## Monday, August 25, 2014

### Planarity Puzzle

I'm obsessed with the untangling of the planar graphs. I'm now able to untangle a 30-vertex planar graph in ~240sec see below (yellow one is the final solution):

## Sunday, August 10, 2014

### Breaking cube into five tets

Picture says it all:

## Sunday, January 05, 2014

### Useful properties of the function $$f(x) = \frac{x}{x-k}$$

The function f(x) = x/(x-k) has some very useful properties -- especially when we restrict x to Integers. Let me start with a simple question about Maxima and Minima of f(x). Applying elementary Calculus may not be useful since f(x) ↦ +∞ as x ↦ k+ (goes to -∞ from the other side and is discontinuous). A quick plot on WolframAlpha reveals the vertical asymptotes [http://wolfr.am/JSXDdV].

Now lets move our attention to the case where x is restricted to Integers. First lets consider the function f(x) when x takes on integers strictly greater than k (i.e. x > k). Notice that f(x) is decreasing the maximum in this case occurs when x = k+1. This gives us the following useful fact:
$$f(n) = \frac{n}{n-k} \leq k+1 \,\,\,\,\,\,\, n \in \{ x\, | \, x > k \vee x \in \cal{I} \}$$

The above inequality is quite handy in your combinatorial analysis tool book. Let me give you a simple application of this inequality by proving that nlog(n) = O(log(n!)) (also recall that log(n!) = O(nlog(n)) which means nlog(n) = Θ(log(n!))).

$$\begin{array}{lcl} && \frac{n}{n-k} \leq k+1 \,\,\,\,\,\, k \geq 1\\ && \\ \rightarrow && n \leq (n-k)\times (k+1) \\ \rightarrow && n < (n-k) \times (k+k) = 2 (n-k)\times k && \\ &&\\ \rightarrow && n < 2(n-1)\times 1\\ \rightarrow && n < 2(n-2)\times 2\\ \rightarrow && n < 2(n-3)\times 3\\ && \ldots \\ \rightarrow && n < 2(n-(n-1))\times (n-1) = 2(1\times (n-1)\\ &&\\ &&\mbox{multiplying all the above n-1 inequalities you get the following} \\ &&\\ \rightarrow&& n^{n-1} < \displaystyle\Pi_{k=1}^{n-1} 2(n-k)\times k = 2^n ((n-1)!)^2 \\ \rightarrow&& n^n < 2^n (n!)^2 \\ \rightarrow&& \log(n^n) < \log(2^n(n!)^2) = 2\log(n!) + n \\ \rightarrow&& n\log(n) < 2\log(n!) + n < 2\log(n!) + \log(n!) = 3\log(n!) \\ \rightarrow&& n\log(n) = O(\log(n!)) \,\, \Box\\ \end{array}$$

## Saturday, November 30, 2013

### Getting around with integer overflows during Combinatorial counting

Many closed form expressions characterizing the number of combinatorial objects involve manipulation of raising and falling factorials (e.g. $$\displaystyle\Pi_{i=1}^{k} (n-i+1)$$). These quantities accumulate very fast we need to use extreme caution during computation. Lets take the example in which we want to compute $${n\choose k} = \frac{n\times (n-1)\ldots (n-k+1)}{k\times k-1\times \ldots 1}$$ which is the ratio of two falling factorials which is always an integer. A naive algorithm which computes numerator and denominator will easily overflow even though the actual value is well within the limits of an integer. If $$n_{max}$$ is the largest value represented by the underlying data-type, the naive algorithm would overflow when: $$n\geq \sqrt[k]{n_{max}}$$. One way to fix this issue is to maintain the ratio implicitly as two integers and knocking off the GCD when ever possible. This reduces the overflow during computation see below for a simple example. Thanks to my friend Chittu for discussing this over lunch.

  public static long computeNchooseK(int n, int k){
assert n >=1 && k<=n : "Invalid n and k to compute";
long num=1, denom=1, factor;
do{
factor = gcd(n,k);
num *= (n--/factor); denom *= (k--/factor);
factor = gcd(num, denom);
num /= factor; denom /= factor;
}while(k>=1);
assert denom == 1 : "Something went wrong in computing NchooseK";
return num;
}


