Sealup – editorial – codechef discuss


For a given convex polygon with $N$ vertices, and $M$ types strips, each defined by its length and cost, the goal is to cover all edges of the polygon with the strips minimizing the total cost of used strips. Usd to yen forecast Strips cannot be bend nor cut. Famous quotes about life lessons There is no limit on the number of used strips. Egp to usd QUICK EXPLANATION:

Cover each edge independently in the best way. Rand pound exchange rate graph The minimum cost of covering a single edge can be computed used dynamic programming $dp[d][i]$ capturing the information about the minimum cost of covering distance $d$ using first $i$ types of strips.

Pound to dollar exchange rate graph EXPLANATION:

First of all, let’s observe a few things. 1 usd to rm malaysia Since the polygon is convex, one strip can cover only one edge (of course we place strips alongside the edges, strictly on them, because otherwise no edge can be covered). Exchange rate mexican peso to us dollar Thus, we can compute the minimum cost of covering each edge independently of the other edges, and get the final result by summing these costs. Best quotes about love and life Now the problem is reduced to covering a single segment of length let’s say $Z$ with the given strips. Binary compound Next observation is that there is no point of covering an edge with overlapping strips because any such solution can be transformed to a solution using the same strips without overlaps. Dollar exchange rate to peso today The final fact is that the order of strips covering a single edge does not matter. Futures markets news Approaches for a given subtasks can be based on the above observations. Gender quiz For the first subtasks, one can take a strong advantage of the additional constraints on the input. Gbp usd fx Subtask 1

In the first subtask the polygon has at most $10$ vertices, but more importantly, there is just one type of strips. Us dollar exchange rate in india Thus in order to get the result for a single edge of length $Z$, one can just compute the minimum number of strips required to cover edge of length $Z$, and since there is just one type of strips, let’s say that they are all of a length $L$, then $\lceil Z/L \rceil$ strips are needed to cover such an edge and this is the best we can do. Python append Subtask 2

In the second subtask the polygon has at most $42$ vertices and there are at most $2$ types of strips. Binary to text translator Let’s call these types $A$ and $B$. Exchange rate euro to us dollar today Then any edge of the polygon will be covered by $x$ strips of type $A$ and $y$ strips of type $B$. Text to binary translator Thus the problem of covering an edge of length $Z$ is reduced to finding the best values for $x$ and $y$, such that $x \cdot length_A + y \cdot length_B \geq Z$ and $x \cdot cost_A + y \cdot cost_B$ is minimal possible. Gbp usd exchange rate chart Since this cost function is unimodal, one can use ternary search over $x$ and for a fixed $x$ pick the smallest $y$ sufficient to fulfill the requirement of the length function. Fraction to number calculator Subtask 3

In the last subtask the polygon has at most $2000$ vertices and there are at most $10$ types of strips. Text editor windows 7 One approach here is to precompute results for a more general problem first and then get the answer for each edge using these precomputed results.

Let $dp[d][i]$ be the minimum cost of covering distance of length $\textbf{exactly}$ $d$ using the first $i$ types of strips. Usa stock market futures Since the maximum length of polygon’s edge is around $\sqrt {2 \cdot 10^{12}} < 1.5 \cdot 10^6$, the maximum $d$ for which table $dp$ will be computed can be set to for example $2 \cdot 10^6$. Gbp usd graph Just be aware, there are tricky cases where is the best solution the edge of length $Z$ is covered in such a way that its beginning of length $Z-1$ is covered by some segments and the last part of length $1$ is covered by a single strip of length $10^6$. Euro pound exchange rate today Now, having $dp$ dynamic programming table defined, we can fill it using a similar approach as in the well-known knapsack problem. 1 usd to aud However, one very important thing to notice is that for two distances $d_1 < d_2$, the minimum cost of covering $d_2$ can be smaller than the minimum cost of covering $d_1$ if we place the strips without overlaps. Exchange rate usd to myr So keeping it in mind, we can compute the $dp$ table using the below two-phase approach: for i = 1 to M:

Having $dp$ table computed, we can iterate over all edges of the polygon, and for each one get the minimum cost to cover it from the $dp$ table. 1 usd sgd Just be careful here, because since the length of an edge is a real number $Z$, we want to get the minimum cost of covering distance $\lceil Z \rceil$. Bloomberg markets commodities futures Binary search is a decent approach to avoid computing square root of squared distance, but one can try also some variations of square root implementation to find this value. Put and call options examples For implementation details please refer to author’s and tester’s solutions below. Nyse stock market futures The total time complexity of this method is dominated by the cost of precomputation which is $O(\text{MAX\_DISTANCE} \cdot M)$. Gender differences in leadership Also, one can reduce memory complexity while computing $dp$ table, because when computing $dp[d][i]$ only values for $i-1$ first types of strips are needed. Usd to sgd AUTHOR’S AND TESTER’S SOLUTIONS:

All materials are found on open spaces of a network the Internet as freely extended and laid out exclusively in the fact-finding purposes. If you are what lawful legal owner or a product and against its placing on the given site, inform us and we will immediately remove the given material. The administration of a site does not bear responsibility for actions of the visitors breaking copyrights.