Graphs. COMPSCI 355 Fall 2016

Transcription

1 Graphs COMPSCI 355 Fall 26

2 Bridges of Königsberg

3 Four Color Problem

4 Graph A set of objects called nodes or vertices. A binary relation (edges) on nodes. Adjacent nodes are related by an edge. 3 4 A path is a sequence of adjacent nodes. 2 A simple path is a path in which each node occurs exactly once. A cycle is a path that starts and ends at the same vertex A simple cycle is a cycle in which each node occurs exactly once (except for the starting and ending node).

5 Graph Mathematical model of a network Internet transportation urban services social networks electronic circuits telecommunications molecular structures oil pipelines and pumping stations semantic structures for knowledge representation 7

6 Directed Graph

7 Weighted Graph

8 World's Most Famous Graph What does it look like? What laws govern the structure and behavior of small-world graphs. Selfish routing and the price of anarchy.

9 Social Networks Paul Erdös Bill Bill Gates Gates Drue Drue Coles Coles

10 Routing Problems 274 BOS 846 ORD SFO LAX JFK 258 DFW 9 2 MIA 2342

11 Conflict Resolution Six professors (A-F) Overlapping class times 3 classrooms (colors) A A FF B B C C E E D D

12 Conflict Resolution Six professors (A-F) Overlapping class times 3 classrooms (colors) A A FF B B C C E E D D

13 State Graphs Goal: move coins from HTH to THT. Move: flip one of the coins middle coin can be flipped at any time. end coin can be flipped only if the other two coins are the same. HHH THH HTH HHT TTH THT HTT TTT

14 Pattern Matching Finite state automaton that accepts binary strings containing as a substring. start accept

15 Planar Graphs Can be drawn on a flat surface without any edges crossing. WA OR ID NV UT CA AZ

16 Planar Graphs Is this a planar graph? 2 3 4

17 Planar Graphs Is this a planar graph?

18 A Non-Planar Graph Water Water Gas Gas Electricity Electricity H H H2 H2 H3 H3

19 Four Color Problem How many colors are needed to color the regions of a map so that no two adjacent regions have the same color?

20 Four Color Problem Three colors are not enough.

21 Four Color Theorem Any map can be colored using only 4 colors in such a way that any two regions with a common border have a different color. First conjectured in 852 Many failed proofs since then Proved in 976 by Appel and Hanken (Univ. of Illinois) Controversy with philosophical implications Simplified in 997 Generally accepted now

22 Euler Circuit A circuit that includes every edge exactly once. BOS ORD SFO JFK LAX DFW MIA

23 Euler Circuits Do these graphs have Euler circuits?

24 Euler's Theorem A graph has an Euler circuit if and only if it is connected and every vertex has even valence.

25 Euler's Theorem

26 Euler's Theorem

27 Hamiltonian Circuits A circuit that includes every vertex exactly once. Many practical applications routing delivery trucks moving an industrial robot arm

28 Hamiltonian Circuits Easy to construct graphs that do not have a Hamiltonian circuit.

29 Hamiltonian Circuits Easy to construct graphs that do not have a Hamiltonian circuit.

30 Hamiltonian Circuits Easy to construct an infinite family of graphs that have no Hamiltonian circuits.

31 Hamiltonian Circuits Easy to construct an infinite family of graphs that have no Hamiltonian circuits.

32 Famous Conjecture There does not exist an efficient algorithm to: find a Hamiltonian circuit in a given graph. decide if a graph has a Hamiltonian circuit. If you prove (or disprove) this conjecture, you will become very famous.

33 Min-Cost Ham Circuits Cleveland St. Louis Minneapolis Chicago

34 Min-Cost Ham Circuits When the graph is complete, searching for a minimum cost Hamiltonian circuit is known as the Traveling Salesman Problem. Cleveland miles miles miles + 54 miles miles St. Louis Minneapolis Chicago

35 Complete Graphs

36 TSP in Disguise Vehicle routing UPS, Dominos, school buses,... Robot arm movement soldering connections drilling holes cutting and welding Job scheduling Computational chemistry

37 Solving TSP Cleveland CHI M 349 S S CL M CL M S CL St. Louis CL 3 S CL M S M Minneapolis CHI Chicago CHI CHI CHI CHI CHI

38 Solving TSP Simple solution: just check every tour and keep track of the one with the lowest cost. But wait, TSP is supposed to be hard...

39 Solving TSP Simple solution: just check every tour and keep track of the one with the lowest cost. But wait, TSP is supposed to be hard... Obviously solvable in O(n!) time. Number of tours: (n-)! / 2

40 Solving TSP Simple solution: just check every tour and keep track of the one with the lowest cost. But wait, TSP is supposed to be hard... Obviously solvable in O(n!) time. Number of tours: (n-)! / 2 Can be solved in O(n22n) time Dynamic programming

41 Adjacency List Representation

42 Adjacency Matrix Representation