Hunting for patterns. Two Competitive Programming Problems

Math is not routine calculations at all

In this blog post I am going to show you my solutions for two competitive programming problems. Particularly, I emphasize patterns which inspired the solutions in first place. It is hoped you appreciate how elegant those problems are. Please, attempt to read the problems and try to work them out on yourself before reading solutions presented here.

PRECAUTION

Note that eventhough formulas and patterns presented here passed UVa’s online-judge empirical test, That does not guarantee the correctness of them. A rigor formal proof is supposed to be established. Frankly, I could not find a one.


UVa 10940 - Throwing cards away II

The problem is found here. Let’s give it a shot.

First Trial

At each step we discard a card away, and move the new card at the top to the bottom of the deck. That suggests we eliminate half of the deck for one iteration. Consider the following deck

1 x
2 ->
3 x
4 ->
5 x
6 ->
7 x
8 ->

Here, We have a deck of eight cards whereby card number one is at the top of the deck. x indicates the card is to be thrown away, while -> indicates the card to be moved to the bottom. Note that in this case, None of the -> cards are going to be thrown away due to later x. It is easy to see that the result is as follows

2
4
6
8

Similarly to the first iteration, the second one would be

2 x
4 ->
6 x
8 ->

yielding

4
8

Finally, Getting card eight as the answer.

So, On each iteration, we divided the dick by half and still got an even number of cards. It is clear that is attributed to the fact that eight is a power of two. Otherwise, On some iteration we would end-up with an odd number of cards. That iteration is not the last one in which we have the last remaining card. You could see if we have a deck whose cards number is some power of two, Then the answer would be the last card at the bottom. In other words, if our cards number is $n = 2^k$ for some k, Then the correct answer of the problem, i.e the last remaining card after discarding cards and moving them to the bottom, is $2^k$.

More Justification of First Trial’s Observation

Let’s try to take a deeper look at why do we always obtain the last bottom card as the remaining one in case the deck is some power of two, As illustrated earlier. Consider our $2^3 = 8$ deck but represented differently

On First Iteration:

$2^0 \times 1$ x
$2^0 \times 2$ ->
$2^0 \times 3$ x
$2^0 \times 4$ ->
$2^0 \times 5$ x
$2^0 \times 6$ ->
$2^0 \times 7$ x
$2^0 \times 8$ ->

On Second Iteration:

$2^1 \times 1$ x
$2^1 \times 2$ ->
$2^1 \times 3$ x
$2^1 \times 4$ ->

On Third Iteration:

$2^2 \times 1$ x
$2^2 \times 2$ ->

Remarkably, Cards multiplied by an odd number gets thrown away, while cards multiplied by an even number gets moved to the bottom and survives to the next iteration. You could also see that $2^{k_0} \times (2k_1) = 2^{k_0+1}$. The even number increases the power of two by one. That correspoinds with our observation that each iteration’s power is increased by one than its predocessor’s power. Clearly, Continuing in this way ends us up with the greatest power of two in the whole deck, which is also the last bottom card.

Generalizing for None Power of Two

Our solution for the generalized case is in fact an extension of the special case of deck’s whose number is a power of two. Let’s begin from where we ended up.

A Deck of Seven Cards

On First Iteration:

1 x
2 ->
3 x
4 ->
5 x
6 ->
7

Note that I intentionally did not mark card seven. Otherwise, The second iteration would have the first card marked as ->, violating consistency of marking among iterations. In case cards number is odd, as in this case, We prefer to avoid marking the last card, and get the second iteration as

7 x
2 ->
4 x
6 ->

We have got here cards of some power of two. Following our illustrated observation in the previous section, We could conclude the last remaining card is card six.

