Algorithm Analysis Graph algorithm. Chung-Ang University, Jaesung Lee
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1 Algorithm Analysis Graph algorithm Chung-Ang University, Jaesung Lee
2 Basic definitions Graph = (, ) where is a set of vertices and is a set of edges Directed graph =<, > where consists of ordered pairs Weighted graph with a weight function ( ) where Degree of a vertex : deg The number of incoming edges to : indeg The number of outgoing edges from : outdeg Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 2
3 Representation of Graphs Considering graph =, where =,,, Adjacency matrix: matrix =, where, =,, 0, h Weighted adjacency matrix:, =,,,, h Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 3
4 Representation of Graphs Considering graph =, where =,,, Adjacency list: an array,, of pointers Adjacency matrix Adjacency list Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 4
5 Paths, Cycles and Subgraphs Path : a sequence of vertices = (,,, ) where,, Simple path: where, Cycle: a sequence of vertices = (,,,, = ) where,, Simple cycle: where, except the pair, Subgraphs: a graph =, is iff and Connected graph: for every pair of vertices,, there exists =,, Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 5
6 Trees and Spanning Trees Tree: a connected acyclic graph Spanning tree: a tree that contains all vertices in and is a subgraph of Let be a spanning tree of a graph. Then Any two vertices in are connected by a unique simple path. If any edge is removed from, then becomes disconnected. If we add any edge into, then the new graph will contain a cycle. Number of edges in is. Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 6
7 Minimum Spanning Trees (MST) Weight of a spanning tree : the sum of weights of all edges in Minimum Spanning Tree: a spanning tree with the smallest possible weight Graph Three (of many possible) spanning trees from Weighted Graph MST of 2 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 7
8 Minimum Spanning Trees (MST) Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 8
9 Examples of MST Network Design Airline routes Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 9
10 MST related algorithms Kruskal s algorithm For a graph with vertices, keep on adding the least cost edge until edges added Prim s algorithm By focusing on vertices instead of edges, yielding a simple algorithm (implementation) Dijkstra s algorithm: The Single Source Shortest-Path problem For example, 2, 3, 3 4, where =, 2, 3, 4, 5 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 0
11 Dijkstra s Algorithm Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee
12 Dijkstra s Algorithm Initial node Y v??? v 2 v Initial node Y v v 5 v 4 v 2 v??? v 3 v??? v 4 v??? v 5 v 5 6 v v v 2 v v 3 v v 5 v 4 v v 5 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 2
13 Algorithm: Description v v 5 v v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 3
14 Algorithm: Description Starting point v v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 4
15 Algorithm: Description Current intersection v v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 5
16 Algorithm: Description Current intersection v v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 6
17 Algorithm: Description Current intersection v min(,7) min(,4) min(,6) min(,) v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 7
18 Algorithm: Description Current intersection v v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 8
19 Algorithm: Description v Current intersection v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 9
20 Algorithm: Description v Current intersection v 5 v Destination v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 20
21 Algorithm: Description v Current intersection v 5 v Destination v 4 5 v Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 2
22 Algorithm: Description v Current intersection min (+,7) min (+,4) min (+,6) min (+0,) v 5 v Destination v 4 5 v Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 22
23 Algorithm: Description v v 5 v Current intersection Destination v 4 5 v 3 2 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 23
24 Algorithm: Description v min (2+3,7) min (2+,4) min (2,2) v 5 v Current intersection Destination v 4 5 v 3 2 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 24
25 Algorithm: Description v v 5 v Current intersection Destination v 4 5 v 3 2 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 25
26 Algorithm: Description v v 5 v Current intersection Destination v 4 5 v 3 2 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 26
27 Algorithm: Pseudocode Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 27
28 Algorithm: Pseudocode Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 28
29 Algorithm using Priority Queue Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 29
30 Algorithm: Procedures Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 30
31 Algorithm: Complexity Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 3
32 Kruskal s Algorithm Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 32
33 Kruskal s Algorithm Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 33
34 Algorithm: Description v v v 5 v 2 v 5 v v 4 v 3 v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 34
35 Algorithm: Description 6 v 4 7 v 5 v v 2 v 5 v v 4 v 3 v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 35
36 Algorithm: Description v 5 6 v v 2 v v 5 3 v 2 2 v 4 v 3 v 4 5 v 3 Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 36
37 Algorithm: Pseudocode Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 37
38 Algorithm: Complexity log ( log ) Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 38
39 Algorithm: Example Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 39
40 Algorithm: Example Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 40
41 Algorithm: Example Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 4
42 Algorithm: Example Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 42
43 Algorithm: Example Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 43
44 Prim s Algorithm Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 44
45 Algorithm: Description Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 45
46 Algorithm: Description Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 46
47 Algorithm: Description Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 47
48 Algorithm: Complexity Minimum edge weight data structure Time complexity in total Adjacency Matrix, searching Binary heap and adjacency List + log = ( log ) Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 48
49 Algorithm: Complexity Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee 49
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![Tutorial. Question There are no forward edges. 4. For each back edge u, v, we have 0 d[v] d[u].](/thumbs/88/114871846.jpg "Tutorial. Question There are no forward edges. 4. For each back edge u, v, we have 0 d[v] d[u].")
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