Thursday, August 20, 2015

Contest Analysis: IUT 6th National ICT Fest 2014, UVa 12830 - 12839

I participated in this contest last year. Our team "NSU Shinobis" managed to solve 5 and placed in top 10 ( failed to find the rank list ). The IUT 7th ICT National Fest 2015 is going to be held next month, so I thought I would try to up solve the remaining problems. I tried to compile detailed hints for the problems here.

UVa 12830 - A Football Stadium

Complexity: $O(N^3)$
Category: Loop, DP, Kadane's Algorithm

Given $N$ points, we have to find the largest area of the rectangle which can fit such that there is no point inside the rectangle. Points on boundaries of the rectangle are allowed.

Now imagine a rectangle which has no point inside it and also no point on its boundary. Now focus on the top boundary of that rectangle. Since there is no point on that boundary, we can extend it up until it hits a point or limit. Same can be said for the bottom, left and right boundary. That is, the largest rectangle will have its boundary aligned with the limits or points.

Now let us fix the top and lower boundary of the rectangle. How many possible values can they take? Each boundary can take values of $y$-coordinate from one of the $N$ points or the limits $0$ and $W$. Using two nested loops, we can fix them.

Next we need to fix the left and right boundary. Lets us sort the points according to $x$ first and then according to $y$. Next we iterate over the points. Initially set the left boundary of rectangle aligned to $y$-axis, that is aligned with the left boundary of Sand Kingdom.

Now, for each point, if the $y$ coordinate of the point is above or below or on the boundary we fixed then we ignore them, cause they don't cause any problem. But if they fall inside, then we need to process this point. We put the right boundary aligned to the $x$ coordinate of this point and calculate this rectangle.

But it's not over. We shift the left boundary over to current right boundary and continue process remaining points.

This is a classical problem of Kadane's Algorithm. With two nested loops to fix upper and lower boundary, and another loop inside iterating over $N$ points makes the complexity $O(N^3)$.

$O(N^2)$ solution is possible, by constructing a histogram for each row ( $O(N)$ ) and the calculating area of histogram in $O(N)$. But that's not necessary here.


UVa 12831 - Bob the Builder

Complexity: $O(\sqrt{V}E)$
Category: Max Flow, Bipartite Matching, Dinic's Algorithm, Path Cover

Another classical problem, though we had a hard time solving this one. Mainly cause we didn't realize this was a path cover problem which can be solved using Max Flow.

Take each number provided and generate all child. Consider each number a node and if it possible to generate $B$ from $A$, then add a directed edge from $A$ to $B$. When you are done, you will have DAG.

Now split all the nodes in two. We are going to create a bipartite graph now. On the left side, we will have all the original nodes and on the right side we will have the copy we got by splitting the nodes. Now, for each edge between $A$ and $B$, add edge in the bipartite graph from $A$ on the left side to $B$ on the right side.

Find the matching ( or maximum flow ) of this graph by running  Dinic's Algorithm. The answer will be $\text{Total Nodes} - \text{Matching}$.


UVa 12832 - Chicken Lover

Complexity: $O(M)$
Category: Expected Value, Probability, DP,  Combination

If a shop can make $N$ different items, and in a single day prepares $K$ items from those $N$ items, then how many different sets of menus ($total$) can they make? $total = C^N_K$. Now, if they decide to make chicken that day for sure, how many sets of menus ( $chicken$ ) can they make now? $chicken = C^{N-1}_{K-1}$. So what is the probability $P$ that if I visit a shop I will get to eat chicken? $P = \frac{chicken}{total}$. And what is the probability that I will not eat chicken? $Q = 1- P$.

So now I know the probability of eating chicken for each shop. How do we find the expected value? We will find it using dynamic programming.

The states of dp only be the shop number. $dp(x)$ will give me the expected number of chicken that I can eat if I visit all shops starting from $x$. Our result will be $dp(1)$.

At each state, I have $P_i$ probability that I will eat chicken and $Q$ probability that I will not. So result for each state will be:

$dp ( pos ) = P \times ( 1 + dp ( pos + 1 ) + Q \times dp ( pos + 1 )$
$dp ( pos ) = P + P \times dp ( pos + 1 ) + Q \times dp ( pos + 1 ) $
$dp ( pos ) = P + dp ( pos + 1 ) \times ( P + Q )$
$dp ( pos ) = P + dp ( pos + 1 )$.

