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1 Fundamentals of Computer Sience and Digital Communications - Logic and Algorithms - Graph Algorithms Johan Larsson Marco Loh

2 Overview What is a Graph? Representations of Graphs Breadth-first Search Depth-first Search Outlook Johan Larsson, Marco Loh

3 What is a Graph? Definition G = ( V, E) V set of vertices E {( v, w) v, w V} b c d Types Undirected / directed Weighted / unweighted Self-loops Cycles Multiple edges a e f g Johan Larsson, Marco Loh

4 Johan Larsson, Marco Loh Logic and Algorithms - Graph Algorithms 4 Adjacency - Matrix Definition weighted graphs accordingly = otherwise E j i if aij, ), (, g f e d c b a g f e d c b a

5 Adjacency - List Definition Array of V lists Adj( v), v V Adj( v) contains all vertices u with ( v, u) E a b c d e f g b e c e f d g f g d Johan Larsson, Marco Loh

6 Graph Algorithms Search Algorithms Minimal Spanning Trees Coloring and Matching Problems Planar Graphs Hamilton and Euler-Graphs Traveling-Salesman Problem N-Connectivity Problems Analytic and Constructive Flow Problems Johan Larsson, Marco Loh

7 Breadth-first Search Classification Single-source graph-search algorithm Unweighted graphs (directed or undirected) Use Finding shortest paths in graphs Building up spanning tree Programming Queue necessary Structure for marking nodes Adjacency list yields better performance Johan Larsson, Marco Loh

8 Example b 2 c 3 d a Q : Ø e f g 2 3 Step : 8 Johan Larsson, Marco Loh

9 Sketch Algorithm Breadth-First Search Initialize data structures Enqueue starting node While Queue not empty Dequeue node v For all descendants u of v, which are not yet discovered Enqueue u Mark u as discovered Build up structure for tree and distances Johan Larsson, Marco Loh

10 Analysis Initialization and queue operations Ο( V ) Adjacency list operations Ο( E ) Total Ο( V ) + Ο( E ) = Ο( V + E ) Johan Larsson, Marco Loh

11 Related problem Diameter Diameter of a graph G=(V,E) is given by max, u v V δ ( u, v), with δ ( u, v) shortest path from u to v Give an efficient algorithm Perform complexity analysis Johan Larsson, Marco Loh

12 Depth-first Search Classification Single-source graph-search algorithm Unweighted graphs (directed or undirected) Use Characterizing graph structure Easily implemented algorithm Programming Use a stack or recursive calls Structure for additional node information Adjacency list gives better performance Johan Larsson, Marco Loh

13 Example /4 a 2/3 b 3/8 c 4/5 d e f g /2 9/ 6/7 after 4 Steps Johan Larsson, Marco Loh

14 Sketch Algorithm Depth-First Search Initialize data structures Call DFS-Visit(starting node) DFS-Visit(v) Mark v discovered For all descendants u of v, that are not discovered Build up structure for tree and time stamps Call DFS-Visit(u) Johan Larsson, Marco Loh

15 Alternative Approach Algorithm Depth-First Search with Stack Initialize data structures PUSH starting node onto stack While Stack not empty While exists undiscovered node u adjacent to TOP(stack) Mark u as discovered PUSH u onto stack POP(stack) Johan Larsson, Marco Loh

16 Analysis Initialization Θ( V ) DFS-Visit calls Θ( E ) Total Θ( V + E ) Johan Larsson, Marco Loh

17 Edge Classification Tree edge Edge followed in depth-first search a Back edge Backward-directed edge along the tree B b F Forward edge (only directed graphs) c More direct connection to a descendant in a tree d Cross edge (only directed graphs) Connection between nodes, which are neither ancestors nor descendants of each other C e Johan Larsson, Marco Loh

18 Related problem Single Connectivity At most one path from u to v for all vertices u,v V Give an efficient algorithm to decide that property Johan Larsson, Marco Loh

19 Outlook Positively-weighted graphs Dijkstra (Heap-Structures) Negatively-weighted graphs (non-negative cycles!) Bellman-Ford All-source shortest paths Floyd / Kleene Algorithm Negative Cycles Checks in Bellman-Ford or Floyd Johnson-Recalibration Heuristic and approximate approaches Johan Larsson, Marco Loh

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