Problem solving & C III Arijit Bishnu arijit@isical.ac.in - - PowerPoint PPT Presentation

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Problem solving & C III Arijit Bishnu arijit@isical.ac.in - - PowerPoint PPT Presentation

Mar 08, 2024 229 likes •1.18k views

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi Problem solving & C III Arijit Bishnu arijit@isical.ac.in Indian Statistical Institute, India. August 5, 2014 Introduction Fibonacci

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Problem solving & C – III

Arijit Bishnu arijit@isical.ac.in

Indian Statistical Institute, India.

August 5, 2014

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Recursion

What is recursion?

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Recursion

What is recursion? A function calling itself with certain properties.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Recursion

What is recursion? A function calling itself with certain properties. There must be a base criteria for which the function does not call itself.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Recursion

What is recursion? A function calling itself with certain properties. There must be a base criteria for which the function does not call itself. After each call, the function must be closer to the base criteria.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Recursion

What is recursion? A function calling itself with certain properties. There must be a base criteria for which the function does not call itself. After each call, the function must be closer to the base criteria. An Example (Can you recognize the expression?) f (n) = n ∗ f (n − 1) if n ≥ 1 = 1 if n = 0

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Fibonacci sequence

The Definition fib(n) = fib(n − 1) + fib(n − 2) if n ≥ 2 = 1 if n = 1 = 1 if n = 0

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Fibonacci sequence

The Definition fib(n) = fib(n − 1) + fib(n − 2) if n ≥ 2 = 1 if n = 1 = 1 if n = 0 Some values fib(2) = 2; fib(3) = 3; fib(4) = 5; fib(5) = 8; fib(6) = 13; fib(7) = 21;fib(8) = 34; fib(9) = 55; fib(10) = 89; fib(11) = 144;

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

A recursive implementation

#include<stdio.h>

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

A recursive implementation

#include<stdio.h> int main(void){ unsigned int m,val,i; printf("\n What is the value of m::> "); scanf("%d",&m); val = f(m); printf("\n Fibonacci(%d) = %d \n",m,val); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

A recursive implementation

#include<stdio.h> unsigned int f(unsigned int num) { } int main(void){ unsigned int m,val,i; printf("\n What is the value of m::> "); scanf("%d",&m); val = f(m); printf("\n Fibonacci(%d) = %d \n",m,val); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

A recursive implementation

#include<stdio.h> unsigned int f(unsigned int num) { if(num==0 || num==1) return 1; } int main(void){ unsigned int m,val,i; printf("\n What is the value of m::> "); scanf("%d",&m); val = f(m); printf("\n Fibonacci(%d) = %d \n",m,val); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

A recursive implementation

#include<stdio.h> unsigned int f(unsigned int num) { if(num==0 || num==1) return 1; else return(f(num-1)+f(num-2)); } int main(void){ unsigned int m,val,i; printf("\n What is the value of m::> "); scanf("%d",&m); val = f(m); printf("\n Fibonacci(%d) = %d \n",m,val); return 0; }

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The recursion tree

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursion tree

f(5) f(3) f(4) f(1) f(2) f(2) f(3) f(0) f(1) f(0) f(1) f(1) f(2) f(0) f(1)

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The recursion tree

f(5) f(3) f(4) f(1) f(2) f(2) f(3) f(0) f(1) f(0) f(1) f(1) f(2) f(0) f(1) How many function calls? f (5) : 1, f (4) : 1, f (3) : 2, f (2) : 3, f (1) : 5, f (0) : 3 Exercise: Find it experimentally and analytically.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

An iterative implementation

#include<stdio.h> int main(void){ unsigned int m,i,f,f_1=1,f_2=1; printf("\n What is the value of n::> "); scanf("%d",&m); printf("\n Fibonacci(%d) = %d \n",m,f); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

An iterative implementation

#include<stdio.h> int main(void){ unsigned int m,i,f,f_1=1,f_2=1; printf("\n What is the value of n::> "); scanf("%d",&m); if(m==0 || m==1){ printf("\n Fibonacci(%d) = %d",m,1); return 0; } printf("\n Fibonacci(%d) = %d \n",m,f); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

