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Algorithm Design and Analysis (ADA)

Algorithm Design and Analysis (ADA). 242-535 , Semester 1 2013-2014. Objective introduce the main kinds of graphs, discuss two implementation approaches, and remind you about trees. 8 . Introduction to Graphs. Overview. Graphs Graph Terminology Implementing Graphs adjency matrix

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Algorithm Design and Analysis (ADA)

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  1. Algorithm Design and Analysis (ADA) 242-535, Semester 1 2013-2014 • Objective • introduce the main kinds of graphs, discuss two implementation approaches, and remind you about trees 8. Introduction to Graphs

  2. Overview • Graphs • Graph Terminology • Implementing Graphs • adjency matrix • adjency list • Trees and Forests • Tree Terminology

  3. 1. Graphs • A graph has two parts (V, E), where: • V are the nodes, called vertices • E are the links between vertices, called edges • Example: • airports and distance between them 849 PVD 1843 ORD 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 1233 LAX 1120 DFW MIA

  4. 1.1.Graph Types • Directedgraph • the edges are directed • e.g., bus cost network • Undirectedgraph • the edges are undirected • e.g., road network

  5. 1.2. Examples • Electronic circuits • Printed circuit board • Integrated circuit • Transportation networks • Highway network • Flight network • Computer networks • Local area network • Internet • Web • Databases • Entity-relationship diagram

  6. Graphs are everywhere

  7. A Calling Graph • A calling graph for a program: main makeList printList mergeSort 4 examples of recursion split merge

  8. Sheet Metal Hole Drilling • Problem: minimise the moving time of the drill over a metal sheet. continued

  9. A Weighted Graph Version • Add edge numbers (weights) for the movement time between any two holes. 8 b a 6 2 6 4 c 3 d 5 9 12 4 e

  10. 2. Graph Terminology V a b h j U d X Z c e i W g f Y • End vertices (or endpoints) of an edge • U and V are the endpoints • Edges incident on a vertex • a, d, and b are incident • Adjacent vertices • U and V are adjacent • Degree of a vertex • X has degree 5 • Parallel edges • h and i are parallel edges • Self-loop • j is a self-loop

  11. Path • sequence of alternating vertices and edges • begins with a vertex • ends with a vertex • each edge is preceded and followed by its endpoints • Simple path • path such that all its vertices and edges are distinct • Examples • P1=(V,b,X,h,Z) is a simple path • P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple V b a P1 d U X Z P2 h c e W g f Y

  12. Cycle • circular sequence of alternating vertices and edges • each edge is preceded and followed by its endpoints • Simple cycle • cycle such that all its vertices and edges are distinct • Examples • C1=(V,b,X,g,Y,f,W,c,U,a) is a simple cycle • C2=(U,c,W,e,X,g,Y,f,W,d,V,a,) is a cycle that is not simple V a b d U X Z C2 h e C1 c W g f Y Graphs

  13. Connectivity • A graph is connected if there is a path between every pair of vertices Connected graph Non connected graph with two connected components

  14. Some Properties Property Sv degree(v)= 2*| E | Proof: each undirected edge is counted twice(called the handshaking lemma) Property In an undirected graph with no self-loops and no multiple edges |E|  |V| (|V| -1)/2 Proof: each vertex has degree at most (|V| -1) Notation Vset of vertices Eset of edges |. . .| the set size degree()degree of a vertex c Example • | V | = 4 • | E | = 6 • degree(a)= 3 d b a

  15. 3. Implementing Graphs • We will typically express running times in terms of |E| and |V| (often dropping the |’s) • If |E|  |V|2 the graph is dense • can also write this as |E| is O(|v2|) • If |E|  |V| the graph is sparse • or |E| is O(|V|) • Dense and sparse graphs are best implemented using two different data structures: • Adjacency matricies: for dense graphs • Adjacency lists: for sparse graphs

  16. Dense Big-Oh • In the most dense graph, a graph of vverticies will have |V|(|V|-1)/2 edges. • In that case, for large n, |E|is O(|V|2) |V| = 5 |E| = (5*4)/2 = 10

  17. Proof that a graph of n nodes has n(n-1)/2 edges. Write as S(n) = n(n-1)/2 • Basis. S(2) = 1. True. • Inductive Case. • assume S(n) = n(n-1)/2 (1) • try to show S(n+1) = (n+1)n/2 (2) • we know: S(n+1) = S(n) + n which is • S(n+1) = n(n-1)/2 + n which is • S(n+1) = (n+1)n/2 which is (2)

