

95, 96, 101

95 (case 3)

90

90 95

95, 96 (deleted)

95, 96, 101

100 (case 3)

90

90 95

90 95 100

95, 96, 101 (deleted)

Once all the input values are treated, the last subsequence would be the longest one.

GitHub: Scoring Weight Loss

FaaS

6th JLTi Code Jam – Aug 2017

Threatened by the JLTi Weight Loss Competition where the participants are lining up in front of Salad shops, and the likes of me, who have entirely given up lunch (hopefully I can continue forever), food court shops who are selling oily, low-fibre and various other kinds of unhealthy food have come up with a novel idea.

Inspired from the software world, and more importantly, to attract the software people who sit in their chairs for long hours and are the primary victims of eating these junk, those food shops have chosen a name for this scheme – Food as a Service (FaaS), borrowed from the likes of SaaS, PaaS, IaaS – whatever that means, if that means anything at all.

Instead of paying on a daily basis, they are asking people to subscribe for food.

For example, without subscription, a set lunch would cost S$ 6, as usual, if you want to pay as you eat, just like as you are doing now. No strings attached.

However, if you subscribe for a week (5 meals, one meal one day, 5 consecutive days, not calendar week, can start at any day), instead of paying S$ 30, you can pay S$ 27.99 for five meals. Of course you have to eat from the same (chain of) shop.

And if you subscribe for a month (20 meals, one meal one day, 20 consecutive days, not calendar month, can start at any day) that they are vying for, you pay only S$ 99.99.

Input: 1, 2, 4, 5, 17, 18

Output: 36

Explanation: Input is a list of day numbers when you want to have a meal. The number can start at 1, and go up to any number.

A certain day number, say, 4, would not come more than once in the input, if it comes at all, assuming one can have only one lunch meal a day.

The above input says – you eat for 6 days. It makes no sense for you to go for a monthly subscription. Well, it also does not make sense to go for a weekly subscription. Paying daily basis for 6 days would be the best cost effective decision for you. You pay: S$ 36.

Input: 3, 4, 5, 6, 7, 17, 18

Output: 39.99

You subscribe for one week (first 5 days) and pay individually for the last 2 days. Your best decision cost you S$ 39.99.

Input: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 24

Output: 105.99

Here, a monthly subscription and S$ 6 for the last day would be the best deal for you.

Task: Given lunch calendar for some days (it can be 3 days, 10 days, 121 days or any number of days) as input, as explained above, I am planning to write a program that would output me the best price. Well, if I can find the best price, I also know what subscription plans etc. are. However, put that aside. Let’s find the best price, as shown and explained above.

Solution – Manipulating Money Exchange

25th Friday Fun Session – 7th Jul 2017

Given a set of currencies and some exchange rates among them, we want to find if there exists an arbitrage; meaning, if it is possible to exploit the discrepancies in the exchange rates and transform one unit of a certain currency to more than one unit of the same, thus making a profit.

This is the solution to JLTi Code Jam – Jun 2017 problem.

Let us walk through an example

Let us take the example as mentioned here. We can start with 1 USD, convert that to SGD (1.380 SGD), then convert that to MYR (1.380 * 3.080 MYR), then convert to INR (1.380 * 3.080 * 15.120 INR), then convert to GBP (1.380 * 3.080 * 15.120 * 0.012 GBP), then convert that back to USD (1.380 * 3.080 * 15.120 * 0.012 * 1.30 = 1.0025503488 USD).

We end up with more than 1 USD. That means, we have an arbitrage in this set of exchange rates. The profit making cycle here is: USD -> SGD -> MYR -> INR -> GBP. And since it is a cycle, we can start from any currency within it. For example, SGD -> MYR -> INR -> GBP -> USD also represents the same cycle.

The transformation

In general, if we have to make a profit, the respective rates in the cycle, when multiplied, should give more than 1, as we have seen in the above example.

Formula

Negative cycle in Bellman-Ford

After some simple transformation of the profit making condition, we see, if we take negative of log rate, and use that as the edge cost/distance, then finding profit making cycle is equivalent to finding negative cycle in the corresponding graph. And we can do so using Bellman-Ford algorithm.

To be precise, each of the currencies would be considered as a vertex. If there exists an exchange rate r between two currencies then there would be a directed edge between the corresponding vertices, and –log r would be the associated cost/distance of that edge.