An iterative implementation

#include<stdio.h> int main(void){ unsigned int m,i,f,f_1=1,f_2=1; printf("\n What is the value of n::> "); scanf("%d",&m); if(m==0 || m==1){ printf("\n Fibonacci(%d) = %d",m,1); return 0; } else { } printf("\n Fibonacci(%d) = %d \n",m,f); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

An iterative implementation

#include<stdio.h> int main(void){ unsigned int m,i,f,f_1=1,f_2=1; printf("\n What is the value of n::> "); scanf("%d",&m); if(m==0 || m==1){ printf("\n Fibonacci(%d) = %d",m,1); return 0; } else { for(i=2;i<=m;i++){ f = f_1+f_2; f_1=f_2, f_2=f; } } printf("\n Fibonacci(%d) = %d \n",m,f); return 0; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

An iterative implementation

#include<stdio.h> int main(void){ unsigned int m,i,f,f_1=1,f_2=1; printf("\n What is the value of n::> "); scanf("%d",&m); if(m==0 || m==1){ printf("\n Fibonacci(%d) = %d",m,1); return 0; } else { for(i=2;i<=m;i++){ f = f_1+f_2; f_1=f_2, f_2=f; } } printf("\n Fibonacci(%d) = %d \n",m,f); return 0; }

Exercise: Compare running times of recursive and iterative versions.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards

The Problem You have to create a singly linked list and then print it backwards without doing any changes on the links.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards

The Problem You have to create a singly linked list and then print it backwards without doing any changes on the links.

NULL head a s p l y

s p l a y

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards

The Problem You have to create a singly linked list and then print it backwards without doing any changes on the links.

NULL head a s p l y

s p l a y

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards

The Problem You have to create a singly linked list and then print it backwards without doing any changes on the links.

NULL head a s p l y

s p l a y Use recursion.

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards using recursion

void PrintArray(data *head) { return; }

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Printing linked list backwards using recursion

void PrintArray(data *head) { /* calling function recursively */ return; }

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Printing linked list backwards using recursion

void PrintArray(data *head) { /*boundary condition to stop recursion*/ /* calling function recursively */ return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards using recursion

void PrintArray(data *head) { /*boundary condition to stop recursion*/ /* calling function recursively */ /* Printing current element*/ return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards using recursion

void PrintArray(data *head) { /*boundary condition to stop recursion*/ /* calling function recursively */ PrintArray(head -> next); /* Printing current element*/ return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards using recursion

void PrintArray(data *head) { /*boundary condition to stop recursion*/ if(head -> next == NULL){ printf(" %d->",head -> x); return; } /* calling function recursively */ PrintArray(head -> next); /* Printing current element*/ return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Printing linked list backwards using recursion

void PrintArray(data *head) { /*boundary condition to stop recursion*/ if(head -> next == NULL){ printf(" %d->",head -> x); return; } /* calling function recursively */ PrintArray(head -> next); /* Printing current element*/ printf(" %d ->",head -> x); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Space Filling Curve

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Space Filling Curve

Z Curve order: 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16

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Space Filling Curve

Z Curve order: 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 Problem

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Space Filling Curve

Z Curve order: 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 Problem Allocate a 2-D matrix of size 2m × 2m.

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Space Filling Curve

Z Curve order: 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 Problem Allocate a 2-D matrix of size 2m × 2m. Fill it with integers 1, . . . , 22m in a row-major fashion.

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Space Filling Curve

Z Curve order: 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 Problem Allocate a 2-D matrix of size 2m × 2m. Fill it with integers 1, . . . , 22m in a row-major fashion. Traverse the matrix in a z-curve order.