## Saturday, September 21, 2013

### $\sum_{k=1}^{n} H_k = (n+1)(H_{n+1} - 1)$

$$\frac{1}{1-z} = 1 + z + z^2 + z^3 \dots \\ \int \frac{1}{1-z}dz = \frac{z}{1} + \frac{z^2}{2} + \frac{z^3}{3} \dots = \sum_{k=1}^{\infty} \frac{z}{k} z^k = -\ln(1-z)\\ \\ H_n = \sum_{k=1}^{n} \frac{1}{k} \hspace{0.5in} \mbox{By definition, also notice they are partial sums}\\ \rightarrow \mbox{O.G.F of H_n is}\hspace{0.02in} \frac{1}{1-z}\times (-\ln(1-z)) = \frac{1}{1-z}\ln(\frac{1}{1-z}) \hspace{0.5in}\mbox{from our previous result on partial sums}\\ \rightarrow \frac{1}{1-z}\ln(\frac{1}{1-z}) = H_1\times z + H_2\times z^2 + H_3\times z^3 \ldots \hspace{0.5in} \mbox{(1)}\\ \rightarrow \frac{d}{dz}(\frac{1}{1-z}\ln(\frac{1}{1-z})) = \frac{1}{(1-z)^2}\ln(\frac{1}{1-z}) + \frac{1}{(1-z)^2} = H_1 + 2H_2\times z + 3H_3\times z^2 \ldots \hspace{0.5in}\mbox{(2)}\\ \mbox{Notice that}\hspace{0.02in} \frac{1}{(1-z)^2} = 1 + 2z + 3z^2 \dots \hspace{0.5in} \mbox{(3)}\\ \rightarrow\mbox{(2) - (3)} = \frac{1}{(1-z)^2}\ln(\frac{1}{(1-z)}) = (H_1-1) + (2H_2-2)\times z + (3H_3 -3) \times z^2 \ldots \\ \rightarrow \frac{1}{(1-z)^2}\ln(\frac{1}{(1-z)}) \hspace{0.02in} \mbox{is the O.G.F of sequence} \hspace{0.02in} (n+1)H_{n+1} - (n+1) = (n+1)(H_{n+1}-1) \hspace{0.5in} \mbox{(4)}\\ \\ \mbox{Applying the result of the partial sums to (1) the O.G.F of sequence}\hspace{0.02in} \sum_{k=1}^{n} H_n \hspace{0.02in} \mbox{is} \hspace{0.02in} \frac{1}{(1-z)}\times {\frac{1}{(1-z)}\ln(\frac{1}{1-z})} = \frac{1}{(1-z)^2}\ln(\frac{1}{(1-z)}) \hspace{0.5in} \mbox{(5)}\\ \mbox{From (4) and (5) the generating functions of both the sequences are same, hence the corresponding terms must be the same}$$

### Simple proof for the O.G.F of partial sums

Let $A(z)$ be the O.G.F of the sequence $S = \{a_0, a_1, a_2 \ldots \}$, then we can derive the O.G.F of the sequence $b_n = \sum_{i=0}^{n} a_i$ (i.e. $B = \{a_0, a_0+a_1, a_0+a_1+a_2 \ldots \}$) as follows.

Consider the sequence $S^1 = \{0 , a_0, a_1, a_2, a_3 \ldots \}$ then O.G.F corresponding to $S^1$ is $0\times z^0 + a_0 \times z + a_1 \times z^2 + a_2 \times z^3 \ldots = z\times A(z)$. Similarly the O.G.F corresponding to the sequence $S^2 = \{0, 0, a_0, a_1, a_3 \ldots \}$ is $z^2 \times A(z)$. Now notice that if we add corresponding terms in the sequences $S^1 , S^2, S^3 \ldots$ then this
results in the sequence $B$. Hence the generating function corresponding to $B$ is $A(z) + z\times A(z) + z^2\times A(z) + z^3\times A(z) \ldots = A(z)\times(1 + z + z^2 + z^3 \ldots) = \frac{A(z)}{1-z}$ $\Box$.

