## Pages

### HackerRank Equal

We have a list of integers, and we want to know in how many rounds we could make them all equal. In each round we add to all the items in the list but one the same number chosen among 1, 2, and 5. Pay special attention when you solve this problem on HackerRank. Currently (April 2018) its enunciation says each increase should be of one, three, or five. However, this is not what the solution is tested for.
It looks like someone decided to change 2 for 3 a few months ago, edited the description and then forgot to modify the testing part. Who knows what is going to happen next. Be ready to be surprised.
Besides, it is inserted in the Algorithms - Dynamic Programming section, but I don't think that is the right approach to follow.

Simplifying the problem

Instead of applying the weird addition process stated in the problem, we could decrease each time a single item. For instance, given in input [2, 2, 3, 7], a solution to the original problem is:
```2, 2, 3, 7
+5 +5 +5  = (1)
7, 7, 8, 7
+1 +1  = +1 (2)
8, 8, 8, 8
```
We could solve it in this way instead:
```2, 2, 3, 7
=  =  = -5 (1)
2, 2, 3, 2
=  = -1  = (2)
2, 2, 2, 2
```
Since we are asked to calculate the number of steps to get to the equal state, in both ways we get the same result.

Base cases

A sort of rough algorithm is already emerging. We get the lowest value in the list, and decrease all the other elements using the available alternatives until we reach it. We start from the biggest value (5) and only when we are forced to, we fall back to the shorter steps.

To better understand how we should manage the last steps, I put the base cases on paper.
If x is at level 1, 2 or 5, we could get zero at cost 1. But if x is at level 3 or 4, it costs 2 to us. Notice that if, instead of getting to level zero, we move both the lowest element and the x element at level -2, we get to the result in the same number of moves. For instance, if our input list is [10, 13]
```10, 13
-2 (1)
-1 (2)
10, 10

10, 13
-2     (1)
-5 (2)
8,  8
```
If the input list is [0, 3, 3] getting down to the bottom from 3 in just one move gives us an advantage:
```10, 13, 13
-2      (1)
-1      (2)
-2  (3)
-1  (4)
10, 10, 10

10, 13, 13
-2          (1)
-5      (2)
-5  (3)
8,  8,  8
```
The algorithm

I think I've got it. Firstly I find the minimum value, M, in the input list. That is a possible target for all the items to be reach. However I have to check other two alternatives, M-1 and M-2.
I loop on all the items in the list. For all the three possible targets, I calculate the difference between it and the current value, count the number of steps to get there, and add it to the total number of steps required for getting to that target.
And then I choose as a result the cheapest target to reach.

The code

Using Python as implementation language, I started with a couple of test cases, and then added a few ones along the way, when I bumped into troubles, and I ended up with this code.
```SHIFT = [0, 1, 2]  # 1

def solution(data):
lowest = min(data)  # 2

results =  * len(SHIFT)  # 3
for item in data:
for i in SHIFT:
gap = item - lowest + i  # 4
results[i] += gap // 5 + SHIFT[(gap%5 + 1) // 2]  # 5
return min(results)  # 6
```
1. Represents the three possible targets, from the minimal value in the list down to -2.
2. Get the minimal value in the list.
3. Buffer for the results when using the different targets.
4. Distance that has to be split.
5. Add the current number of steps to the current buffer. Firstly I get the number of long moves dividing gap by 5. Now I am in the base case, as showed in the picture above. Notice that the cost of moving from X to target is [0, 1, 1, 2, 2] for gap in [0..5], if we take gap, apply modulo five, increase it and then divide by two, we get the index in SHIFT to the number of steps actually needed. Admittedly an obscure way to get there, if this was production code, I would have probably resisted temptation to use it.
6. Get the minimal result and return it.

All tests passed in hackerrank, python script and test case pushed to GitHub.

