## Posts Tagged ‘**Knuth**’

## Knuth–Morris–Pratt Algorithm (KMP)

Knuth–Morris–Pratt algorithm is the most popular linear time algorithm for string matching. It is little difficult to understand and debug in real time contests. So most programmer’s have a precoded KMP in their kitty.

To understand the algorithm, you can either read it from Introduction to Algorithms (CLRS) or from the wikipedia page. Here’s a sample C++ code.

void preKMP(string pattern, int f[]) { int m = pattern.length(),k; f[0] = -1; for (int i = 1; i<m; i++) { k = f[i-1]; while (k>=0) { if (pattern[k]==pattern[i-1]) break; else k = f[k]; } f[i] = k + 1; } } bool KMP(string pattern, string target) { int m = pattern.length(); int n = target.length(); int f[m]; preKMP(pattern, f); int i = 0; int k = 0; while (i<n) { if (k==-1) { i++; k = 0; } else if (target[i]==pattern[k]) { i++; k++; if (k==m) return 1; } else k=f[k]; } return 0; }

NJOY!

-fR0DDY

## Evaluation of Powers

This is a very interesting problem with a lots of history. Anyways we will not wonder into it. We shall see how fast can we calculate *x*^{n}, given x and n where n is a positive integer. The brute force method would be to run a loop from 2 to n and calculate in n-1 steps. We will discuss three methods to do it quickly.

**Binary Method**

This the most common method used in programs today. It is also called the Dynamic Programming method. Here is an C++ implementation :

int binary(int x,int n) { if (n==1) return x; if (n%2==0) { int t=binary(x,n/2); return t*t; } else return x*binary(x,n-1); }

But does this method give the minimum number of multiplications. The answer is **NO**.

The smallest counterexample is n=15. The binary method takes six multiplications but the best method can do it in five multiplications. We can calculate y = *x*^{3} in two multiplications and *x*^{15} = *y*^{5} in three more, needing only five multiplications in total.

**Factor Method**

Let n=pq where p is smallest prime factor of n. We can calculate n by first calculating p and then raising this quantity to q-th power. If n is prime we calculate *x*^{n-1} and multiply by x. For example to calculate *x*^{55}, we first calculate y = *x*^{5} = *x*^{4}x = *( x^{2})*

^{2}x; then we form

*y*

^{11}=

*y*

^{10}y =

*(*

*y*^{2})^{5}y. The whole process takes eight multiplications.

But does this method give the minimum number of multiplications. The answer is

**NO**.

The smallest counterexample is n=33. The factor method takes seven multiplications but the best method(binary method) can do it in six multiplications.

**The “Power Tree Method”**

Lets first see the power tree.

Figure above shows the first few levels of the “power tree.” The (k + 1)-st level of this tree is defined as follows, assuming that the first k levels have been constructed: Take each node n of the lath level, from left to right in turn, and attach below it the nodes

n + 1, n + *a*_{1}, n + *a*_{2}, . . . , n + *a*_{k-1} = 2n

(in this order), where 1, *a*_{1}, *a*_{2}, . . . , *a*_{2} is the path from the root of the tree to n;but discard any node that duplicates a number that has already appeared in the tree. The C++ implementation of the above can be :

/** LINKU[j], LINKR[j] for 0 <= j <= 2^r; point upwards and to the right, respectively, if j is a number in the tree **/ int LINKU[2050]={0},k=0,LINKR[2050]={0},q,s,nm,m,i,n; LINKR[0]=1; LINKR[1]=0; //11 being number of level while (k < 11) { n=LINKR[0];m=0; do { q=0,s=n; do { if (LINKU[n+s]==0) { if (q==0) nm=n+s; LINKR[n+s]=q; LINKU[n+s]=n; q=n+s; } s=LINKU[s]; }while (s!=0); if (q!=0) { LINKR[m]=q; m=nm; } n=LINKR[n]; }while (n!=0); LINKR[m]=0; k=k+1; }

But does this method give the minimum number of multiplications. The answer is **NO**.

The smallest counterexample is n=77.Other examples are 154 and 233.

**Some Analysis**

- The first case for which the power tree is superior to both the binary method and the factor method is 23.
- The first case for which the factor method beats the power tree method is 19879 = 103*193. Such cases are rare, only 6 for n<
*10*.^{5} - Power tree never gives more multiplications for the computation of
*x*^{n}than the binary method.

If You are not interested in Maths stop reading here.