For eight cards deck, The second iteration was <2, 4, 6, 8>. Removing card eight from the deck resulted in having card seven as a remainder from the first iteration, and shifting <2, 4, 6, 8> one position to the right. As a result, We have card six as the last one in second iteration

For eight cards deck, The last remaining card was eight. For seven cards deck, the last remaining card is six. Removing one card from the eight cards deck yielded the same remaining card but subtracted by two. In other words, $8 - 1$ cards deck yields the last remaining card sevenCardsAnswer = eightCardsAnswer - (2 * removedCards) = 8 - (2 * 1) = 6. Let’s try more trials and see how they relate with the case of eight cards deck

A Deck of Six Cards

On First Iteration:

1 x
2 ->
3 x
4 ->
5 x
6 ->

On Second Iteration:

2 x
4 ->
6 x

So, we end-up with card four. Again, $8 - 2$ cards deck yields the last remaining card sixCardsAnswer = eightCardsAnswer - (2 * removedCards) = 8 - (2 * 2) = 4. Here, unlike the case of seven cards deck, There is no remainder from the first iteration so that we end up with four cards in the second iteration. As two cards are removed from eight cards deck, we have three cards in second iteration rather than four. card six here is in an odd position, so it gets thrown away. The last remaining card is card four. In other words, It seems removing two cards from eight cards deck shifted our <2, 4, 6, 8> a position to the right in addition to removing the last card.

A Deck of Five Cards

On First Iteration:

1 x
2 ->
3 x
4 ->
5

As in the case of seven cards deck, We do not mark card five in the first iteration. Recall the the last card is not marked whenever we have an odd number of cards in an interation.

On Second Iteration:

5 x
2 ->
4 x

Again, $8 - 3$ cards deck yields the last remaining card fiveCardsAnswer = eightCardsAnswer - (2 * removedCards) = 8 - (2 * 3) = 2. It seems removing two cards shifted <2, 4, 6, 8> on position to the right, and removing an additional card shifted it another position to the right but with a remainder, namely card five. So, we get card two as the answer.

Magical Formula

The illustrated reasoning SEEMS to work on not just $2^3 = 8$ but any power of two. For an arbitrary number of deck cards $n$, We find the power of two greater than or equal to $n$, Then compute the answer for $n$ by nCardsAnswer = Power2CardsAnswer - (2 * (Power2Cards - nCards)). So, How do find the power of two equal or greater than $n$? Here is a trick: $2^{ceil(log_2(n))}$. So, The final formula would be

$$2^{ceil(log_2(n))} - (2 \times (2^{ceil(log_2(n))} - n)) = 2 \times (n-2^{ceil(log_2(n))-1})$$

Accepted Source Code on UVa

#include <cstdio>
#include <math.h>

using namespace std;

int main() {
  int n, res;

  scanf("%d", &n);

  res = 2*(n-pow(2, (ceil(log2(n))-1)));
  
  if (n == 1) printf("1\n");
  else printf("%d\n", res);
  
  return 0;
}


UVa 120004 - Bubble Sort

The problem is found here.

Restructuring The Problem, More Conveniently

Before tackling a solution, We need to reformulate the given problem. You might consider this a reduction to a form which is more convenient to solve. The problem states We are given an array of size n whose elements are {1, 2, .., n} and are distinct. That concludes the given array a is a permutation of {1, 2, .., n}. If we listed all these permuations and computed bubbleCounts on each, Then taken their average, That would be the answer to UVa’s problem. Running findSwaps() infinitely is just a fancy way of describing our

Definition: Average bubble counts of all permutations.

Observations

On n = 2,

bubbleCount
1 2 0
2 1 1

average = $\frac{0+1}{2}$ = $\frac{1}{2}$

On n = 3,

bubbleCount
1 2 3 0
1 3 2 1
2 1 3 1
2 3 1 2
3 1 2 2
3 2 1 3

average = $\frac{0+1+1+2+2+6}{6}$ = 2

On n = 4,

bubbleCount
1 2 3 4 0
1 2 4 3 1
1 3 2 4 1
1 3 4 2 2
1 4 2 3 2
1 4 3 2 3
2 1 3 4 1
2 1 4 3 2
2 3 1 4 2
2 3 4 1 3
2 4 1 3 3
2 4 3 1 4
3 1 2 4 2
3 1 4 2 3
3 2 1 4 3
3 2 4 1 4
3 4 1 2 4
3 4 2 1 5
4 1 2 3 3
4 1 3 2 4
4 2 1 3 4
4 2 3 1 5
4 3 1 2 5
4 3 2 1 6

average = $\frac{0+1+1+2+2+3+1+2+2+3+3+4+2+3+3+4+4+5+3+4+4+5+5+6}{24}$ = 3

Symmetry

Consider the case of $n=3$. Notice that the least bubbleCount is the first one accounting for zero, and the greatest bubbleCount is the last one accounting for 3. You could see that for each permutation of bubbleCount 1, There is a corresponding permutation of bubbleCount 2. The sum of 1 and 2 is also 3 !

