Module 4. l Graph Algorithms-MST. l HW3. l OO class point. C++ for C Programmers by Ira Pohl

Transcription

1 Module 4 Graph Algorithms-MST HW3 OO class point

2 Homework Week 3 Already built a graph (Abstract Data Type) Used Dijkstra algorithm to generate graph based on density and size Computed average length

3 Minimum Spanning Tree Assume a graph is connected with arbitrary edge costs Cij. Find a spanning tree of least cost. A spanning tree is a set of edges within the graph that connects all the nodes and where there is no cycle.

4 Extra Credit : Color idea Assume each edge has one of three colors Red, Blue, Green Find MST when edges must be of a specified color or pair of colors.

5 Quiz: Enum color enum color{ RED, BLUE, GREEN}; Why were the colors capitalized? What int value is GREEN? How can we overload << to print RED

6 Answers Convention to use all caps for enumeration constants GREEN is 2 out& operator<<(ostream& out, color c) { ---what goes here return??? }

7 C++11 enum classes Strongly typed enums - enum classes enum class Color{RED, GREEN, BLUE}; enum class Traffic{RED, YELLOW, GREEN}; class indicates that each enum type is different. and not comparable to other enum types. Color::RED!= Traffic::RED

8 Jarnik-Prim MST ### This algorithm was developed in 1930 by the Czech mathematician Vojtěch Jarník and rediscovered by Robert Prim in Prim s algorithm starts with a single vertex and repeatedly joins the minimum-weight edge that joins the tree to a non-tree vertex. At each step the tree is a minimum spanning tree of the vertices that have been processed thus far.

9 Minimum Spanning Tree

10 Minimum Spanning Tree Start at A Nearest is E Pick nearest next node until All nodes in graph or no more nodes to connect

11 Quiz If all edges have unit cost and a graph has 10 nodes and it has an MST what will be its cost? Same question for n nodes?

12 Answer 10 nodes : MST 9 edges : therefore 9 N nodes: therefore N-1

13 Minimum Spanning Tree 2

14 Prim pseudocode Given a graph, G, with edges E of the form (v1, v2) and vertices V dist : array of distances from the source to each vertex edges: array indicating, for a given vertex, which vertex in the tree it is closest to i : loop index MST : list of finished vertices U : list or heap unfinished vertices /* Initialization: set every distance to INFINITY until we discover a way to link a vertex to the spanning tree */ for i = 0 to V - 1 dist[i] = INFINITY edge[i] = NULL C++ for C Programmers end by Ira Pohl

15 Prim pseudocode pick a vertex, s, - say vertex 0 dist[s] = 0 while(mst is missing a vertex) pick the vertex, v, in U with the shortest edge to the group of vertices in the spanning tree add v to MST for each edge of v, (v1, v2) if(length(v1, v2) < dist[v2]) dist[v2] = length(v1, v2) edges[v2] = v1 possibly update U, depending on implementation end if end for end while

16 Prim is greedy Prim (and Dijkstra) are called greedy algorithms they take a locally best next element and this leads to a globally best solution greedy algorithm when optimal tend to be very efficient when non-optimal they still might be heuristically okay

17 Kruskal Pseudocode ### Create a forest F (a set of trees), where each vertex in the graph is a separate tree. Create a set S containing all the edges in the graph. While S is nonempty and F is not yet spanning remove an edge with minimum weight from S if that edge connects two different trees, then add it to the forest, combining two trees into a single tree otherwise discard that edge. At the termination of the algorithm, the forest has only one component and forms a minimum spanning tree of the graph.

18 Kruskal simulate

19 Quiz How would you change algorithm to find a maximum spanning tree ie greatest cost spanning tree.

20 Answer Same structure but pick largest cost edges.

21 Further C++ ideas Ch6 ### Operator overloading Friend functions

22 OOP Principle User defined types should be indistinguishable from native types Operator overloading and conversion allows us to implement this principle

23 Point and conversions class point{ public: point (double u):x(u), y(0.0){}. private double x, y; };

24 Convert double to point point s; double d = 3.5; s = d; //implicit conversion s = static_cast<point>(d); //explicit

25 Convert point to double Can t get at definition of native type but C++ can do as follows: point ::operator double() { return sqrt(x*x +y*y);} d = s; //this would now work??? Not always a good idea maybe opaque

26 C++11 explicit suppress conversion explicit keyword in front of a constructor of one argument turns off its ability to be used implicitly for conevrsion explicit point::point(double d) {.. // code };

