  # Résolution de problèmes algorithmiques

café Fluctuat Nec Mergitur, July 9th, 2016, 9:00 To train for the programming contest

• read many problems, to train quick translation into formal statement with all constraints
• get to know your classical algorithms
• know how to implement them, and also how to implement algorithmic easy problems

## Hints

### Teams

There are n ways to choose the team leader and then $2^{n-1}$ to choose the remaining team members among the n-1 other persons. So you have to output $n\cdot 2^{n-1}$. Use fast exponentiation for that.

### Mirror Clock

The main “difficulty” is to print integers with 2 digits. For this use a format string

### Travel Company

Basically you are given a directed graph, where every arc $(A_i,B_i)$ has the weight $I_i-P*E_i$. We need to detect a negative cycle. Use Bellman-Ford or even Floyd-Warshall.

### Fun with Strings

Basically you have to solve a linear equation system with two variables and two equations. The coefficients are Fibonacci numbers. In order to compute those numbers in short time use fast exponentiation on the matrix

$M = \left( \begin{array}{cc}1&1\\1&0\end{array}\right)$ which gives $M(a,b) = (a+b,a)$.

### Kisu Pari Na 2

Funny text, it starts with unrelated topic, so you cannot see immediately that this is a graph problem.

Given a forest and an integer K we want to know what is the length of the shortest path that can visit K distinct vertices. Clearly each tree forms an independent subproblem. It is not very clear from the examples whether a path can traverse a same vertex several times. But querying the uDebug website tells us that such paths have to be considered as well. If you think about, for a given tree the first vertex is covered for free, the next ones need a single or two edges. Let P be the longest path in the tree. Its length is called the diameter of the tree. Then if K≤len(P), the answer is just K-1. For every additional vertex two edges are necessary to cover it from P, hence the answer is min(K, len(P)) -1 + 2 max(0, K-len(P)) if K is not more than the number of vertices in the tree, otherwise the answer is ∞.

To answer a query K, simply choose the tree with the largest diameter among all trees of size at least K. This tree can be found using binary search and some preprocessing (sorting).

### Pizza cutting

The answer is n(n+1)/2. Simply think of an arrangement of lines such that no lines are parallel and no 3 lines intersect at the same point. Let all intersection points be inside the pizza. When you remove a line among the n lines, then it will merge n+1 pairs of pizza pieces. If you continue removing you reach the claimed bound.

### What is the Card?

Boring simulation. But go ahead and code it quickly.