Note also that the corresponding permutation is exactly like the other one but inversed. For instance permutation <3, 1, 2> is the inversed in order of <2, 1, 3>.

So, we could divide our list of permutations into two halves such that a pair’s sum equals $min(bubbleCount) + max(bubbleCount)$. Clearly, There are a total of $n!$ permutations. The number of those pairs is half of total permutations. Hence, total sum of bubble counts is $\frac{n!}{2} \times (min(bubbleCount) + max(bubbleCount))$. Now we divide that total sum on total number of permutations to get the average of all bubbleCounts. So, The formula is now $\frac{(min(bubbleCount) + max(bubbleCount))}{2}$. Clearly, least bubbleCount is always zero, As we have the permutation which is already sorted. What about the greatest one? The worst case is the permutation sorted inversely. In such case, The first iteration, i.e outer loop, accounts for $(n-1)$ bubbles. The second itertaion accounts for $(n-2)$, and so on untill an iteration accounts for exactly one bubble. So, $max(bubbleCount)$ = (n-1) + (n-2) + .. + 1 = $\frac{n \times (n-1)}{2}$. Hence, Our conjectured formula is

$$\frac{0 + \frac{n(n-1)}{2} }{2} = \frac{n(n-1)}{4}$$

Check this for more information about gaussian’s famous equation.

More Justification on Symmetry

We have shown that least bubbleCount and greatest bubbleCount among all permutations are equal to zero and (n-1) + (n-2) + .. + 1, respectively. Let’s take a deeper and more general look on why we could divide our permutations list into two halfs whereby each pair’s sum is equal to greatest bubbleCount. That pair’s permutations are also inverse of each other.

For the case of $n=3$, Pick up two permutations which are inverse of each other and try to run bubbleSort algorithm on both of them. You shall find for a permutation, The bubble counted on some pair of numbers, is not counted in the other corresponding permutation. For instance, permutation <1, 3, 2> needs one bubble swap in <3, 2> pair. For the permutation’s inverse <2, 3, 1>, There’s no need to bubble swap <2, 3> pair. That saves us one bubble swap out of three which is the maximum bubbleCount. 3 - 1 = 2, The bubbleCount of <2, 3, 1>. The same applies for any two pairs of permutations which are inverse of each other.

Accepted Source Code on UVa

#include <cstdio>

#define ll long long
#define ull unsigned ll

using namespace std;

bool checkIthBit (int n, int i) {
  if( n & (1 << i) )
    return true;
  return false;
}


int main() {
  int t, cou = 1;

  scanf("%d", &t);

  while (cou <= t) {
    int n;
    ull numerator; int denominator;

    scanf("%d", &n);

    numerator = ((ull)n*(ull)(n-1));
    denominator = 4;

    // check if nume is div by 2, and simplify rational form
    for (int i=0; i<2; ++i) {
      if (!checkIthBit(numerator, 0)) {
	numerator = numerator/2;
	denominator = denominator/2;
      }
    }

    if (denominator == 1) printf("Case %d: %llu\n", cou, numerator);
    else printf("Case %d: %llu/%d\n", cou, numerator, denominator);

    cou++;
  }
  return 0;
}

Many of those who do not appreciate math, think of it as a routine where you just follow a systematic order of operations on numbers. If you are one those, I hope this article changed, at least doubted, how you perceive it. There are a whole deep and elegant adventures still awaiting you if you delved more deeply.

Mostafa Touny
Mostafa Touny
Software Engineering Undergrad