Source of Bellman-Ford

The next question comes: using which vertex as source shall we run the Bellman-Ford? Let us see the below graph.

Souce.png

Suppose, we have a single profit making cycle here: GBP-> AUD -> CAD. In that case, if we start with USD as source vertex, we will never detect this cycle.

Add extra currency as source

To solve this problem, we need to add an extra currency, and then create edges from it to all the existing currencies with cost 0. Now using this extra vertex (EXT) as source we have to run Bellman-Ford and that would ensure that we can detect a cycle, if there exist one.

Extra Souce

GitHub: Manipulating Money Exchange

Scoring Weight Loss

5th JLTi Code Jam – Jul 2017

Now that weight loss competition is back, some people are more than excited about it. And why not? After all, only by running 10 km in the last 3 days, they can effortlessly shed 15 kg!

On the other hand, slim people, by any global standard, like me have to starve an entire month and still have to win this competition only in dream, in some rainy days.

Since the enthusiastic participants approached me to participate, I am thinking of a new scoring system that would remove this inherent bias in the existing scoring system – deducting final weight from first day weight.

So here I propose a new scoring system that would otherwise value the sustained effort and success of the participants, ignoring the total/final/absolute loss.

Input: 73, 72.9, 72.8, 72.8, 72.9, 72.7, 72.0, 71.6, 73, 72.5, 72.4, 71.3, 73.5, 74

Output: 7

Explanation: The above is an estimation of my performance, if the competition runs for 14 days. You can clearly see I start with 73 and end up with 74. In the old standard, I gain weight and penalized. In the new scoring system I score 7. How? Well, it computes how long I keep on decreasing weight, without seeing how much. In the above example, the longest stretch where I continue to lose weight (a value in a sequence must be smaller than its immediate predecessor) is shown below.

73, 72.9, 72.8, 72.7, 72.0, 71.6, 71.3

Let us also find the approximate score of the last time winner. A possible set of weights of him might look like the below:

Input: 95, 94, 97, 89, 99, 100, 101, 102, 103, 104, 105, 100, 95, 90

Output: 4

His success story lies in the last 3 days of blitzkrieg (the first weight in the input does not necessarily need to be the first value in the sequence): 105, 100, 95, 90

Let us also talk about a hypothetical participant who misunderstands this to be a weight gain competition and eats cheese all along.

Input: 53, 53.1, 53.2, 53.4, 53.5, 53.6, 53.9, 54, 54.1, 54.2, 54.2, 54.7, 55.8, 56

Output: 1

The scorer takes note of 53 and it never goes towards the right direction.

Task: A good scoring system indeed – nobody gets zero or below. And I am sure all of you would agree with this. Now let us quickly write a small program that takes an array of weights and computes the score.

Maximum Subarray Problem

21st Friday Fun Session – 9th Jun 2017

Maximum subarray finds the contiguous subarray within a one-dimensional array having the largest sum.

Visualizing the divide and conquer solution

For the time being, let us forget about maximum subarray problem and focus on the divide and conquer solution that we discussed in the last session.

If we visualize the tree, we see that from the left subtree the smallest value is propagated upwards.  On the way up, it is treated as the buy value and the right side values are treated as sell values. This way profits are calculated and maximum among them is retained. So we see two themes of processing as we go from left to right of the array:

  1. Retain the minimum value and treat it as the buy value.
  2. Calculate profit by treating each value seen as we go right and retain the maximum profit.

DP1.png

The above table shows day number in first row and the corresponding stock prices in second row. Third row shows the minimum value seen so far. The fourth row shows the profit had we sold on this day, buy price being the minimum value seen so far (shown in green).

The intuition

The intuition being, when we see a new lower value than the one already seen, we treat that as the new buy value. For example, when we see the new lower value 1 on day 5, onward we treat that as the new buy value and calculate profits considering each of the following days as sell days. This is because the new lower value (lowest till now) would give a better profit when the following days are treated as potential sell days. By treating the previous lower value 2 that was found on day 1, we already considered all possible profits prior to 5th day and retained the best among them. On 5th day, the utility of the previous lower value, which is 2, stops.

From divide and conquer to dynamic programming

Now let us now consider the dynamic programming (DP) point of view. In dynamic programming we make use of the result of an already solved overlapping subproblem.

