CSCE 210 Data Structures and Algorithms Prof. Amr Goneid AUC Part 10. Graphs Prof. Amr Goneid, AUC

Graphs Prof. Amr Goneid, AUC

Graphs Basic Definitions Paths and Cycles Connectivity Other Properties Representation Examples of Graph Algorithms: Graph Traversal Shortest Paths Minimum Cost Spanning Trees Prof. Amr Goneid, AUC

1. Basic Definitions A graph G (V,E) can be defined as a pair (V,E) , where V is a set of vertices, and E is a set of edges between the vertices E = {(u,v) | u, v  V}. e.g. V = {A,B,C,D,E,F,G} E = {( A,B),(A,F),(B,C),(C,G),(D,E),(D,G),(E,F),(F,G)} If no weights are associated with the edges, an edge is either present(“1”) or absent (“0”). A F G B E D C Prof. Amr Goneid, AUC

Basic Definitions A graph is like a road map. Cities are vertices. Roads from city to city are edges. You could consider junctions to be vertices, too. If you don't want to count them as vertices, a road may connect more than two cities. So strictly speaking you have hyperedges in a hypergraph. If you want to allow more than one road between each pair of cities, you have a multigraph, instead. Prof. Amr Goneid, AUC

Basic Definitions Adjacency: If vertices u,v have an edge e = (u,v) | u, v  V then u and v are adjacent. A weighted graph has a weight associated with each edge. Undirected Graph is a graph in which the adjacency is symmetric, i.e., e = (u,v) = (v,u) A Sub-Graph: has a subset of the vertices and the edges A F G B E D C 3 2 2 1 5 4 1 2 Prof. Amr Goneid, AUC

Basic Definitions 3 Directed Graph: is a graph in which adjacency is not symmetric, i.e., (u,v)  (v,u) Such graphs are also called “Digraphs” Directed Weighted Graph: A directed graph with a weight for each edge. Also called a network. A F G B E D C 3 2 2 1 5 4 1 2 Prof. Amr Goneid, AUC

2. Paths & Cycles Path: A list of vertices of a graph where each vertex has an edge from it to the next vertex. Simple Path: A path that repeats no vertex. Cycle: A path that starts and ends at the same vertex and includes other vertices at most once. A F G B E D C A F G B E D C Prof. Amr Goneid, AUC

Directed Acyclic Graph (DAG) Directed Acyclic Graph (DAG): A directed graph with no path that starts and ends at the same vertex A F G B E D C Prof. Amr Goneid, AUC

Hamiltonian Cycle Hamiltonian Cycle: F G B E D C Hamiltonian Cycle: A cycle that includes all other vertices only once, e.g. {D,B,C,G,A,F,E,D} Named after Sir William Rowan Hamilton (1805 –1865) The Knight’s Tour problem is a Hamiltonian cycle problem Icosian Game Prof. Amr Goneid, AUC

Hamiltonian Cycle Demo A Hamiltonian Cycle is a cycle that visits each node exactly once. Here we show a Hamiltonian cycle on a 5-dimensional hypercube. It starts by completely traversing the 4-dimensional hypercube on the left before reversing the traversal on the right subcube. Hamiltonian cycles on hypercubes provide constructions for Gray codes: orderings of all subsets of n items such that neighboring subsets differ in exactly one element. Hamilitonian cycle is an NP-complete problem, so no worst-case efficient algorithm exists to find such a cycle. In practice, we can find Hamiltonian cycles in modest-sized graphs by using backtracking with clever pruning to reduce the search space. Prof. Amr Goneid, AUC

Hamiltonian Cycle Demo http://www.cs.sunysb.edu/~skiena/combinatorica/animations/ham.html Hamiltonian Cycle Demo Prof. Amr Goneid, AUC

Euler Circuit Euler Circuit: A cycle that includes every edge once. Leonhard Euler Konigsberg Bridges (1736) (not Eulerian) Euler Circuit: A cycle that includes every edge once. Used in bioinformatics to reconstruct the DNA sequence from its fragments Prof. Amr Goneid, AUC