In order to print the result in $\frac{A}{B}$ form, we need to avoid $double$ and use integer arithmetic in all places. I implemented my own fraction class for that purpose.


UVa 12833 - Daily Potato

Complexity: $O(26 \times N)$
Category: String, Manacher's Algorithm

For each query, we have to count the number of palindromes which are substring of $S$, starts and ends with given $C$ and has exactly $X$ occurrences of $C$ in it. Since it deals with palindrome, perhaps it has something to do with Manacher's Algorithm?

With Manacher's Algorithm, we can find out the maximum length of palindrome in $O(N)$. But what's more, we can actually generate an array $M$ which gives us, for each center in the extended string $E$, ( $aba$ when extended becomes $\text{^#a#b#a#\$} $ ) the maximum length of palindrome with that particular center. How can we use this knowledge to solve our problem?

Let us consider the character $'a'$ only. We can easily extend it for other characters by repeating the whole process $26$ times.

Suppose we are working with center $x$. It has a palindrom from it with length $y$. Therfore, in extended string of manacher, the palindrome starts from $x-y$ and ends in $x+y$. Now, how many times does $'a'$ occurs in this palindrome? Using Cumulative Sum it is possible to answer in $O(1)$. Let that occurance be $f = \text{# of times 'a' occurs in palindrome with center x}$. Let us mark this in another array $freq$. That is we mark it like $\text{freq[f]++}$, meaning we have $freq[f]$ palindromes where $'a'$ occurs $f$ times. But wait, what if the palindrome does not start and end with $'a'$? Simple, we keep on throwing the leading and trailing character until it starts and ends with $'a'$ and it will still have $f$ occurances of $'a'$ in it.

So we repeat this for all possible center. Now, if the query is find number of palindromes that starts and ends with $'a'$ and $'a'$ occurs exactly $X$ times, how do we solve it?

First of all, our result will contain $res = freq[X]$ in it. What's more, our result will also contain $res \text{+=}  freq[X+2] + freq[X+4] + freq[X+6] + ...$. Why is that? Take any palindrome that contains more than $X$ occurances of $'a'$. Since they start and end with $'a'$, we can just throw them out of that palindrome and reduce the occurance of $'a'$ in it by $2$. After that, we keep on trimming down head and tail of that palindrome until we reach $'a'$ again. That is, a palindrome with $Y$ occurrences of $'a'$ can be trimmed down to palindrome with $Y-2$, $Y-4$, $Y-6$, $...$ occurrences of $'a'$.

Instead of getting the result $res = freq[X] + freq[X+2] + freq[X+4] + ... $ we can just use cumulative sum again to find it in $O(1)$ here. Just find cumulative sum of alternate terms.


UVa 12834 - Extreme Terror

Complexity: $O(N\: logN )$
Category: Adhoc

This was the easiest problem. Each shop gives me some money ($income$) and then for that shop I have to give Godfather some cut ($expense$). So for each shop I get $profit = income - expense$. So I calculate profit for each shop and then sort them. I can now skip $K$ shops. I will of course skip shops for which profit is negative as long as I am allowed to skip.


UVa 12835 - Fitting Pipes Again

Complexity: $O(N!\: N^2)$
Category: Geometry, Trigonometry, Permutation, Packing Problem

It's a packing problem. At first glance it seems like a tough problem but it's not. Let us first define few variables first.

This is the polygon. Each polygon has two horizontal lines through it. We will call them low and top line. Each side of the polygon has length of $x$. The radius of the polygon is $r = \frac{x}{2} + y$.

We are given height of each polygon. But first we will need to calculate the value of $x$ and $y$. We can find their values using trigonometry.

$y^2 + y^2 = x^2$
$2y^2 = x^2$
$x = \sqrt{2y^2}$

We also know that $h = y + x + y$. From that we can derive:

$y + x + y = h$
$2y+x = h$
$2y + \sqrt{2y^2} = h$
$2y + y\sqrt{2} = h$
$y( 2 + sqrt{2} ) = h$
$y = \frac{h}{2} +\sqrt{2}$

With the above, we can find the value of $y = \frac{h}{2} +\sqrt{2}$ and $x = \sqrt{2y^2}$ But why did we find their values?