  18. 3.1. Adjacency Matrix a b a b c d e a 0 1 0 0 1 b 1 0 1 0 1 c 0 1 1 0 1 c d 0 0 0 0 1 e 1 1 1 1 0 d e Graph Adjacency Matrix

  19. Properties • An adjacency matrixrepresents the graph as a V * Vmatrix A: • A[i, j] = 1 if edge (i, j)  E = 0 if edge (i, j)  E • The degree of a vertex v (of a simple graph) = sum of row v or sum of column v • e.g. vertexa has degree 2 since it is connected to b and e • An adjacency matrix can represent loops • e.g. vertex c on the previous slide continued

  20. An adjacency matrix can represent parallel edgesif non-negative integers are allowed as matrix entries • ijth entry = no. of edges between vertex i and j • The matrix duplicates information around the main diagonal • the size can be easily reduced with some coding tricks • Properties of graphs can be obtained using matrix operations • e.g. the no. of paths of a given length, and vertex degree

  21. The No. of Paths of Length n • If an adjacency matrix A is multiplied by itself repeatedly: • A, A2, A3, ..., An Then the ijth entry in matrix An is equal to the number of paths from i to j of length n.

  22. Example a b a b c d e A = a 0 1 0 1 0 b 1 0 1 0 1 c c 0 1 0 1 1 d 1 0 1 0 0 e 0 1 1 0 0 d e

  23. a b c d e 0 1 0 1 0 0 1 0 1 0 a 2 0 2 0 1 1 0 1 0 1 1 0 1 0 1 b 0 3 1 2 1 A2 = = c 0 1 0 1 1 0 1 0 1 1 2 1 3 0 1 1 0 1 0 0 1 0 1 0 0 d 0 2 0 2 1 0 1 1 0 0 0 1 1 0 0 e 1 1 1 1 2

  24. Why it Works... • Consider row a, column c in A2: c a-b-c b d 0 a-d-c ( 0 1 0 1 0 ) a 1 b = 0*0 + 1*1 + 0*0 + 1*1 + 0*1= 2 0 d 1 1 continued

  25. A non-zero product means there is at least one vertex connecting verticiesa and c. • The sum is 2 because of: • (a, b, c) and (a, d, c) • 2 paths of length two

  26. The Degree of Verticies • The entries on the main diagonal of A2 give the degrees of the verticies (when A is a simple graph). • Consider vertex c: • degree of c == 3 since it is connected to the edges (c,b), (c,d), and (c,e). continued

  27. In A2 these become paths of length 2: • (c,b,c), (c,d,c), and (c,e,c) • So the number of paths of length 2 for c = the degree of c • this is true for all verticies

  28. Coding Adjacency Matricies • #define NUMNODESnint arcs[NUMNODES][NUMNODES]; • arcs[u][v] == 1 if there is an edge (u,v); 0 otherwise • Storage used: O(|V|2) • The implementation may also need a way to map node names (strings) to array indicies. continued

  29. If n is large then the array will be very large, with almost half of it being unnecessary. • If the nodes are lightly connected then most of the array will contain 0’s, which is a further waste of memory.

  30. Representing Directed Graphs • A directed graph: 0 1 3 2 4

  31. Its Adjacency Matrix • Not symmetric; all the array may be necessary. • Still a waste of space if nodes are lightly connected. finish 0 1 2 3 4 0 1 1 1 0 0 0 0 0 1 0 1 2 1 1 0 0 1 start 0 0 1 0 1 3 0 1 0 0 0 4

  32. When to use an Adjacency Matrix • The adjacency matrix is an efficient way to store dense graphs. • But most large interesting graphs are sparse • e.g., planar graphs, in which no edges cross, have |e| = O(|v|) by Euler’s formula • For this reason the adjacency list is often a better respresentation than the adjacency matrix

  33. Euler’s Formula Characteristic • Euler (1752) proved that for any connected graph, where: F = no. of facesE = no. of edgesV = no. of verticies/nodes then the formula holds: F = E – V + 2 F = 5; E = 9; V = 6

  34. 3.2. Adjacency List • Adjacency list: for each vertex v  V, store a list of vertices adjacent to v • Example: • adj[0] = {0, 1, 2} • adj[1] = {3} • adj[2] = {0, 1, 4} • adj[3] = {2, 4} • adj[4] = {1} • Can be used for directed and undirected graphs. 0 1 3 2 4

  35. 0 1 0 1 2 4 3 1 4 2 • An implementation diagram: adj[] 0 1 2 3 means NULL size of array = no. of vertices (|V|) 4 no. of cells == no. of edges (|E|)

  36. Data Structures • struct cell { /* for a linked list */ Node nodeName; struct cell *next;};struct cell *adj[NUMNODES]; • adj[u] points to a linked list of cells which give the names of the nodes connected to u.