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The main program

int main(void){ int m,N, i,j,k=1,**matrix; printf(" \n What is the value of m::>"); scanf("%d",&m); N=(int)(pow(2.0,m)); matrix = allocate2D(N,N); }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The main program

int main(void){ int m,N, i,j,k=1,**matrix; printf(" \n What is the value of m::>"); scanf("%d",&m); N=(int)(pow(2.0,m)); matrix = allocate2D(N,N); /*Fill up and print the matrix */ for(i=0;i<N;i++) for(j=0;j<N;j++){matrix[i][j]=k; k++;} for(i=0;i<N;i++){ printf("\n"); for(j=0;j<N;j++) printf("%d \t",matrix[i][j]); } }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The main program

int main(void){ int m,N, i,j,k=1,**matrix; printf(" \n What is the value of m::>"); scanf("%d",&m); N=(int)(pow(2.0,m)); matrix = allocate2D(N,N); /*Fill up and print the matrix */ for(i=0;i<N;i++) for(j=0;j<N;j++){matrix[i][j]=k; k++;} for(i=0;i<N;i++){ printf("\n"); for(j=0;j<N;j++) printf("%d \t",matrix[i][j]); } printf("\n \nReading the matrix in z-curve fashion::>"); zee(0,N-1,0,N-1,matrix); printf("\n"); return 0; }

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Allocating the 2-D matrix

int** allocate2D(int row,int col){ int **S,k; S=(int **)calloc(row,sizeof(int *)); if(S==NULL){printf("\n No space \n"); exit(0);} for(k=0;k<row;k++){ S[k]=(int *)calloc(col,sizeof(int)); if(S[k]==NULL){ printf("\n No space \n"); exit(0);} } return S; }

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The recursive function

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The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee( ); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee( ); zee( ); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee( ); zee( ); zee( ); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee( ); zee( ); zee( ); zee( ); return; }

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The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee(r_b,(r_e+r_b)/2,c_b,(c_e+c_b)/2, matrix); zee( ); zee( ); zee( ); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee(r_b,(r_e+r_b)/2,c_b,(c_e+c_b)/2, matrix); zee(r_b,(r_e+r_b)/2,(c_e+c_b)/2+1,c_e, matrix); zee( ); zee( ); return; }

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The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee(r_b,(r_e+r_b)/2,c_b,(c_e+c_b)/2, matrix); zee(r_b,(r_e+r_b)/2,(c_e+c_b)/2+1,c_e, matrix); zee((r_e+r_b)/2+1,r_e,c_b,(c_e+c_b)/2, matrix); zee( ); return; }

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The recursive function

void zee(int r_b,int r_e,int c_b,int c_e, int **matrix){ if((r_b==r_e)&&(c_b==c_e)){ printf(" %2d ",matrix[r_b][c_b]); return; } zee(r_b,(r_e+r_b)/2,c_b,(c_e+c_b)/2, matrix); zee(r_b,(r_e+r_b)/2,(c_e+c_b)/2+1,c_e, matrix); zee((r_e+r_b)/2+1,r_e,c_b,(c_e+c_b)/2, matrix); zee((r_e+r_b)/2+1,r_e,(c_e+c_b)/2+1,c_e, matrix)); return; }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

Outline

1 Introduction 2 Fibonacci sequence 3 Printing link list backwards 4 Space Filling Curve 5 Towers of Hanoi

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Towers of Hanoi

The Problem

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Towers of Hanoi

The Problem Three pegs A, B and C are given and on peg A there are n disks with decreasing size.

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Towers of Hanoi

The Problem Three pegs A, B and C are given and on peg A there are n disks with decreasing size. We have to move disks from peg A to peg C using peg B as an auxiliary peg following some rules.

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Towers of Hanoi

The Problem Three pegs A, B and C are given and on peg A there are n disks with decreasing size. We have to move disks from peg A to peg C using peg B as an auxiliary peg following some rules.

The top disk on any peg can be moved to any other peg.

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Towers of Hanoi

The Problem Three pegs A, B and C are given and on peg A there are n disks with decreasing size. We have to move disks from peg A to peg C using peg B as an auxiliary peg following some rules.

The top disk on any peg can be moved to any other peg. A larger disk can never be placed on a smaller disk.

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Towers of Hanoi

The Problem Three pegs A, B and C are given and on peg A there are n disks with decreasing size. We have to move disks from peg A to peg C using peg B as an auxiliary peg following some rules.