On the first day, we can buy but cannot sell. After all, no profit would be made selling on the first day with the same price as the buy price. Also note that we have to buy and only then we can sell. So on day 1, profit is 0. Now if we want to find the best profit on day 2, can we use the solution of the previously solved overlapping subproblem? What is that already solved overlapping subproblem at day 2? Well, it is the best profit found for day 1, which is 0. How can we make use of the previous solution to find the best profit at day 2? Well, we have to consider two things:

  1. If we have to make the most profit by selling today, then we have to buy using the lowest price seen so far.
  2. If the profit calculated above is better than the best seen on previous day, then this is the new best. Else previous day’s best is still the best for today.

For example, on day 2 we realize that we can make a profit of (8-0) = 8 and it is better than the profit at day 1, which is 0. Hence, the best profit for day 2 is updated to 8. On day 3, we find we can make a profit of 3 but the best profit till day 2 is better than this. So, we retain day 2’s best profit as day 3 best profit.

So we realize, what we found by visualizing and transforming the divide and conquer solution is nothing but this dynamic programming. In fact, this is possibly one of the simplest forms of dynamic programming.

The below code would find the solution. For brevity buy day and sell day is not tracked that is easy to accommodate.

void StockDpN(double price[], int n, double &maxProfit)
{
  double minPriceSoFar = price[0]; 
  maxProfit = 0;
  
  for(int i=1; i<n; i++)  
  { 
    if(price[i] - minPriceSoFar > maxProfit) 
      maxProfit = price[i] - minPriceSoFar;

    if(price[i] < minPriceSoFar) 
     minPriceSoFar = price[i]; 
  }
}

The reverse can also be used. If we start from right and move leftwards, we have to keep track of the maximum value seen so far and that is the sell value. As we go left, we see new values and they are buy values. The associated code is not shown here.

Moving to maximum subarray problem

Suppose we buy a stock at one day and then sell it on the following day. For example, buy at day 1 and then sell on day 2. Buy at day 2 and then sell on day 3 and so on. Each day we make a profit, incur a loss and sometimes it is neural, meaning no profit or loss (buy value and sell value being the same). The third row of the below table shows the same (loss shown in red).

MSData.png

The optimal solution to our stock profit problem with our example set is to buy on day 1 at price 2 and sell it on day 4 at price 12, thus making a profit of 10. It is the same as saying:

  1. We buy at day 1 and sell at day 2 making profit 8 and then
  2. Buy at day 2 and sell at day 3 making loss 5 and then
  3. Buy at 3 and sell at day 4 making profit 7 and then
  4. Add all profits/losses made in our buy/sell operations that started by buying on day 1 and ended by selling on day 4. The final and best profit is: 8 + (-5) + 7 = 10.

Thus we have transformed the previous stock profit problem to a maximum subarray problem. As explained earlier, we are interested to find contiguous portion of array that gives the maximum sum. In the above 8 values that we have, we got two such subarrays each giving a sum of 10. They are showed in colored boxes.

Kadane’s algorithm

Kadane’s algorithm also deploys DP to solve this. Once again in DP, we have to make use of already solved overlapping subproblems. Here it is done by this way:

  1. Maximum subarray ending in position i+1 includes already solved maximum subarray ending at i, if doing so increases the sum for subarray ending at i+1
  2. Else maximum subarray ending in position i+1 will only have itself.

MSDP

Maximum subarray at day 1: day 1 value which is 0.

Maximum subarray at day 2: since adding the subarray sum for day 1, which is 0, is not increasing the sum for day 2, maximum subarray at day 2 will have only day 2 value itself, meaning 8.

Maximum subarray at day 3: subarray sum at day 2 is positive, which is 8, and helping day 3, so subarray at day 3 includes day 2. Subarray sum at day 3 = 8 + (-5) = 3.

It boils down to a simple thing. If the previous sum is positive then take it forward else not. The red color in the Maximum subarray sum row (4th row) shows the cases where it does not include the (immediately) prior subarray. In two cases it happens (8 at day 2 and 2 at day 6) because the prior sums (0 and -1 respectively) are not more than zero.