  37. Storage Needs • How much storage is required? • The degree of a vertex v == number of incident edges • directed graphs have in-degree, out-degree values • For directed graphs, the number of items in an adjacency lists is out-degree(v) = |E| • This uses (V + E) storage

  38. For undirected graphs, the number of items in the adjency list is  degree(v) = 2*|E| (the handshaking lemma) • Why? If we mark every edge connected to every vertex, then by the end, every edge will be marked twice • This also uses (V + E) storage • In summary, adjacency lists use (V+E) storage

  39. 3.3. Running Time: Matrix or List? • Which representation is better for graphs? • The simple answer: • dense graph – use a matrix • sparse graph – use an adjcency list • But a more accurate answer depends on the operations that will be applied to the graph. • We will consider three operations: • is there an edge between u and v? • find the successors of u (in a directed graph) • find the predecessors of u (in a directed graph) continued

  40. Is there an edge (u,v)? • Adjacency matrix: O(1) to read arcs[u][v] • Adjacency list: O(1 + E/V) // forget the |...| • O(1) to get to adj[u] • length of linked list is on average E/V • if a sparse graph (E<<V): O(1+ E/V) => O(1) • if a dense graph (E ≈ V2): O(1+ E/V) => O(V)

  41. Find u’s successors (u->v) • Adjacency matrix: O(V) since must examine the entire row for vertex u • Adjacency list: O(1 + (E/V)) since must look at entire list pointed to by adj[u] • if a sparse graph (E<<V): O(1+ E/V) => O(1) • if a dense graph (E ≈ V2): O(1+ E/V) => O(V)

  42. Find u’s predecessors (t->u) • Adjacency matrix: O(V) since must examine the entire column for vertex u • a 1 in the row for ‘t’ means that ‘t’ is a predecessor • Adjacency list: O(E) since must examine every list pointed to by adj[] • if a sparse graph (E<<V): O(E) is fast • if a dense graph (E ≈ V2): O(E) is slow

  43. Summary: which is faster? • Operation Dense Graph Sparse GraphFind edge Adj. Matrix EitherFind succ. Either Adj. listFind pred. Adj. Matrix Either • As a graph gets denser, an adjacency matrix has better execution time than an adjacency list.

  44. 3.4. Storage Space: Matrix or List? • The size of an adjacency matrix for a graph of V nodes is: • V2 bits (assuming 0 and 1 are stored as bits) continued

  45. An adjacency list cell uses: • 32 bits for the integer, 32 bits for the pointer • so, cell size = 64 bits • Total no. of cells = total no. of edges, e • so, total size of lists = 64*E bits • successors[] has V entries (for V verticies) • so, array size is 32*V bits • Total size of an adjacency list data struct: 64*E + 32*V

  46. Size Comparison • An adjacency list will use less storage than an adjacency matrix when: 64*E + 32*V < V2 which is: E < V2/64 – V/2 When V is large, ignore the V/2 term: E < V2/64 continued

  47. V2 is (roughly) the maximum number of edges. • So if the actual number of edges in a graph is 1/64 of the maximum number of edges, then an adj. list representation will be smaller than an adj. matrix coding • but the graph must be quite sparse

  48. 4. Trees and Forests • A (free) tree is an undirected graph T such that • T is connected • T has no cycles This definition of tree is different from the one of a rooted tree • A forest is an undirected graph without cycles • The connected components of a forest are trees Tree Forest Graphs

  49. Uses of Trees President Vice-President for Academics Vice-President for Admin. Dean of Engineering Dean of Business Planning Officer Purchases Officer . . . . . . . . Head of CoE Head of EE Head of AC. . . . .

  50. Saturated Hydrocarbons H • Non-rooted (free) trees • a free tree is a graph with no cycles H H H C H H C C C H H C H H H C H H H C H H Isobutane H C H H Butane H

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