Okay, what happens when we put to polygon side by side? In order to solve this problem we need to be able to find the distance between the center of two polygon $A$ and $B$.

Now if two polygons are of same height and they are placed side by side, then the difference between their centers will be $d = r_a + r_b$. What happens when two polygons of arbitrary height come beside each other?

$3$ things can happen when two polygon $A$ and $B$ are placed side by side. $A$ is on left of $B$.

  1. Height of bottom line of $A$ is higher than height of top line of  $B$. In this case, $A$ is so big that $B$ slides inside the radius of $A$.
  2. Height of bottom line of $B$ is higher than height of top line of $A$. In this case $A$ is so small that it slides inside the radius of $B$.
  3. Nobody can slide inside each others radius. So $d = r_a + r_b$.
We need to calculate the value of $d$ for step $1$ and $2$. That can also be easily done using trigonometry. Try to draw some diagram yourself.

So we used $x$ and $y$ to find the value of $d$ between two polygon. How do we use this to find the minimum width of box? 

Suppose we are trying to pack the polygons in some order. Which order? We don't know which order will give us minimum width so we try them all. There will be $N!$ order. 

So for each order, what will be the minimum of width. First lets take the first polygon, and put it inside the empty box such that it touches the left side of the box. We will calculate the center of each polygon relative to the left side of the box. The first box will have center at $r_0$.

Now take the second box. First imagine there is no polygon in the box. There where will be the center of the second polygon? Ar $r_1$. Now, since there is a polygon, we will try to put it beside that one. Where will be the center now? $r_0 + d$. We will take the maximum possible center.
Repeat this for the third polygon. Empty box, beside first polygon, beside second polygon. Take the maximum again. Repeat this for all polygon.

From all polygon, find the one with maximum center position. Add radius of that polygon to it's center to find the width.

Take the minimum width from all permutation. 

UVa 12836 - Gain Battle Power

Complexity: $O(N^2)$
Category: Interval DP, LIS, Knuth Optimization

First we need to calculate the power of each deatheater. We can do that by calculating LIS from both direction and then adding them.

Once we calculate the power, we run an interval dp between $1$ and $N$. The dp will have states $start$ and $end$. Inside the dp a loop between start and end will run, choosing different cut sections. We will take the one with minimum value.

But this results in $O(N^3)$ solution. Using Knuth's optimization we can reduce it to $O(N^2)$. 

UVa 12837 - Hasmot Ali Professor

Complexity: $O(100 \times |S|^2 )$
Category: String, Trie, Data Structure

 We will create two trie trees. The first one will contain all the queries. Take a query and concatanate them in a special way. Take the first string of the query and add a $\text{'#'}$ and then reverse the second string of the query and attach it to result. That is, if we have $abc$ and $pqr$ as query, then special string will be $\text{abc#rqp}$. Insert all special strings for each query in the first tries.

Now, let us process the main string $S$. We will take each of its suffix and insert into the second trie. Now, when inserting the suffixes, each time a new node is created, we can say that we found a unique substring. 

Each time we find a unique substring we will process it further. Take the unique substring, and using brute force generate all possible special strings ( the one we made using query strings ) with the first and last characters of the unique string. We don't need to take more than $10$ characters from each end.

For each of those special string we made from unique substring, we will query the first trie with it and find the node where it ends. We will add one to that node number in a global array.

Once we finish processing all nodes in the second trie, we will simply traverse the first trie according to each query and print the result found in that node.

UVa 12838 - Identity Redemption

Complexity: Unknown
Category: Matching on General Graph

I didn't manage to solve this yet. It looks like matching on general graph.

UVa 12839 - Judge in Queue

Complexity: $O(N\:logN)$
Category: Data Structure, Heap

We want to minimize the waiting time for each person. So first we will sort the people in descending order. Next we will create a priority queue, which will contain all the service center along with the information about when that service center will be free. Initially all service centers are free.

Now, we take each person and take out the service center that will get free at the earliest time. The person had to originally wait  $x$ minutes and it took $y$ minutes for the service to get free again. So the person had to wait $x+y$ minutes. We insert the service again inside the heap and update its free time by the time it takes to serve one person.

Repeat and keep track of the highest time a person had to wait.

I hope the details are clear enough for everyone to understand. Let me know if you find any mistakes.

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