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Object Oriented Data StructuresPowerPoint Presentation

Object Oriented Data Structures

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Object Oriented Data Structures

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

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Stack FramesTime

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

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1 if n = 0 and x not 0

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

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Recursive DefinitionsKruse/Ryba ch05

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

- 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

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

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Legal Knight Moves

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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

- 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

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8 4 2 1

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Cell Description0

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Application: Depth- And Breadth-First SearchKruse/Ryba ch05

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

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

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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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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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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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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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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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

- 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 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 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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