The document discusses different data structures and their implementations and applications. It covers arrays, linked lists, stacks, queues, binary trees, and binary search. The key points are:
- Arrays allow fast access but have fixed size; linked lists can grow dynamically but access is slower.
- Binary trees allow fast (O(log n)) search, insertion, and deletion operations due to their hierarchical structure.
- Stacks and queues are useful for modeling LIFO and FIFO data access with applications like function calls and job scheduling.
- Binary search runs in O(log n) time by recursively dividing the search space for sorted data.
8. Arrays (Cont.) Multi-dimensional Array A multi-dimensional array of dimension n (i.e., an n -dimensional array or simply n -D array) is a collection of items which is accessed via n subscript expressions. For example, in a language that supports it, the (i,j) th element of the two-dimensional array x is accessed by writing x[i,j]. m x i : : : : : : : : : : : : : : : 2 1 0 n j 10 9 8 7 6 5 4 3 2 1 0 C o l u m n R O W
60. Graph Terminology A E D C B F a c b d e f g h i j Vertices A and B are endpoints of edge a
61. Graph Terminology A E D C B F a c b d e f g h i j Vertex A is the origin of edge a
62. Graph Terminology A E D C B F a c b d e f g h i j Vertex B is the destination of edge a
63. Graph Terminology A E D C B F a c b d e f g h i j Vertices A and B are adjacent as they are endpoints of edge a
64.
65. Graph Terminology U Y X W V Z a c b d e f g h i j Edge 'a' is incident on vertex V Edge 'h' is incident on vertex Z Edge 'g' is incident on vertex Y
66. Graph Terminology U Y X W V Z a c b d e f g h i j The outgoing edges of vertex W are the edges with vertex W as origin {d, e, f}
67. Graph Terminology U Y X W V Z a c b d e f g h i j The incoming edges of vertex X are the edges with vertex X as destination {b, e, g, i}
68.
69. Graph Terminology U Y X W V Z a c b d e f g h i j The degree of vertex X is the number of incident edges on X. deg(X) = ?
70. Graph Terminology U Y X W V Z a c b d e f g h i j The degree of vertex X is the number of incident edges on X. deg(X) = 5
71. Graph Terminology U Y X W V Z a c b d e f g h i j The in-degree of vertex X is the number of edges that have vertex X as a destination. indeg(X) = ?
72. Graph Terminology U Y X W V Z a c b d e f g h i j The in-degree of vertex X is the number of edges that have vertex X as a destination. indeg(X) = 4
73. Graph Terminology U Y X W V Z a c b d e f g h i j The out-degree of vertex X is the number of edges that have vertex X as an origin. outdeg(X) = ?
74. Graph Terminology U Y X W V Z a c b d e f g h i j The out-degree of vertex X is the number of edges that have vertex X as an origin. outdeg(X) = 1
75.
76. Graph Terminology U Y X W V Z a c b d e f g h i j We can see that P1 is a simple path. P1 = {U, a, V, b, X, h, Z} P1
77. Graph Terminology U Y X W V Z a c b d e f g h i j P2 is not a simple path as not all its edges and vertices are distinct. P2 = {U, c, W, e, X, g, Y, f, W, d, V}
86. Depth First Search Algorithim DFS() Input graph G Output labeling of the edges of G as discovery edges and back edges for all u in G.vertices() setLabel(u, Unexplored) for all e in G.incidentEdges() setLabel(e, Unexplored) for all v in G.vertices() if getLabel(v) = Unexplored DFS(G, v).
87. Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G as discovery edges and back edges setLabel(v, Visited) for all e in G.incidentEdges(v) if getLabel(e) = Unexplored w <--- opposite(v, e) if getLabel(w) = Unexplored setLabel(e, Discovery) DFS(G, w) else setLabel(e, BackEdge)
88. Depth First Search A A Unexplored Vertex Visited Vertex Unexplored Edge Discovery Edge Back Edge
111. Breadth First Search Algorithm BFS(G) Input graph G Output labeling of the edges and a partitioning of the vertices of G for all u in G.vertices() setLabel(u, Unexplored) for all e in G.edges() setLabel(e, Unexplored) for all v in G.vertices() if getLabel(v) = Unexplored BFS(G, v)
112. Algorithm BFS(G, v) L 0 <-- new empty list L0.insertLast (v) setLabel(v, Visited) i <-- 0 while( ¬L i.isEmpty()) L i+1 <-- new empty list for all v in G.vertices(v) for all e in G.incidentEdges(v) if getLabel(e) = Unexplored w <-- opposite(v) if getLabel(w) = Unexplored setLabel(e, Discovery) setLabel(w, Visited) Li+1.insertLast (w) else setLabel(e, Cross) i <-- i + 1
149. A(0) D C B F E 8 4 2 1 7 5 9 3 2 Add starting vertex to cloud.
150. A(0) D C B F E 8 4 2 1 7 5 9 3 2 We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices.
151. A(0) D(4) C(2) B(8) F E 8 4 2 1 7 5 9 3 2 We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices.
152. A(0) D(4) C(2) B(8) F E 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).
153. A(0) D(3) C(2) B(8) F(11) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}
154. A(0) D(3) C(2) B(8) F(11) E(5) 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).
155. A(0) D(3) C(2) B(8) F(8) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}.
156. A(0) D(3) C(2) B(8) F(8) E(5) 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).
157. A(0) D(3) C(2) B(7) F(8) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}.
158. A(0) D(3) C(2) B(7) F(8) E(5) 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).
159. A(0) D(3) C(2) B(7) F(8) E(5) 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).