The top disk on any peg can be moved to any other peg. A larger disk can never be placed on a smaller disk.

Exercise Try with n = 3, 4, . . . and start thinking recursively.

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Towers of Hanoi

Peg A Peg B Peg C

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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Towers of Hanoi

Peg A Peg B Peg C

1 2 3

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The algorithm

The recursive method

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The algorithm

The recursive method Move the top n − 1 disks from peg A to peg B using peg C as auxiliary.

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The algorithm

The recursive method Move the top n − 1 disks from peg A to peg B using peg C as auxiliary. Move the top disk from peg A to peg C.

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The algorithm

The recursive method Move the top n − 1 disks from peg A to peg B using peg C as auxiliary. Move the top disk from peg A to peg C. Move the top n − 1 disks from peg B to peg C using peg A as auxiliary.

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The main program

int main(void) { int n; printf("Enter the number of disks : "); scanf("%d",&n); printf("The Tower of Hanoi involves the moves :\n \n"); return 0; }

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The main program

int main(void) { int n; printf("Enter the number of disks : "); scanf("%d",&n); printf("The Tower of Hanoi involves the moves :\n \n"); towers(n,’A’,’C’,’B’); return 0; }

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The main program

int main(void) { int n; printf("Enter the number of disks : "); scanf("%d",&n); printf("The Tower of Hanoi involves the moves :\n \n"); towers(n,’A’,’C’,’B’); printf("\n \n"); return 0; }

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The recursive function

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The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ }

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The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ }

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The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ /* Move top n-1 disks from A to B, using C as auxiliary */ }

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The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ /* Move top n-1 disks from A to B, using C as auxiliary */ /* Move remaining disks from A to C*/ }

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Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ /* Move top n-1 disks from A to B, using C as auxiliary */ /* Move remaining disks from A to C*/ /* Move n-1 disks from B to C using A as auxiliary */ }

slide-87
SLIDE 87

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ if(n==1) { } /* Move top n-1 disks from A to B, using C as auxiliary */ /* Move remaining disks from A to C*/ /* Move n-1 disks from B to C using A as auxiliary */ }

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SLIDE 88

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ if(n==1) { printf("\n Move disk 1 from peg %c to peg %c", frompeg,topeg); return; } /* Move top n-1 disks from A to B, using C as auxiliary */ /* Move remaining disks from A to C*/ /* Move n-1 disks from B to C using A as auxiliary */ }

slide-89
SLIDE 89

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ if(n==1) { printf("\n Move disk 1 from peg %c to peg %c", frompeg,topeg); return; } /* Move top n-1 disks from A to B, using C as auxiliary */ towers(n-1,frompeg,auxpeg,topeg); /* Move remaining disks from A to C*/ /* Move n-1 disks from B to C using A as auxiliary */ }

slide-90
SLIDE 90

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ if(n==1) { printf("\n Move disk 1 from peg %c to peg %c", frompeg,topeg); return; } /* Move top n-1 disks from A to B, using C as auxiliary */ towers(n-1,frompeg,auxpeg,topeg); /* Move remaining disks from A to C*/ printf("\n Move disk %d from peg %c to peg %c", n,frompeg,topeg); /* Move n-1 disks from B to C using A as auxiliary */ }

slide-91
SLIDE 91

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The recursive function

void towers(int n,char frompeg,char topeg,char auxpeg){ /* If only 1 disk, make the move and return */ if(n==1) { printf("\n Move disk 1 from peg %c to peg %c", frompeg,topeg); return; } /* Move top n-1 disks from A to B, using C as auxiliary */ towers(n-1,frompeg,auxpeg,topeg); /* Move remaining disks from A to C*/ printf("\n Move disk %d from peg %c to peg %c", n,frompeg,topeg); /* Move n-1 disks from B to C using A as auxiliary */ towers(n-1,auxpeg,topeg,frompeg); }

slide-92
SLIDE 92

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The Last One Questions???

slide-93
SLIDE 93

Introduction Fibonacci sequence Printing link list backwards Space Filling Curve Towers of Hanoi

The Last One Questions??? Thanks!!!