Lecture Notes, 6 Dec 2010

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Index to all lecture notes
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§9.1;
§9.5 to p. 640;
§9.6, esp. pp. 653–655:

Graphs (cont'd)

1. Graphs can be used to model:
   1. computer networks
      1. including the Internet:
         • nodes = computers
           arcs = network connections
           or

         • nodes = local networks
           arcs = network connections to the Internet

      2. and the World Wide Web:
         • nodes = webpages (URLs)
           arcs = directed links between webpages

         • See Hendler et al. 2008
           desJardins et al. 2008
           Horrocks 2008

      3. and social networks
         • See Hoffmann 2009

   2. tournaments:
      • a•→•b for: a beats b

   3. road maps

   4. words and their meanings
      • Let each vertex represent a word,
        and let each edge connect a word with the words in its definition

      • See Massé 2008

   5. the human brain (or the mind (via a "semantic network"))
      • Let vertices represent neurons (or else ideas)
      • Let edges represent synapses (or else associations between ideas)
        • On representing the mind, see:
          Shapiro, Stuart C., & Rapaport, William J. (1987),
          "SNePS Considered as a Fully Intensional Propositional Semantic Network",
          in Nick Cercone & Gordon McCalla (eds.),
          The Knowledge Frontier:
          Essays in the Representation of Knowledge
          (New York: Springer-Verlag): 262–315.
          • shorter version

   6. anything where ∃ binary relation:
      • (which, according to Prof. Dipert (PHI), is everything!)

      • For more on graphs, see Graphs and Trees

   ---

2. Euler Paths & Circuits:
   • pronounced "OILer"
   1. Read about the 7 bridges of Königsberg:
      • The K-Bridge graph:
        • Let nodes represent land, arcs represent bridges:
        

   2. Defs:
      Let G be a graph.
      Then:
      1. A path =def
         a sequence of edges between successive vertices.

      2. An Euler path =def a path that:
         1. is "Eulerian":
            i.e.) contains every edge at least once.
            and
         2. is "simple":
            i.e.) does not contain any edge more than once

      3. An Euler circuit =def
         an Euler path that is a "circuit"
         • i.e.) starts & ends at the same vertex

   3. More Defs:
      1. A connected graph =def
         a graph such that ¬∃ 2 subgraphs with no common edge.
         • i.e.) ∃ a path between any 2 vertices.

         • e.g.) the graph at Lecture Notes for 12/3/10 at I.B.1 is connected

         • e.g.) the graph at Lecture Notes for 12/3/10 at II.B is not connected

      2. A multigraph =def a graph with duplicate edges
         • i.e.) some pairs of vertices are connected by ≥2 edges

      3. Let G=(V,E) be a graph.
         Let v ∈ V.
         Then degree(v) =def
         the number of edges with v as an endpoint
         • i.e.) degree(v) = |{e ∈ E | ∃v′[e = {v,v&prime}]}|

   4. Thm 1:
      A connected multigraph has an Euler circuit
      ↔
      each vertex has even degree.
      1. e.g.)
         

      2. ∴ the Königsberg Bridge graph has no Euler circuit!

      3. Neither does:
         

   5. Thm 2:
      A connected multigraph has an Euler path
      (but no Euler circuit)
      ↔
      it has exactly 2 vertices of odd degree
      • ∴ the K-Bridge graph doesn't even have an Euler path!
      • But the graph at I.B.4.c, above, does!

   ---

3. Traveling Salesman Problem:
   1. Salesman wants to visit n cities exactly once each
      & return to starting point by shortest path
      (e.g., to save gas or time)
      (possibly traveling over same road >1 times)

      1. This is called a Hamiltonian circuit:
         • travel through every vertex exactly once
           & return to starting point

         • inspired by a wooden puzzle devised by Hamilton

      2. Compare Euler circuit:
         • travel through every edge exactly once, & return

      3. See articles by Brian Hayes, especially Hayes 2008.

   2. Fact 1:
      There are no known necessary & sufficient conditions for Hamiltonian circuits
      (as there are for Euler circuits)

   3. Fact 2:
      Let graph G have n vertices.
      Let v₀ be the starting vertex.
      ∴ ∃ ≤ (n–1)! different H-circuits to examine.
      • ∃ ≤ n–1 choices for v₁
      • ∃ ≤ n–2 choices for v₂, etc.

      But can travel H-circuit in reverse order:

      • same trip, only backwards

      ∴ "only" need to examine ≤ (n–1)!/2 H-circuits to find shortest.

      But n=25 → need to examine 24!/2 ≈ 3.1*10²³ circuits (in the worst case)!

      At 1 nanosecond/circuit (= 10^–9sec/circuit), need ∼10,000,000 years!!

      • (actually, 9,837,144 years, 321 days, 4 hours, 48 minutes,
        but who's counting?)

   4. This is computable in theory
      • You could easily write a program to find all Hamiltonian circuits and then find the shortest.

      but not computable in practice

      • What does that mean exactly? Next time!

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