Euler Circuit Demo An Euler circuit in a graph is a traversal of all the edges of the graph that visits each edge exactly once before returning home. A graph has an Euler circuit if and only if all its vertices are that of even degrees. It is amusing to watch as the Euler circuit finds a way back home to a seemingly blocked off start vertex. We are allowed (indeed required) to visit vertices multiple times in an Eulerian cycle, but not edges. Prof. Amr Goneid, AUC

Euler Circuit Demo http://www.cs.sunysb.edu/~skiena/combinatorica/animations/euler.html Euler Circuit Demo Prof. Amr Goneid, AUC

3. Connectivity Connected Graph: An undirected graph with a path from every vertex to every other vertex A Disconnected Graph may have several connected components Tree: A connected Acyclic graph A F G B E D C A F G B E D C Prof. Amr Goneid, AUC

Connected Components Demo What happens when you start with an empty graph and add random edges between vertices? As you add more and more edges, the number of connected components in the graph can be expected to drop, until finally the graph is connected. An important result from the theory of random graphs states that such graphs very quickly develop a single ``giant'' component which eventually absorbs all the vertices. Prof. Amr Goneid, AUC

Connected Components Demo http://www.cs.sunysb.edu/~skiena/combinatorica/animations/concomp.html Randomly Connected Graph Demo Prof. Amr Goneid, AUC

Connectivity Articulation Vertex: if removed with all of its edges will cause a connected graph to be disconnected, e.g., G and D are articulation vertices A F G B E D C A F B E D C Prof. Amr Goneid, AUC

Connectivity Degree Of a vertex, the number of edges connected to it. Degree Of a graph, the maximum degree of any vertex (e.g. B has degree 2, graph has degree 3). In a connected graph the sum of the degrees is twice the number of edges, i.e A F G B E D C Prof. Amr Goneid, AUC

Connectivity In-Degree/Out-Degree: the number of edges coming into/emerging from a vertex in a connected graph (e.g. G has in-degree 3 and out-degree 1). A F G B E D C Prof. Amr Goneid, AUC

Connectivity Complete Graph: There is an edge between every vertex and every other vertex. In this case, the number of edges is maximum: Notice that the minimum number of edges for a connected graph ( a tree in this case) is (V-1) Prof. Amr Goneid, AUC

Density (of edges) Density of a Graph: Dense Graph: Number of edges is close to Emax = V(V-1)/2. So, E = O(V2) and D is close to 1 Sparse Graph: Number of edges is close to Emin = (V-1). So, E = O(V) Prof. Amr Goneid, AUC

4. Other Properties Planar Graph: A graph that can be drawn in the plain without edges crossing A D C B A D C B Non-Planar Prof. Amr Goneid, AUC

Other Properties Graph Coloring: To assign color (or any distinctive mark) to vertices such that no two adjacent vertices have the same color. The minimum number of colors needed is called the Chromatic Order of the graph (G). For a complete graph, (G) = V. 1 2 4 3 Prof. Amr Goneid, AUC

Other Properties Slot Courses 1 (red) 2 (Green) 3 (Blue) CS An Application: Find the number of exam slots for 5 courses. If a single student attends two courses, an edge exists between them. EE Slot Courses 1 (red) CS 2 (Green) EE, Econ, Phys 3 (Blue) Math CS Math Econ Phys Prof. Amr Goneid, AUC

5. Representation A B C D 1 Adjacency Matrix: V x V Matrix a(i,j) a(i,j) = 1 if vertices (i) and (j) are adjacent, zero otherwise. Usually self loops are not allowed so that a(i,i) = 0. For undirected graphs, a(i,j) = a(j,i) Matrix is symmetric A B C D 1 A D C B Prof. Amr Goneid, AUC

Adjacency Matrix For weighted undirected graphs, a(i,j) = a(j,i) = wij A D C B A B C D 2 1 3 5 1 2 5 3 Prof. Amr Goneid, AUC

Adjacency Matrix For weighted directed graphs, a(i,j) ≠ a(j,i) Matrix is not symmetric A D C B A B C D 2 1 3 5 2 1 5 3 Prof. Amr Goneid, AUC

Representation The adjacency matrix is appropriate for dense graphs but not compact for sparse graphs. e.g., for a lattice graph, the degree of a vertex is 4, so that E = 4V. Number of “1’s” to total matrix size is approximately 2 E / V2 = 8 / V. For V >> 8, the matrix is dominated by zeros. Prof. Amr Goneid, AUC

