SPOJ GUANGGUN

Problem link: SPOJ Problem Set (classical): 9952. 111…1 Squared

Interesting and fun problem. This is one of those problems that you can sense a pattern from miles away. Just do some bruteforce and print the patterns. You can use pen and paper too, but Python is surely a better option when you have it. All I did is write some bruteforce in my python interpreter, and the pattern is instantly visible, here is the output of f(n)² for n = 1 to 50:

1
121
12321
1234321
123454321
12345654321
1234567654321
123456787654321
12345678987654321
1234567900987654321
123456790120987654321
12345679012320987654321
1234567901234320987654321
123456790123454320987654321
12345679012345654320987654321
1234567901234567654320987654321
123456790123456787654320987654321
12345679012345678987654320987654321
1234567901234567900987654320987654321
123456790123456790120987654320987654321
12345679012345679012320987654320987654321
1234567901234567901234320987654320987654321
123456790123456790123454320987654320987654321
12345679012345679012345654320987654320987654321
1234567901234567901234567654320987654320987654321
123456790123456790123456787654320987654320987654321
12345679012345679012345678987654320987654320987654321
1234567901234567901234567900987654320987654320987654321
123456790123456790123456790120987654320987654320987654321
12345679012345679012345679012320987654320987654320987654321
1234567901234567901234567901234320987654320987654320987654321
123456790123456790123456790123454320987654320987654320987654321
12345679012345679012345679012345654320987654320987654320987654321
1234567901234567901234567901234567654320987654320987654320987654321
123456790123456790123456790123456787654320987654320987654320987654321
12345679012345679012345679012345678987654320987654320987654320987654321
1234567901234567901234567901234567900987654320987654320987654320987654321
123456790123456790123456790123456790120987654320987654320987654320987654321
12345679012345679012345679012345679012320987654320987654320987654320987654321
1234567901234567901234567901234567901234320987654320987654320987654320987654321
123456790123456790123456790123456790123454320987654320987654320987654320987654321
12345679012345679012345679012345679012345654320987654320987654320987654320987654321
1234567901234567901234567901234567901234567654320987654320987654320987654320987654321
123456790123456790123456790123456790123456787654320987654320987654320987654320987654321
12345679012345679012345679012345679012345678987654320987654320987654320987654320987654321
1234567901234567901234567901234567901234567900987654320987654320987654320987654320987654321
123456790123456790123456790123456790123456790120987654320987654320987654320987654320987654321
12345679012345679012345679012345679012345679012320987654320987654320987654320987654320987654321
1234567901234567901234567901234567901234567901234320987654320987654320987654320987654320987654321
123456790123456790123456790123456790123456790123454320987654320987654320987654320987654320987654321

Beautiful, isn't it?

SPOJ DCEPC206

Problem link: SPOJ Problem Set (classical): 10810. Its a Murder!

Given a sequence of integers (at most 10⁵ integers ranging from 0 to 10⁶. You have to go from left to right, and for each position, you have to add the sum of all the numbers before current position which are strictly smaller than the number in current position.

Well, if the numbers where sorted, then we could do it more easily, i.e. keep an array for cumulative sum, and then calculate the answer in O(n) time. However, this is not the case. So, what can we do? According the problem, you have to maintain the cumulative sum anyway. So for each number in the sequence, if we keep adding them one by one, we can break down the problem in two parts. Let's think of a data structure which keeps cumulative sums, and we can add numbers anywhere and still maintain the sum. Segment tree / or binary indexed tree could come handy in this situation, where, the later one is more straight forward and less memory hungry. Now the steps are:

1 ⇒ Query: When a number is to be added, what is the sum of numbers smaller than current one? Note, as we are adding one by one in the order they appear, so the question whether there are any numbers which came after current number is eliminated. It is guaranteed that, when you are adding the i^th number, all the numbers already present in the data-structure came before position i.

2 ⇒ Update: Once you have taken the partial sum for current number, it is now time to add the number in proper place, and update the cumulative sum structure. Here, clearly, appropriate position means, the position where the number would be placed when sorted. Now, we can do two things here. (a), we can sort the number and give each number an unique id to indicate its position in the sorted array. And (b), as the numbers are up to 10⁶, we can use the numbers as their indices or positions in the sorted array. It is faster and the memory overhead is not that of a big issue here.