The code shown below implements this. Note that the input array profit contains the profit and loss unlike the earlier DP function where we passed the stock prices. It is also noteworthy that if all values are positive then the whole array is the maximum subarray. After all, adding all of them would give the highest sum.

void StockKadaneDpN(double profit[], int n, double &maxProfit)
{  
  double curProfit = 0; maxProfit = 0;
  
  for(int i=1; i<n; i++) 
  { 
    curProfit = curProfit > 0 ? curProfit + profit[i] : profit[i]; 
    if(curProfit > maxProfit) 
      maxProfit = curProfit; 
  }
}

If we observe closely, we see that this DP is essentially the same as the one we discussed earlier in this post.

Backtrace

At the end, when we find the maximum subarray sum 10 at day 4, we will do what is called backtrace, typical of DP to find the path, in this case, the maximum subarray. We know that at day 4, we included the subarray ending at day 3. At day 3, we included the subarray ending at day 2. At day 2, we did not include prior subarray. So the maximum subarray starts at day 2 and ends at day 4. It could be easily tracked/stored as we went ahead in the computation using appropriate data structure and would not require a come back.

Map maximum subarray solution to stock profit

If we want to map this solution back to our stock profit problem, then we know the profit at start day of the maximum subarray, that is day 2, is essentially found by buying stock at the previous day that is day 1. So the solution is: buy at day 1 and sell at the last day of the maximum subarray that is day 4. And the profit would be the maximum subarray sum that is 10.

The transformations

This is an interesting problem to observe as we started with a O(n) brute force accumulator pattern, moved to O(n log n) divide and conquer that we optimized later to O(n). Finally, we transformed that to a O(n) DP solution only to find that it is interchangeable to O(n) maximum subarray problem that is also a DP solution.

Can we do better than O(n)? Well, that is not possible. After all, we cannot decide the best solution unless we read all the data at least once. Reading the data once is already O(n).

Where is pattern recognition here?

Maximum subarray essentially gives the brightest spot in a one-dimensional array. Finding this brightest spot is one kind of pattern recognition. Note that we just solved a problem that reads like this: given the profit/ loss made by a company over the period find the longest duration(s) when the company performed the best. The answer here is: from day 2 to day 4 or from day 6 to day 7.

Even though we focused on finding the single brightest spot, it is also possible to find, k brightest spots.

Again, maximum subarray considers only one dimension. In real life, data sets typically contain more than one dimension. For example, a problem involving two dimensions might read like: can you find the largest segment of the customers buying product x based on age and income? A potential answer might be: customer from age 30 to 40 years with income range $3000 – $6000. There are other algorithms to deal with multi-dimensional data.

GitHub: Stock Profit Kadane Code

Manipulating Money Exchange

4th JLTi Code Jam – Jun 2017

Input:

1 USD = 1.380 SGD

1 SGD = 3.080 MYR

1 MYR = 15.120 INR

1 INR = 0.012 GBP

1 GBP = 1.30 USD

I CAD = 0.57 GBP

Explanation: Now that we realize, we have to wait substantial amount of time to make any meaningful gain from stock market, we change our focus to money exchange. I am particularly very excited after collecting the above exchange rates using Google search today at 8th Jun 2017. If I start with 10, 000 USD, then convert them to MYR, then convert them to INR and then to GBP and then back to USD, I realize I will end up with 10, 025.50 USD, making USD 25.50 on the same spot within minutes.

Output: USD -> SGD -> MYR -> INR -> GBP -> USD

The next step that I am going to take is to get a list of all money exchanges available to me, somehow collect their exchange rates daily, and push it to a program that will tell me when it sees there is a chance to make some profit.

I might not always be lucky. To check whether my adrenaline rush is justified I looked at the same rates in an online money exchange. The rates from it look like below:

Input:

1 USD = 1.38295 SGD

1 SGD = 3.08614 MYR

1 MYR = 15.0996 INR

1 INR = 0.0119755 GBP

1 GBP = 1.295 USD

As you realize, we will end up losing money.

Output: No luck here

Task: I also realize, given a few hundred currencies and thousands of exchange rates among them, there is a possibility of having a number of ways we can make money. For example, given a set of rates (not as shown in the above example), we could make USD 10, starting with USD 10, 000, using this route: USD -> SGD -> MYR -> USD. Again, we could possibly make SGD 50, starting with SGD 10, 000, using another route: SGD -> MYR -> INR -> SGD. I am happy with just one such route, not necessarily the one providing the most profit.