27 Overloading and function selection Exact match to signature Try standard promotions - eg int to double Try standard type conversions eg int to short Try user defined conversions Use a match to ellipsis if found ( ) Read pp on rational

28 Friend functions and classes Googly with Nappie having access to their Private parts

29 Friend function class point{ }; private x,y; //restricted to class methods ostream& <<operator(ostream& out, point p) { out << ( << x <<, <<y << ) ; return out;} FAILS because x and y are private

30 Friend is important for << class point{ friend ostream& operator<<(ostream& ); }; private x,y; //restricted to class methods By having this be a class friend we allow it access priveleges and unlike a class method it does not have a dot argument

31 Operator+ class point{ public: point operator+(point p) { return point(x+p.x, y+p.y);}. private double x, y; };

32 Typical friend operator + (type arg1, type arg2) Versus classname : : operator + (type arg 1) // here this provides arg1

33 Point binary + point a, b(2, 1), c(0.5, 6);.. a = b + c; // b.operator+(c) Now if we used a friend function which is preferred we would get a = b + c; // operator+(b, c) Now both arguments subject to symmetrical semantics as in function arg conversions

34 Operator overloading Need to be very careful Need to conform to rules from the math domain - eg point, polynomial, rational but not window Need to be comprehensive + defined, so would expect -, *, /

35 Quiz: Not all operators in C++ can be overloaded; One of these is scope resolution; Another is the only ternary operator in C++; What is it and why is not overloadable?

36 Answer:?: is ternary It would be ugly to overload it and would not have a natural semantics. Bye?:

37 Standard Template Library ### STL STL C++11 -new

38 Three Legged St oo l STL Containers Algorithms Iterators

39 Standard Template Library Three Legged Stool: Containers: vector, list, map Iterators: forward, backward, random access Algorithms: sort, permute

40 Quiz Name two standard sorting algorithms What is their expected time? Worst case time?

41 Answer Bubblesort, Quicksort Bubblesort n*n Quicksort n*ln(n) - worst case n*n

42 Example ### // Simple STL vector program #include <iostream> #include <vector> using namespace std; int main () { vector<int> v(100); //allocate space **vector is a template and is a random access container replace array in C++

43 Example code for (int i = 0; i < 100; ++i) v[i] = i;//index thru like an array for (vector<int>::iterator p = v.begin(); p!= v.end(); ++p) cout << *p << '\t'; cout << endl; } **second for used iterators note the declaration, and it is safer no off by 1 errors.

44 begin() end() methods Here we use a vector and its associated iterator. Think of p as conceptually a pointer. It has two endpoints that define its range: (v.begin(), v.end()) where v.begin() stores a value, but v.end() does not.

45 C feature use auto ### for (vector<int>::iterator p = v.begin(); cout << *p << '\t'; cout << endl; ) for (auto p = v.begin(); ) auto is used to declare types that can be deduced from context and is no longer a storage specifier.

46 More auto examples auto i = 3; //typical would be int auto d = 3.7; //would be double auto c = d; //would be double

47 Quiz deduce type auto c = B ;

48 Answer: auto c = B ; c is a char type

49 Vector methods v.size() get current size v.resize(int size) resize the vector Constructors vector<t> v; empty sequence vector<t> v(n); size n with default constructor T() called for each element vector<t> v(n, value) ; value is of type T Or convertible to T and is value of each element

50 Vector + file IO // Simple STL vector program with file IO #include <fstream> //file io #include <iostream> #include <vector> using namespace std;

51 Read from a file int main () { vector<int> v(10); ifstream ifp("data", ios::in); int i = 0; while(!ifp.eof() ) { ifp >> v[i++]; if (i%9 ==0) v.resize(v.size() +10); }

52 Simple file input ifstream ifp("data", ios::in); Input file pointer opened for reading data not a full path name so it must exist and be in the local directory

53 More code v.resize(i-1); //get rid of extra elements for (vector<int>::iterator p = v.begin(); p!= v.end(); ++p) cout << *p << '\t'; cout << endl; }

54 Iterator idiom for (vector<int>::iterator p = v.begin();p!= v.end(); ++p) *** declare iterator and intialize to begin *** ++p auto increment iterator(pointer) to advance cout << *p << '\t'; *** dereference iterator to get T value of object being pointed at