Representation Adjacency List: A B C D An array of vertices with pointers to linked lists of adjacent nodes, e.g., The size is O(E + V) so it is compact for sparse graphs. A D C B A B C B A C C A B D D C Prof. Amr Goneid, AUC

Edge List An edge (u,v,w) existing between nodes (u) and (v) with weight (w) is modeled as a structure (Edge). Space requirements is O(E) u A B C v D w 2 1 3 5 A D C B 1 2 5 3 Prof. Amr Goneid, AUC

6. Examples of Graph Algorithms Graph Traversal Shortest Paths Minimum Cost Spanning Trees Prof. Amr Goneid, AUC

6.1 Graph Traversal Depth-First Search (DFS) Visits every node in the graph by following node connections in depth. Recursive algorithm. An Array val[v] records the order in which vertices are visited. Initialized to “unseen”. Any edge to a vertex that has not been seen is followed via the recursive call. Prof. Amr Goneid, AUC

Algorithm void DFS() { int k; int unseen = -2; // Assume order = 0 initially void DFS() { int k; int unseen = -2; // Initialize all to unseen for (k = 1; k <= v; k++) val[k] = unseen; // Follow Nodes in Depth for (k = 1; k <= v; k++) if (val[k] == unseen) visit(k); } Prof. Amr Goneid, AUC

Algorithm (continued) void visit(int k) { int t; val[k] = ++order; for (t = 1; t <= v; t++) if (a[k][t] != 0) if (val[t] == unseen) visit(t); } Prof. Amr Goneid, AUC

Example A B C D E F G 1 start 1 A B C F G D E VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 1 A B C F G 2 D E VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 3 1 A B C F G 2 3 D E VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 3 4 1 A B C 2 3 4 F G D E VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 3 5 4 1 A B C 2 3 4 F G D E 5 VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 3 5 6 4 1 A B C 4 F G 2 3 D E 5 6 VAL[K] Prof. Amr Goneid, AUC

Example A B C D E F G 1 2 3 5 6 4 7 1 A 7 B C 4 F G 2 3 D E 5 6 VAL[K] Prof. Amr Goneid, AUC

Exercises Show how DFS can be used to determine the number of connected components in a graph. Explore the non-recursive version of the DFS using a stack. What will happen if you use a queue instead of the stack? Prof. Amr Goneid, AUC

Non-Recursive DFS // Assume unseen = -2 and hold = -1 // Val[ ] is set to “unseen” initially , order = 0 Stact<int> S; // A stack of integers void DFS(int k) { int t; S.push(k); while (! S.stackIsEmpty()) S.pop(k); val[k] = ++order; for (t = v; t >= 1; t--) // Scan from right to left if (a[k][t] != 0) if (val[t] == unseen) { S.push(t); val[t] = hold;} } Prof. Amr Goneid, AUC

Breadth-First Search (BFS) // Replacing the stack by a queue, gives the BFS algorithm Queuet<int> Q; // A queue of integers void BFS(int k) { int t; Q.enqueue(k); while (! Q.queueIsEmpty()) Q.dequeue(k); val[k] = ++order; for (t = 1; t <= v; t++) // Scan from left to right if (a[k][t] != 0) if (val[t] == unseen) { Q.enqueue(t); val[t] = hold;} } Prof. Amr Goneid, AUC

Example BFS A B C D E F G 1 5 3 6 7 4 1 A B C 4 F 5 G 2 3 D E 6 7 Prof. Amr Goneid, AUC

DFS and BFS of a Tree For a tree structure: DFS is equivalent to Pre-order traversal BFS is equivalent to Level-order traversal DFS Demo BFS Demo Prof. Amr Goneid, AUC

Graph Traversal Demos http://www.cs.sunysb.edu/~skiena/combinatorica/animations/search.html http://www.cosc.canterbury.ac.nz/people/mukundan/dsal/GraphAppl.html Prof. Amr Goneid, AUC

Exercise Model the shown maze as a graph. Show how DFS can be in Model the shown maze as a graph. Show how DFS can be used to find the exit. out Prof. Amr Goneid, AUC

