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Object Oriented Data Structures. Recursion Introduction to Recursion Principles of Recursion Backtracking: Postponing the Work Tree-Structured Programs: Look Ahead in Games. Recursion. Recursion. Recursion. Recursion. Recursion. Recursion. Recursion. Recursion. C. C. C. A. A. A. A.

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slide1

Object Oriented Data Structures

RecursionIntroduction to RecursionPrinciples of RecursionBacktracking: Postponing the WorkTree-Structured Programs: Look Ahead in Games

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Recursion

Recursion

Recursion

Recursion

Recursion

Recursion

Recursion

Recursion

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stack frames

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Stack Frames

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tree of subprogram calls

Finish

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Tree of Subprogram Calls

Start

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recursive definitions

1 if n = 0

n*(n-1)! if n > 0

n! =

1 if n = 0 and x not 0

x*(xn-1) if n > 0 and x not 0

xn =

Recursive Definitions

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designing recursive algorithms
Designing Recursive Algorithms
  • Find the key step
  • Find a stopping rule (base case)
  • Outline your algorithm
  • Check termination
  • Draw a recursion tree

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tail recursion
Tail Recursion
  • The very last action of a function is a recursive call to itself
  • Explicit use of a stack not necessary
  • Reassign the calling parameters to the values specified in the recursive call and then repeat the function

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backtracking
Backtracking

An algorithm which attempts to complete a search for a solution to a problem by constructing partial solutions, always ensuring that the partial solutions remain consistent with the requirements. The algorithm then attempts to extend a partial solution toward completion, but when an inconsistency with the requirements of the problem occurs, the algorithm backs up (backtracks) by removing the most recently constructed part of the solution and trying another possibility.

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knight s tour
Knight\'s Tour

Legal Knight Moves

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knight s tour1
Knight\'s Tour

Legal Knight Moves

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knight s tour2
Knight\'s Tour

Legal Knight Moves

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knight s tour3
Knight\'s Tour

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Legal Knight Moves

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hexadecimal numbers
Hexadecimal Numbers
  • Hexadecimal – Base 16 numbering system 0-F
  • Decimal - Base 10 numbering system 0-9
  • Octal - Base 8 numbering system 0-7
  • Hexadecimal Digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
  • Decimal Digits 0,1,2,3,4,5,6,7,8,9
  • Ocal Digits 0,1,2,3,4,5,6,7
  • What’s . ?

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hexadecimal digit values
Hexadecimal Digit Values
  • 3210 Oct Dec Hex
  • 0000 0 0 0
  • 0001 1 1 1
  • 0010 2 2 2
  • 0011 3 3 3
  • 0100 4 4 4
  • 0101 5 5 5
  • 0110 6 6 6
  • 0111 7 7 7
  • 1000 10 8 8
  • 1001 11 9 9
  • 1010 12 10 A
  • 1011 13 11 B
  • 1100 14 12 C
  • 1101 15 13 D
  • 1110 16 14 E
  • 1111 17 15 F

3 2 1 0

23,22,21,20

8 4 2 1

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cell description

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application depth and breadth first search1

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Application: Depth- And Breadth-First Search

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class cell maintain
Class Cell Maintain:
  • A number. This is an integer value used to identify the cell.
  • Cells are numbered consecutively from left to right and top to bottom. (Order is important!)
  • A list of neighboring cells. Each cell will have an entry in this list for all other neighbor cell that can be reached.
  • A Boolean value, named visited, that will be used to mark a cell once it has been visited.
  • Traversing a maze often results in dead ends, and the need to back up and start again.
  • Marking cells avoids repeating effort and potentially walking around in circles.

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class description
Class Description

classcell

{public: cell(intn) : number(n), visited(false) {}

void addNeighbor(cell * n){neighbors.push_back(n);}voidvisit (deque<cell *> &);

protected:intnumber;boolvisited; list <cell *> neighbors;

};//end class cell

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class maze
Class Maze

classmaze

{public: maze(istream&);voidsolveMaze();

protected: cell * start;boolfinished;deque<cell *> path; // used to hold the path // or paths currently // being traversed

};//end class maze

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slide21

maze::maze(istream & infile)// initialize maze by reading from file{intnumRows, numColumns;int counter = 1; cell * current = 0;

infile >> numRows >> numColumns;

vector <cell *> previousRow (numRows, 0);

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slide22

for(inti = 0; i < numRows; i++)for(int j=0; j<numColumns; j++)

{ current = new cell(counter++);int walls;infile >> walls;

if((i>0) && ((walls & 0x04)==0))

{ current->addNeighbor(previousRow[j])previousRow[j]->addNeighbor(current);}

if((j>0> && ((walls & 0x08) == 0))

{ current->addNeighbor(previousRow[j-1]);previousRow[j-1]->addNeighbor(current);}previousRow[j] = current;} start = current; finished = false;

}//end maze()

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previousRow[2]

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slide26

void maze::solveMaze()// solve the maze puzzle

{ start->visit(path);while ((!finished) && (! path.empty()))

{ cell * current = path.front();path.pop_front(); finished = current->visit(path);}if ( ! finished)cout << “no solution found\n”;}//end solveMaze()

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slide27

bool cell::visit(deque<cell *> & path) { //depth firstif(visited) // already been herereturn false; visited = true; // mark as visitedcout << “visiting cell “ << number << endl;if (number == 1)

{cout << “puzzle solved\n”;return true;} list <cell *>:: iterator start, stop; start = neighbors.begin(); stop = neighbors.end();for ( ; start != stop; ++start)if (! (*start)->visited)path.push_front(*start);return false;

}

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slide28

bool cell::visit(deque<cell *> & path){// breadth firstif(visited) // already been herereturn false; visited = true; // mark as visitedcout << “visiting cell “ << number << endl;if (number == 1)

{cout << “puzzle solved\n”; return true;} list <cell *>:: iterator start, stop; start = neighbors.begin(); stop = neighbors.end();for ( ; start != stop; ++start)if (! (*start)->visited)path.push_back(*start);return false;

}

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depth first vs breadth first
Depth First vs. Breadth First
  • Because all paths of length one are investigated before examining paths of length two, and all paths of length two before examining paths of length three, a breadth-first search is guaranteed to always discover a path from start to goal containing the fewest steps, whenever such a path exists.

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depth first vs breadth first1
Depth First vs. Breadth First
  • Because one path is investigated before any alternatives are examined, a depth-first search may, if it is lucky, discover a solution more quickly than the equivalent breadth-first algorithm.

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depth first vs breadth first2
Depth First vs. Breadth First
  • In particular, suppose for a particular problem that some but not all paths are infinite, and at least one path exists from start to goal that is finite.
  • A breadth-first search is guaranteed to find a shortest solution.
  • A depth-first search may have the unfortunate luck to pursue a never-ending path, and can hence fail to find a solution.

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slide32

Chapter 5 Ripples Away

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