55 Quiz: Avoiding bounds errors for(int i =0; i<=upper_bound; ++i) sum = sum + data[i]; What goes wrong?

56 Answer: <= upper_bound is usually not right Off-by one

57 C++11 New for statement ### for (declaration : expression) Statement This is the range for statement A simple example that takes care of the bounds error problem in arrays for(double d : data) sum +=d; So data is an array of doubles such as double data[upper_bound]

58 Range for The expression can be an array, a sequence container, a string. They are conceptually expected to have begin and end members. Another example: For(auto &element : data) //auto infers type element = element + 2; //allows mutate

59 STL input file example ### //read a series of words into a vector #include<iostream> #include<fstream> #include<algorithm> #include<vector> using namespace std; void getw(string& t, ifstream& in){ in>> t; }

60 Read in words int read_string(vector<string> &words, ifstream& in) { int i = 0; while(!in.eof())//sentinel getw(words[i++], in); return (i - 1); }

61 main int main() { ifstream in("text"); vector<string> w(1000); //hope 1000 is enough int howmany = read_string(w, in); w.resize(howmany); sort(w.begin(), w.end()); for( auto str : w) cout << str << "\t" ; }

62 Sort algorithm sort(w.begin(), w.end()); The sort() routine is found in the STL algorithm library. It is an efficient form of quicksort [Hoare]. If you don t think so try to write your own and compare by timing it on a large number of elements. sort() uses two iterator positions to sort the elements in the vector w. This sort routine expects that the elements pointed at can be randomly accessed.

63 Iterators categories ###

64 Iterator categories In STL there are five iterator categories: input (weakest), -read, single pass output, - write, single pass forward, read and write multipass -1D bidirectional, multi-pass + 2 directions Random access (strongest), anywhere

65 Algorithms and Iterators In STL the design principle is to use the weakest iterator that accomodates the most efficient algorithm. Quicksort requires random access. As a consequence it cannot be used on a nonrandom access container such as a list.

66 Input Iterator The iterator hierarchy: 1.input iterator: The input iterator must allow the elements to be searched in a forward direction one-by-one. This requires that operator ++ be defined ("advance the iterator") and dereferencing operator *.

67 Input it Canonically accessing elements sequentially in an input stream requires this minimal set of operations. Similarly using a find(elem) in a collection searched sequentially has these requirements. Also input iterators have == and!= defined.

68 Example input it //use an input iterator #include<iostream> #include<fstream> #include<numeric> using namespace std;

69 Example int main() { ifstream myin("data"); istream_iterator<int> in(myin); istream_iterator<int> eos; cout << "Sum of data is " << accumulate(in, eos, 0) << endl; }

70 accumulate() The algorithm accumulate() is from the <numeric> library accumulate(input_it, input_it, value) value is usually 0 accumulate over the specified range

71 Quiz: int data[4] = {1,2,3,4}; cout << accumulate(data, data + 4, 5); What gets printed

72 Answer:

73 Output a random graph ### #include <iostream> #include <ctime> #include <cstdlib> #include <fstream> using namespace std; { } double prob() return (static_cast<double>(rand())/rand_max);

74 Will represent graph with matrix { int main(void) int size = 15; double density ; cout << "graph size?" << endl; cin >> size; cout << "graph density (0, 1)?" << endl; cin >> density; bool** graph; int** color; int** cost; srand(time(0)); //seed random number generator

75 Create matrices off heap } graph = new bool*[size]; color = new int*[size]; cost = new int*[size]; for(int i = 0; i <size; ++i){ graph[i] = new bool[size]; color[i] = new int[size]; cost[i] = new int[size];

76 Generate graph for (int i = 0; i < size; ++i) //generate undirected edges for(int j = i; j < size; ++j) if (i == j) graph[i][j]= false; //no loops else graph[i][j] = graph[j][i] = (prob() < density); for (int i = 0; i < size; ++i) //generate costs and color for(int j = i; j < size; ++j) if (graph[i][j]){ color[i][j] = color[j][i] = rand()% 3; cost[i][j] = cost[j][i] = prob()* 30; }

77 Create output file ofstream outp("graph_data.txt"); outp << size << "\n"; for (int i = 0; i < size; ++i) for(int j = 0; j < size; ++j){ if ( graph[i][j]) outp << i <<'\t' << j << '\t' << cost[i][j] << '\t' << color[i][j] << '\t'; }

78 Next Section Second Half of the Class More advanced C++ with new C++11 features Build a Game