Let l(n) denote the minimum number of multiplications for a given number n.

λ(n) = floor(lg n), where floor(x) is the largest integer not greater than x.

v(n) = number of 1s in the binary representation of n.

They follow the following recurrence relations :

λ(1)=0, λ(2n) = λ(2n+1) = λ(n) + 1;

v(1)=1, v(2n)=v(n), v(2n+1) = v(n)+1.

The binary method requires λ(n) + v(n) – 1 steps.

So we have following theorems

- l(n)<= λ(n) + v(n) -1.
- l(n)>= ceiling(lg n), where ceiling(x) is the smallest integer not less than x.
- l(
*2*^{A}) = A. - l(
*2*^{A}+*2*^{B}) = A+1 if A>B. - l(
*2*^{A}+*2*^{B}+*2*^{C}) = A+2 if A>B>C.

**Conjecture**

- l(2n) = l(n) +1; It fails since l(191) = l(382) = 11.
- The smallest four values for which l(2n) = l(n) are n = 191,701,743,1111.

**Conclusion**

The table of l(n) may be prepared for 2<=n<=1000 by using the formula l(n) = min(l(n-1)+1 , *l*_{n}) – *δ*_{n},

where *l*_{n} = ∞ if n is prime, otherwise *l*_{n} = l(p)+l(n/p) if p is the smallest prime dividing n; and *δ*_{n} = 1 for n in Table below and 0 otherwise.

23 | 163 | 229 | 319 | 371 | 413 | 453 | 553 | 599 | 645 | 707 | 741 | 813 | 849 | 903 |

43 | 165 | 233 | 323 | 373 | 419 | 455 | 557 | 611 | 659 | 709 | 749 | 825 | 863 | 905 |

59 | 179 | 281 | 347 | 377 | 421 | 457 | 561 | 619 | 667 | 711 | 759 | 835 | 869 | 923 |

77 | 203 | 283 | 349 | 381 | 423 | 479 | 569 | 623 | 669 | 713 | 779 | 837 | 887 | 941 |

83 | 211 | 293 | 355 | 382 | 429 | 503 | 571 | 631 | 677 | 715 | 787 | 839 | 893 | 947 |

107 | 213 | 311 | 359 | 395 | 437 | 509 | 573 | 637 | 683 | 717 | 803 | 841 | 899 | 955 |

149 | 227 | 317 | 367 | 403 | 451 | 551 | 581 | 643 | 691 | 739 | 809 | 845 | 901 | 983 |

Phew!!! That was long.

-fR0D

(notes from TAOCP written by Donald E Knuth)

Links :

http://www.research.att.com/~njas/sequences/a003313.txt

http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html

## Binary GCD Algorithm

Most of us know that to find the GCD of two numbers, Euclid Algorithm is the best algorithm. But thanks to **Josef Stein**, there is the Binary GCD Algorithm which is touch faster than the Euclid’s Algorithm.

The Euclids Algorithm implemented is :

long long gcd(long long a, long long b) { if(a==0) return(b); return(gcd(b%a, a)); }

The Binary GCD Algorithm as given in the The Art of Computer Programming written by D.E.Knuth is as follows :

Given positive integers u and v, this algorithm finds their greatest common divisor. A1. [Find power of 2.] Set k←0, and then repeatedly set k← k + 1, u ← u/2,v ← v/2, zero or more times until u and v are not both even. A2. [Initialize.] (Now the original values of u and v have been divided by 2^k,and at least one of their present values is odd.) If u is odd, set t ← -v, and go to A4. Otherwise set t ← u. A3. [Halve t.] (At this point, t is even, and nonzero.) Set t ← t/2. A4. [Is t even?] If t is even, go back to A3. A5. [Reset max(u, v).] If t > 0, set u ← t; otherwise set v← -t. (The larger of u and v has been replaced by |t|, except perhaps during the first time this step is performed.) A6. [Subtract.] Set t ← u-v. If t != 0, go back to A3. Otherwise the algorithm terminates with u* 2^k as the output.

Implementation in C++ would look something like this :

int binarygcd(int u,int v)

{

int k=0,t=0,i;

while (!(u&1) && !(v&1))

{

k++;

u>>=1;

v>>=1;

}

if (u&1)

t=u;

else

t=-v;

do

{

while (!(t&1))

t>>=1;

if (t>0)

u=t;

else

v=-t;

t=u-v;

}while (t);

for (i=0;i