## Sunday, June 09, 2013

### An iterative method to compute the integer linear combination of Greatest Common Divisor

Euclid's iterative method to compute the GCD is very well known. You might have also heard that $GCD(a,b)$ can be expressed as integer linear combination of $a,b$ (i.e. $GCD(a,b) = \alpha\times a + \beta \times b \,\, \alpha,\beta \in \mathcal{I}$). The following is an iterative method -- I wrote on a flight between AUS -> DFW just for fun -- which computes this integer linear combination. Runs in $O(\log(\max\{a,b\}))$ time.

template
static inline Unit gcd_integer_linear_combination(const Unit& a, const Unit& b, Unit& alpha, Unit& beta ){
Unit divd = a, divs = b, remd=0, quot;
Unit a_divd[2] = {1,0}; //linear combination of the divident//
Unit a_divs[2] = {0,1}, temp[2] = {0,0}; //linear combination of the divisor//
bool b_flip = false;
if(divd < divs){
divd = b; divs = a;
b_flip = true;
}
//Invariant's: divd = a_divd[0]*a + a_divd[1]*b //
//      divs = a_divs[0]*a + a_divs[1]*b //
while(divs){
remd = divd%divs; quot = (divd-remd)/divs;
divd = divs; divs = remd;
temp[0] = a_divs[0]; temp[1] = a_divs[1];
a_divs[0] = a_divd[0] - (quot * a_divs[0]);
a_divs[1] = a_divd[1] - (quot * a_divs[1]);
a_divd[0] = temp[0]; a_divd[1] = temp[1];
}
if(!b_flip){
alpha = a_divd[0]; beta = a_divd[1];
}else{
alpha = a_divd[1]; beta = a_divd[0];
}
return divd;
}


The following are some examples $GCD(a,b) = (\alpha)*a + (\beta)*b$.
PASSED: 2 = (1613)*42 + (-8)*8468
PASSED: 1 = (-1684)*6335 + (1641)*6501
PASSED: 5 = (344)*9170 + (-551)*5725
PASSED: 1 = (-1949)*1479 + (308)*9359
PASSED: 1 = (967)*6963 + (-1508)*4465
PASSED: 2 = (858)*5706 + (-601)*8146
PASSED: 6 = (181)*3282 + (-87)*6828
PASSED: 2 = (121)*9962 + (-2450)*492
PASSED: 1 = (-596)*2996 + (919)*1943
PASSED: 1 = (-2348)*4828 + (2085)*5437
PASSED: 1 = (283)*2392 + (-147)*4605
PASSED: 1 = (-61)*3903 + (1546)*154
PASSED: 1 = (122)*293 + (-15)*2383
PASSED: 1 = (-1528)*7422 + (1301)*8717
PASSED: 1 = (615)*9719 + (-604)*9896
PASSED: 1 = (-304)*5448 + (959)*1727
PASSED: 1 = (-139)*4772 + (431)*1539
PASSED: 1 = (493)*1870 + (-93)*9913
PASSED: 4 = (628)*5668 + (-565)*6300
PASSED: 1 = (3461)*7036 + (-2461)*9895
PASSED: 4 = (-60)*8704 + (137)*3812
PASSED: 1 = (77)*1323 + (-305)*334
PASSED: 3 = (662)*7674 + (-1089)*4665
PASSED: 2 = (3)*5142 + (-2)*7712
PASSED: 1 = (1840)*8254 + (-2211)*6869
PASSED: 1 = (2027)*5548 + (-1471)*7645
PASSED: 1 = (929)*2663 + (-897)*2758
PASSED: 2 = (-301)*38 + (4)*2860
PASSED: 2 = (-823)*8724 + (737)*9742
PASSED: 1 = (-3)*7530 + (29)*779
PASSED: 1 = (-1499)*2317 + (1144)*3036
PASSED: 1 = (662)*2191 + (-787)*1843
PASSED: 1 = (10)*289 + (-27)*107
PASSED: 1 = (4289)*9041 + (-4336)*8943


## Friday, October 05, 2012

### Friday fun with a dynamic program.

Its been almost an year I wrote my personal blog. I missed so many fun things I used to do as student. Well I'm a pursuit of relentless improvement. Today I just wanted to write a dynamic program --  which I have missed for a while -- so I randomly pick one of the hardest DIV-1 problems and got the dynamic program working. This is a very interesting dynamic program -- a two level one. You can read the problem -- because I cannot' post here-- from this link http://community.topcoder.com/stat?c=problem_statement&pm=1996&rd=4710 . Following is my code which passed all the 88 system tests.