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We are given n nodes and a (possibly huge) number of edges. We are also given the cost of building a library in a city (i.e. a node) and a road (i.e. an edge). Based on these data we want to minimize the cost of creating a forest of graphs from the given nodes and edges, with the requirement that each graph should have a library on one of its nodes. This is a HackerRank problem on Graph Theory algorithms, and I am about to describe my python solution to it.

If a library is cheaper than a road, the solution is immediate. Build a library on every node.
```def solution(n, library, road, edges):
return n * library

# ...
```
Otherwise, we want to create a minimum spanning forest, so to minimize the number of roads, keeping track of the number of edges used and how many graphs are actually generated. I found natural using an adapted form of the Kruskal MST (Minimum Spanning Tree) algorithm, that looks very close to our needs.

Kruskal needs a union-find to work, and this ADT is not commonly available out of the box. So, I first implemented a python UnionFind class, see previous post for details.
Then, while working on this problem, I made a slight change to it. My algorithm was simpler and faster if its union() method returned False when nothing was actually done in it, and True only if it led to a join in two existing graph.

Using such refactored UnionFind.union(), I wrote this piece of code based on Kruskal algorithm:
```uf = UnionFind(n)

for edge in edges:
if uf.union(edge - 1, edge - 1):  # 2
if uf.count == 1:  # 5
break
```
1. The union-find object keeps track of the numbers of disjointed graphs in the forest, but not of edges. This extra variable does.
2. I need to convert the edges from 1-based to 0-based convention before use them. If the two nodes get connected by this call to union(), I have some extra work to do.
4. An edge has been used by union(), keep track of it.
5. If union() connected all the nodes in a single graph, there is no use in going on looping.

Now it is just a matter of adding the cost for roads and libraries to get the result.
```return road_count * road + uf.count * library
```

Complete python code for problem, union-find, and test case on GitHub.

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### Union Find

I have a (possibly huge) bunch of edges describing a forest of graphs, and I'd like to know how many components it actually has. This problem has a simple solution if we shovel the edges in a union-find data structure, and then just ask it for that piece of information.

Besides the number of components, our structure keeps track of the id associated to each node, and the size of each component. Here is how the constructor for my python implementation looks:
```class UnionFind:
def __init__(self, n):  # 1
self.count = n
self.id = [i for i in range(n)]  # 2
self.sz = [1 for i in range(n)]  # 3
```
1. If the union-find is created for n nodes, initially the number of components, named count, is n itself.
2. All the nodes in a component have the same id, initially the id is simply the index of each node.
3. In the beginning, each node is a component on its own, so the size is initialized to one for each of them.

This simple ADT has two operations, union and find, hence its name. The first gets in input an edge and, if the two nodes are in different components, joins them. The latter returns the id of the passed node.

Besides, the client code would check the count data member to see how many components are in. Pythonically, this is exposure of internal status is not perceived as horrifying. A more conservative implementation would mediate this access with a getter.

Moreover, a utility method is provided to check if two node are connected. This is not a strict necessity, still makes the user code more readable:
```def connected(self, p, q):
return self.find(p) == self.find(q)
```
The meaning of this method is crystal clear. Two nodes are connected only if they have the same id.

In this implementation, we connect two nodes making them share the same id. So, if we call union() on p and q, we'll change the id of one of them to assume the other one. Given this approach, we implement find() in this way:
```def find(self, p):
while p != self.id[p]:
p = self.id[p]
return p
```
If the passed p has id different from its default value, we check the other node until we find one that has its original value, that is the id of the component.

We could implement union() picking up randomly which id keep among the two passed, but we want keep low the operational costs, so we work it out so to keep low the height of the tree representing nodes in a component, leading to O(log n) find() complexity.
```def union(self, p, q):
i = self.find(p)
j = self.find(q)
if i != j:  # 1
self.count -= 1  # 2
if self.sz[i] < self.sz[j]:  # 3
self.id[i] = j
self.sz[j] += self.sz[i]
else:
self.id[j] = i
self.sz[i] += self.sz[j]
```
1. If the two nodes are already in the same component, there is no need of doing anything more.
2. We are joining two components, their total number in the union-find decrease.
3. This is the smart trick to keep low the cost of find(). We decide which id to keep as representative for the component accordingly with the sizes of the two merging ones.