8. A Simple Graph Class To represent a weighted undirected graph with a maximum of Vmax vertices and Emax = Vmax(Vmax-1)/2 edges. The verices are numbered 0,1,...V-1. The graph is assumed to be on a text file in the form of an adjacency matrix. The weights on the edges are assumed to be positive integers with zero weight indicating the absence of an edge. When loaded from the text file, the weights are stored in a 2-D array (AdjMatrix) representing the adjacency matrix. Another array (edges) stores the non-zero edges in the graph. An edge (u,v,w) existing between nodes (u) and (v) with weight (w) is modeled as a class (Edge). Prof. Amr Goneid, AUC

Edge Class // File: Edge.h // Definition of Edge class #ifndef EDGE_H #define EDGE_H typedef int weightType; // weights are positive integers class Edge { public: int u,v; weightType w; bool operator < (const Edge &e) { return (w < e.w); } bool operator <= (const Edge &e) { return (w <= e.w); } }; // end of class Edge declaration #endif // EDGE_H Prof. Amr Goneid, AUC

Graph Class // File: Graphs.h // Graph library header file #ifndef GRAPHS_H #define GRAPHS_H #include <string> #include "Edge.h" using namespace std; const int Vmax = 50; // Maximum number of vertices const int Emax = Vmax*(Vmax-1)/2; // Maximum number of edges Prof. Amr Goneid, AUC

Graph Class class Graphs { public: Graphs(); // Constructor ~Graphs(); // Destructor // Map vertex number to a name (character) char Vname(const int s) const; void getGraph(string fname); // Get Graph from text File (fname) void dispGraph( ) const; // Display Ajacency Matrix int No_of_Verices( ) const; // Get number of vertices (V) int No_of_Edges( ) const; // Get Number of Non-zero edges (E) void dispEdges( ) const; // Display Graph edges void DFS( ); // Depth First Search Traversal (DFS) Prof. Amr Goneid, AUC

Graph Class private: int V, E; // No.of vertices (V) and edges (E) weightType AdjMatrix[Vmax][Vmax]; // Adjacency Matrix Edge edges[Emax]; // Array of non-zero edges int order; // Order of Visit of a node in the DFS int val[Vmax]; // Array holding order of traversal void getEdges(); // Get edges from adjacency matrix void printEdge(Edge e) const; // Output an edge (e) void visit(int k); // Node Visit Function for DFS }; #endif // GRAPHS_H #include "Graphs.cpp" Prof. Amr Goneid, AUC

6.2 Shortest Paths (General) In a graph G(V,E), find shortest paths from a single source vertex (S) to all other vertices. Edsger Dijkstra published an algorithm to solve this problem in 1959. Prof. Amr Goneid, AUC

Shortest Paths: Dijkstra’s Algorithm Dijkstra’s Algorithm for V vertices: Uses three arrays: - Distance[i]: holds distance from (S) to vertex (i). - Processed[i]: to flag processed vertices. - Via[i]: holds index of vertex from which we can directly reach vertex (i). Prof. Amr Goneid, AUC

Initialization Prof. Amr Goneid, AUC

Method closest to S adjacent to j not yet processed j Distance[j] Already Processed Wij S i Distance[i] z x y adjacent to j Prof. Amr Goneid, AUC

Dijkstra’s Algorithm Repeat Find j = index of unprocessed node closest to (S) Mark (j) as now processed For each node (i) not yet processed: if (i) is adjacent to (j) then { new_distance = Distance[j] + Wij if new_distance < Distance[i] then { Distance[i] = new_distance ; Via[i] = j ; } } Until all vertices are processed Prof. Amr Goneid, AUC

Example A 20  35 15 Initial Source = A A B C D E No yes Dist A 15 20 No yes 20  35 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A 20  35 15 j = B A B C D E No yes Dist A 15 20 B E 35 Via 40 No yes 20  35 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B 20 55 35 15 j = B i = D A B C D E No yes Dist A 15 20 B E No yes 20 55 35 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B 20 55 35 15 j = E A B C D E yes No Dist A 15 20 B E 35 Via yes No 20 55 35 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B E 20 55 30 15 j = E i = C A B C D E yes No Dist A 15 20 B yes No 20 55 30 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B E 20 55 30 15 j = C A B C D E yes No Dist A 15 20 B E 35 yes No 20 55 30 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B E 20 55 30 15 j = C i = D A B C D E yes No Dist A 15 20 B yes No 20 55 30 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B E 20 55 30 15 j = D A B C D E yes Dist A 15 20 B E 35 Via yes 20 55 30 15 Dist A 15 20 B Processed E 35 Via 40 10 D C 35 Prof. Amr Goneid, AUC