/*
* Solution to  problem: http://community.topcoder.com/stat?c=problem_statement&pm=1996&rd=4710 (SRM-178, DIV-1, hard)
*
* Vamsi K. Kundeti 10/05/2012
*
**/
#include
#include
#include
#include
#include

using namespace std;
#define MAX_ROWS 50
#define MAX_STROKES 3000
class MinPaint {
private:
/*Optimal number of mismatches to color a row 'i' till 'j' with a
*stoke ending black using 'k' strokes : 'm_aB[j%2][i][k]'
*/
size_t m_aB[2][MAX_ROWS][MAX_STROKES];
size_t m_aW[2][MAX_ROWS][MAX_STROKES];
/*Optimal number of mismatches in the image upto row 'i' using 'k'
*strokes*/
int m_aOPT[2][3001];
//optimal value is stored in m_aB[0][i][j]
vector m_aImg;

public:
MinPaint(): m_aB(), m_aW(), m_aImg() {}

void initRow(size_t idx, size_t maxStrokes){
for(size_t j=0; j<=maxStrokes; j++){
m_aB[0][idx][j] = !j ? 1 : m_aImg[idx][0] == 'W';
m_aW[0][idx][j] = !j ? 1 : m_aImg[idx][0] == 'B';
}
}
size_t findMin(size_t a, size_t b){
return a < b ? a : b;
}
void dynamicProgramPerRow(size_t idx, size_t maxStrokes){
initRow(idx, maxStrokes);
for(size_t i=1; i < m_aImg[idx].size(); i++){
for(size_t j=0; j <= maxStrokes; j++){
if(m_aImg[idx][i] == 'W'){
m_aW[i%2][idx][j] = !j ? i+1 :
findMin(m_aW[(i-1)%2][idx][j], m_aB[(i-1)%2][idx][j-1]);
m_aB[i%2][idx][j] = !j ? i+1 :
findMin(m_aB[(i-1)%2][idx][j]+1, m_aW[(i-1)%2][idx][j-1]+1);
}else{
m_aB[i%2][idx][j] = !j ? i+1 :
findMin(m_aB[(i-1)%2][idx][j], m_aW[(i-1)%2][idx][j-1]);
m_aW[i%2][idx][j] = !j ? i+1 :
findMin(m_aW[(i-1)%2][idx][j]+1, m_aB[(i-1)%2][idx][j-1]+1);
}
}
}
//choose the min between m_aB and m_aW
size_t nidx = m_aImg[idx].size()-1;
for(size_t i=0; i<=maxStrokes; i++){
m_aB[nidx%2][idx][i] = findMin(m_aB[nidx%2][idx][i],
m_aW[nidx%2][idx][i]);
}
}

int leastBad(const vector &Img, const int &maxStrokes){
m_aImg = Img;
for(size_t i=0; i< Img.size(); i++){
dynamicProgramPerRow(i, (size_t)maxStrokes);
}

//now find the optimal across rows//
for(int i=0; i<=maxStrokes; i++){
m_aOPT[0][i] = m_aB[(Img[0].size()-1)%2][0][i];
}

for(size_t j=1; j<Img.size(); j++){
/*choose number of strokes from [1, maxStrokes-j] to paint row 'j'*/
size_t rsize_idx = m_aImg[j].size()-1;
for(int usedStrokes=0;  usedStrokes<=maxStrokes; usedStrokes++){
int min = numeric_limits::max();
int current_min;
int min_k=0;
for(int k=0;  k<=usedStrokes; k++){
//use 'k' strokes to color current row 'j' and 'usedStrokes-j' for the
//array below 'j'.
current_min =
m_aOPT[(j-1)%2][usedStrokes-k] + (m_aB[(rsize_idx)%2][j][k]);
if(current_min < min){
min = current_min;
min_k = k;
}

}
m_aOPT[j%2][usedStrokes] = min;
}
}
return m_aOPT[(Img.size()-1)%2][maxStrokes];
}

size_t OptPerRow(size_t idx, size_t maxStrokes){
return m_aB[(m_aImg[idx].size()-1)%2][idx][maxStrokes-1];
}
};
/*INPUT FILE FORMAT:
*
*
* X Y
* string1
* string2
* ...
* ...
* ...
* stringX
*
*X is the number of rows
*Y is the maximum number of strokes
**/
int main(int argc, char **argv){
char buf[4096];
vector input;
size_t imgSize, maxStrokes;
MinPaint X;

while(scanf("%lu%lu", &imgSize, &maxStrokes)==2){
input.clear();
while(imgSize-- && scanf("%s", &buf) == 1){
input.push_back(string(buf));
}