As example, consider this:
```uf = UnionFind(10)
uf.union(4, 3)
uf.union(3, 8)
uf.union(6, 5)
uf.union(9, 4)
uf.union(2, 1)
uf.union(5, 0)
uf.union(7, 2)
uf.union(6, 1)
```
I created a union-find for nodes in [0..9], specifying eight edges among them, from (4, 3) to (6, 1).
As a result, I expect two components and, for instance, to see that node 2 and node 6 are connected, whilst 4 and 5 not.

I based my python code on the Java implementation provided by Robert Sedgewick and Kevin Wayne in their excellent Algorithms, 4th Edition, following the weighted quick-union variant. Check it out also for a better algorithm description.

I pushed to GitHub full code for the python class, and a test case for the example described above.

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We are given two lists of integers. The first one is monotonically decreasing and represent the scores of the topmost players in a leaderboard. The second one is monotonically increasing and contains the score history of Alice, a player who rapidly climbed the board.
Following the dense ranking convention, we want to get back a list containing the history of rank positions for Alice.
This is a HackerRank Algorithm Implementation problem, and I am going to show you how I solved it, using Python as implementation language.

I noticed that the first list, scores, is already sorted, we just have to get rid of duplicates to have a matching between the position and the score Alice has to get to achieve that ranking.

Bisect solution

I just have to do the matching. First idea jumped to my mind, was performing a binary search on scores to do it. It helps that Python provides for the job a well known library, bisect. There's just a tiny issue, bisect expects the underlying list to be sorted in natural order, so we need to reverse our scores.

It looks promising, let's implement it.

A pythonic way to get our ranking would be this:
```ranking = sorted(list(set(scores)))
```
I get the original list, convert to a set to get rid of duplicates, than back to list, so that I can sort it in natural order. Nice, but in this problem we are kind of interested in performance, since we could have up to 20 thousand items in both lists. So we want to take advantage of the fact that the list is already sorted.

So, I ended up using this rather uncool code:
```ranking = [scores[-1]]
for i in range(len(scores)-2, -1, -1):
if scores[i] > ranking[-1]:
ranking.append(scores[i])
```
I initialize the ranking list with the last item in scores, then I range on all the other indices in the list from right to left. If the current item is bigger than the latest one pushed in ranking, I push it too.

Now I can just rely on the bisect() function in the bisect python module, that would find which position the current item should be inserted in the list. With a caveat, I have reverted the order, so I have to adjust the bisect() result to get the result I'm looking for:
```results = []
last = len(ranking) + 1
for score in alice:
results.append(last - bisect(ranking, score))
```
This code pass all the tests, job done.

However. Do I really need to pay for the bisect() search for each element of alice?

Double scan solution

Well, actually, we don't. Since we know that both list are sorted, we can use also the ordering in alice to move linearly in ranking.

Since we are not using anymore bisect, we don't need to revert the sorting order in ranking, and the duplicate cleanup is getting a bit simpler:
```ranking = [scores]
for score in scores[1:]:
if score < ranking[-1]:
ranking.append(score)
```

Now we compare each item in alice against the items in ranking moving linearly from bottom to head:
```results = []
for score in alice:
while ranking and score >= ranking[-1]:
ranking.pop()
results.append(len(ranking) + 1)
```
We don't have to be alarmed by the nested loops, they don't have a multiplicative effect on the time complexity, since we always move forward on both lists, the result is a O(M + N) time complexity.

Is this a better solution than the first one? Well, it depends. We should know more on the expected input. However, for large and close values of N and M, it looks so.

I pushed the python script for both solutions and a test case to GitHub.

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