Example A B E 20 55 30 15 A→B A→E→C A→B→D A→E Final A B C D E yes Dist yes 20 55 30 15 Dist A 15 20 B Processed E 35 Via 40 10 A→B A→E→C A→B→D A→E D C 35 Prof. Amr Goneid, AUC

How to print the path? /* The function Vname(k) maps a vertex number to a name (e.g a character A, B,.. etc). Given the via[ ] array resulting from Dijkstra’s algorithm, the following recursive function prints the vertices on the shortest path from source (s) to destination (i). */ void Graphs::printPath(int s, int i) const { if (i == s) cout << Vname(s); else { printPath(s,via[i]); cout << Vname(i); } } Prof. Amr Goneid, AUC

Demo http://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.html Shortest Paths Demo1 Shortest Paths Demo2 Prof. Amr Goneid, AUC

6.3 Minimum Cost Spanning Trees (a) Spanning Tree Consider a connected undirected graph G(V,E). A sub-graph S(V,T) is a spanning tree of the graph (G) if: V(S) = V(G) and T  E S is a tree, i.e., S is connected and has no cycles Prof. Amr Goneid, AUC

Spanning Tree S(V,T): V = {A,B,C,D,E,F,G} T = {AB,AF,CD,DE,EF,FG} F E Prof. Amr Goneid, AUC

Spanning Tree Notice that: |T| = |V| - 1 and adding any edge (u,v) T will produce a cycle so that S is no longer a spanning tree (e.g. adding (G,D)) F E G A D C B Prof. Amr Goneid, AUC

One Graph, Several Spanning Trees Prof. Amr Goneid, AUC

Spanning Forest F E G A D C B For a connected undirected graph G(V,E), a spanning forest is S(V,T) if S has no cycles and TE. S(V,T) will be composed of trees (V1,T1), (V2,T2), …, (Vk,Tk), k ≤ |V| F E G A D C B Prof. Amr Goneid, AUC

(b) Minimum Cost Spanning Tree (MST) Consider houses A..F connected by muddy roads with the distances indicated. We want to pave some roads such that: We can reach a house from any other house via paved roads. The cost of paving is minimum. This problem is an example of finding a Minimum Spanning Tree (MST) 4 G 4 E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Minimum Spanning Tree (MST) Cost: For a weighted graph, the cost of a spanning tree is the sum of the weights of the edges in that tree. Minimum Spanning tree: A spanning tree of minimum cost For the shown graph, the minimum cost is 22 4 G 4 E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Kruskal’s Algorithm for MST The algorithm was written by Joseph Kruskal in 1956 A Greedy Algorithm: Builds the MST edge by edge into a set of edges (T). At a given stage, chooses an edge that results in minimum increase in the sum of costs included so far in (T). The set (T) might not be a tree at all stages of the algorithm, but it can be completed into a tree iff there are no cycles in (T). Prof. Amr Goneid, AUC

Kruskal’s Algorithm for MST Builds up forests , then joins them in a single tree. Constraints: - The graph must be connected. - Uses exactly V-1 edges. - Excludes edges that form a cycle. Prof. Amr Goneid, AUC

-Form Set E of edges in increasing order of costs. Abstract Algorithm -Form Set E of edges in increasing order of costs. - Set MST T = empty -Repeat Select an edge (e) from top. Delete edge from E set. Add edge (e) to (T) if it does not form a cycle, otherwise, reject. -Until we have V-1 successful edges. Prof. Amr Goneid, AUC

Example u v w 1 E F 2 A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F Edge u v w accept 1 E F 2 A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C Edge u v w accept 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C Edge u v w accept 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C Edge u v w accept 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B NO 4 4 G E A 9 7 2 3 3 Edge u v w accept 1 E F 2 yes A C 3 4 NO 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B NO 4 4 G E A 9 7 2 3 3 Edge u v w accept 1 E F 2 yes A C 3 4 NO 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B NO 4 4 G E A 9 7 2 3 3 Edge u v w accept 1 E F 2 yes A C 3 4 NO 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B NO 4 4 G E A 9 7 2 3 3 Edge u v w accept 1 E F 2 yes A C 3 4 NO 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