Now, here are two more things to be taken care about, (a), there can be duplicate numbers, and (b), 0 can be present in the input sequence, which can cause some trouble if you use BIT. However, (a) is handled naturally, because the problem wanted you to add only strictly smaller numbers, so it doesn't matter if you add same numbers in the same positions. There will be no query points in-between the equal numbers. And for (b), there is really no need to worry about 0, cause it contributes nothing at all, just ignore 0.

So, the overall algorithm looks like the following:

for each number ai:
    if ai > 0:
        ans += BIT_Query(ai - 1)
        BIT_Update(ai, ai)

Here: BIT_Query(pos) returns the sum up to position pos and BIT_Update(pos, val) adds val in pos in the BIT structure. Solutions involving the segment tree is also similar. Just keep the tree of total sums, whenever you are trying to add a number, read the partial sum, and then update the structure.

Happy coding..

SPOJ ABCLOG

Problem link: SPOJ Problem Set (classical): 17659. Time to get a job

This problem should go to the Riddle section or what-so-ever, not worthy for classical. Just reverse the bit string.

SPOJ POLCONST

Problem link: SPOJ Problem Set (classical): 17707. Constructible Regular Polygons

As stated in the problem description, you are required to utilize this page. As there are only 5 Fermat Primes, you can easily do a pre-processing up to 10⁶ in constant time (2⁵×5) and answer each query in O(1).

SPOJ HOLI

Problem link: SPOJ Problem Set (classical): 9942. Holiday Accommodation

Given a weighted tree, consider there are N people in N nodes. You have to rearrange these N people such that everyone is in a new node, and no node contains more than one people and maximizes the cost. Here cost is the distance travelled for each person.

First observation is, in order to maximize cost, all edges will be used to travel around. So, if we can calculate how many times each edge will be used, we can calculate the cost.

Now for any edge E_i, we can partition the whole tree into two subtrees, if one side has n nodes, the other side will have N - n nodes. Also, note that, min(n, N-n) people will be crossing the edge from each side. Because if more people cross the edge, then by pigeon-hole principle in one side, we will get more people than available node which is not allowed in the problem statement. So, E_i will be used a total of 2 * min(n, N-n) times.

Thus the final result is:

cost = ∑ 2*min(ni, N-ni)*wieght(Ei) | for each edge Ei

Here, n_i is the count for nodes in one side of the edge E_i and N-n_i on the other side. So, we can use simple DFS algorithm to solve the problem. During the traversal, just count number of nodes in the subtree for an edge u⇒v and calculate result for each edge in the process.

SPOJ PRIMIT

Problem link: SPOJ Problem Set (classical): 211. Primitivus recurencis

Given a list of directed edges or features, we have to find the smallest possible sequence of nodes where all the given edges are present without violating the direction.

Although at first this may seem to be a bipartite matching problem or problem related to strongly connected component. But I am not sure whether this particular problem can be solved using any of those techniques. Here we have two things to observe.

1 ⇒ Ordering for each weakly connected component is independent. So, we can add up the results for each weakly connected component as we can arrange them in any order, because they don't have any node in common.

2 ⇒ The minimum length will be the length of eulerian circuit for each component. We can break the circuit at suitable places to make it a chain. Considering [1], we can break the chain anywhere we want. Because other components have nothing to do with it.

So, for each weakly connected component, it boils down to finding minimum number of edges need to be added to make the component an euler circuit. For each component G_i, we can find the minimum number of additional edges by the following formula (Note, the formula will also handle the components which are already an euler circuit).

Number of additional edges = ∑max( cnt(Gi) / 2, 1 ) for all i

Note: if you already have an euler circuit, i.e. cnt(G_i) = 0, then you still need 1 additional term in the sequence. The simplest example is two edges, 1-2 and 2-1. You will need either 1-2-1, or 2-1-2 sequence.

And to find the value of cnt() for a component G_i, we just add the absolute difference of in and out degree count for each node in that component.

cnt(Gi) = ∑|d+(u) - d-(u)| for each u ∈ Gi
Here, d+(u) ⇒ indegree count for u, and d-(u) ⇒ outdegree count for u.

So, the overall solution outline is something like this, first use disjoint set data structure, or commonly known as union-find to mark the components. You can also use simple DFS / BFS as well. Ofcourse we are ignoring direction in this stage as we want to generate weakly connected components. Then for each node, calculate total indegree and outdegree counts. Now for each component, use the above formula to find additional edges x_add, thus the final answer is N + x_add, here x is the given number of edges.