Example u v w 1 E F 2 yes A C 3 4 5 D 6 G 7 8 B NO 4 4 G E A 9 7 2 3 3 Edge u v w accept 1 E F 2 yes A C 3 4 NO 5 D 6 G 7 8 B 4 4 G E A 9 7 2 3 3 C B F 6 4 5 D Prof. Amr Goneid, AUC

How to test for Cycles? Simple Union- Simple Find Given a set (1,2,..,n} initially partitioned into n disjoint sets (each element is its own parent): Find (i): return the sub-set that (i) is in (i.e. return the parent set of i). Union (k,j): combine the two sub-sets that (j) and (k) are in (make k the child of j) Union builds up a tree, while find will search for the root of the tree. Cost is O(log n) Prof. Amr Goneid, AUC

How to test for Cycles? Represent vertices as Disjoint Sets Initially, each node is its own parent (-1) A B C D E F G Let the parent set of u be p(u) A B C D E F G 1 2 3 4 5 6 7 -1 Prof. Amr Goneid, AUC

How to test for Cycles? A B C D E F G 1 2 3 4 5 6 7 -1 A B C D E F G When edge (E,F) is selected, we find that p(E)  p(F). So, we make a union between p(E) and p(F), i.e., p(F) becomes the child of p(E). Same for edge (A,C) A B C D E F G Parent table after selecting (E,F) then (A,C) A B C D E F G 1 2 3 4 5 6 7 -1 Prof. Amr Goneid, AUC

How to test for Cycles? A B C D E F G 1 2 3 4 5 6 7 -1 D A E F G B C When (C,E) is selected, we find that P(C)  P(E). This means that the edge will be accepted. Select (A,E), Find will give P(A) = P(E) (cycle, reject). D A E F G B C Parent table after accepting (E,F) then (A,C), then (C,E), then rejecting (A,E) A B C D E F G 1 2 3 4 5 6 7 -1 Prof. Amr Goneid, AUC

How to test for Cycles? A B C D E F G 1 2 3 4 5 6 7 -1 G D A E F B C When (C,D) is selected, we find that P(C)  P(D). This means that the edge will be accepted. Select (E,G), Find will give P(E)  P(G) (accept). G D A E F B C A B C D E F G 1 2 3 4 5 6 7 -1 Parent table after 5th accepted edge Prof. Amr Goneid, AUC

How to test for Cycles? A B C D E F G 1 2 3 4 5 6 7 -1 G D A E F B C When (D,F) is selected, we find that P(D) = P(F). This means that the edge will be rejected. Select (B,D), Find will give P(B)  P(D) (accept). G D A E F B C A B C D E F G 1 2 3 4 5 6 7 -1 Parent table after 6th accepted edge (Final MST) Prof. Amr Goneid, AUC

Kruskal’s Algorithm Insert edges with weights into a minimum heap Put each vertex in a separate set i = 0 ; While ((i < V-1) && (heap not empty)) { Remove edge (u,v) from heap Find set (j) of connected vertices having (u) Find set (k) of connected vertices having (v) if ( j != k ) { i++; MST [i].u = u; MST [i].v = v; MST [i].w = w; Make a Union between set (j) and set (k); } } Prof. Amr Goneid, AUC

Kruskal’s Algorithm Demo http://www.cs.sunysb.edu/~skiena/combinatorica/animations/mst.html Kruskal's algorithm at work on a graph of distances between 128 North American cities. Almost imperceptively at first, short edges get added all around the continent, slowly building forests until the tree is completed. MST (Kruskal) Demo1 MST (Kruskal) Demo2 Prof. Amr Goneid, AUC

Learn on your own about: Directed Graphs or Digraphs Edge-List representation of graphs Prim’s algorithm for Minimum Spanning Trees. The algorithm was developed in 1930 by Czech mathematician Vojtěch Jarník and later independently by computer scientist Robert C. Prim in 1957 and rediscovered by Edsger Dijkstra in 1959. Therefore it is also sometimes called the DJP algorithm, the Jarník algorithm, or the Prim–Jarník algorithm. Prof. Amr Goneid, AUC