MA/CS 109 Lecture 2. The Konigsberg Bridge Problem And Its generalizabons

Transcription

1 MA/CS 109 Lecture 2 The Konigsberg Bridge Problem And Its generalizabons

2 All Course informabon at hdp:// Remark: This page will be conbnually updated. First Homework to be posted soon. Due Tues Sept 19 IN CLASS Discussions meet Monday (and Tuesday)

3 Template for Doing MathemaBcs Problem Model Repeat Modify Examples/Conjectures Model Proof- - - Did we answer the quesbon?- - - No Yes Fame + $$$

4 Is it possible to walk through the city, crossing each bridge exactly once and ending where you began?

5 Don t care about the building, only the brigdes.

6 Doesn t mader how wide the bridges are or how curvy or how wide the river is.

7 Finally, the water doesn t mader and how big the land masses are doesn t mader and the color of the picture certainly doesn t mader So all we really need to talk about the 7 bridges of Konigsberg is the picture

8 Networks A network is a collecbon of dots called nodes, and a collecbon of curves that start and end at nodes. The curves are called edges. [NOTE: Nodes are somebmes called verbces and networks are somebmes called graphs but we will use Network to avoid confusion with graphs with horizontal and verbcal axes ]

9 We can spedivy a network by giving a list of nodes and a list of edges and saying what the end points of each edga are. The Konigsberg netowrk has four nodes (1,2,3,4) and seven edges (a,b,c,d,e,f,g). a goes 1 to 2 b goes 1 to 2 c goes 2 to 3 d goes 2 to 3 e goes 1 to 4 f goes 2 to 4 g goes 3 to 4 a c b d f g e 4

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11 A path on a network is a list of edges such that each edge of the list ends at a node where the next edge begins. So on the network below, starbng at node 1, abacgje is a path, but startgin at node 1, abfc is not. 1 e a c 2 3 b d f g 4

12 A circuit is a path that starts and ends at the same node. So starbng at node 1, abacge is a circuit, but starbng at node 1, afgc is a path that is not a circuit (it is a path, but ends at node 2). a c b d f g e 4

13 An Euler circuit (pronounced Oil- er ) is a circuit that uses each edge of the graph exactly once (so a path that starts and ends at the same node that uses each edge exactly once). a c b d f g e 4

14 Leonhard Euler: Dead, white, male, European mathemabcian in a funny hat Eu

15 Konigsberg Bridge Problem: Does the network below have an Euler circuit? 1 e a 2 b f 4 c d g 3

16 Why bother The Konigsberg bridge problem is a toy problem a starter problem that helps us to think about what is important in a network. The study of paths and circuits in networks has lots of applicabons efficient delivery problems, communicabon in terrorist networks,

17 Other applicabons: Spread of disease: Nodes = individuals, edge represents Has had sex with paths indicate how sexually transmided disease can spread Computer communicabon: Nodes = individual computers (connected to the internet), edges represent exchanged rumors spread along paths. Internet traffic: Nodes = web pages, edges are hyperlinks. Paths indicate possibility of surfing from on page to another.

18 NavigaBon: Nodes = intersecbons on a map, edges are roads connecbng intersecbons. A path is literally a path decorate paths with lengths or Bme of travel and ask for shortest paths. Ecology: Nodes = species, edge is the relabonship eats or is eaten by (we may want to add arrows to indicate eats giving a directed network like a map with one way streets). Paths indicate the possibility of impact of eliminabng one species on the populabon of another.

19 Etc., etc., A good abstract idea can have lots of applicabons. Figuring out something about the abstract problem tells us something about all the applicabons (That is the power of abstract thinking. Even though it simplifies the real world, it also extends the applicability of our work in unexpected ways.) Back to Konigsberg and Euler circuits

20 Go the problem- made the model Problem Model Repeat Modify Examples/Conjectures Model Proof- - - Did we answer the quesbon?- - - No Yes Fame $$$

21 Examples: Always start simple! Given a network, how can you tell if it has an Euler circuit or not? Example of a network without an Euler circuit This network is disconnected, no way to get from one piece to the other.

22 First restricbon: We say a network is strongly connected if for every pair of edges, there is a path that uses both edges (i.e., it is possible to walk from every bridge to every other bridge.) Since this is required for Euler circuits, we will just add it to our model. From now on, all networks we consider will be strongly connected.

23 Also, for convenience, we will not allow edges that start and end at the same node that is, the picture Is not allowed real bridges don t do this. (Building the model is a powerful is a powerful step you can create the world you want.)

24 (We are actually modifying the model here adding restricbons to avoid cases we don t want to think about. This means our work won t be quite so general, but it will be a lidle easier and we make the trade off.)

25 Three simple networks 1 a 2 d b 3 Does the ler one have an Euler circuit?

26 Three simple networks 1 a 2 d b 3 The ler one has an Euler circuit. We know this ( prove this) since we can say what the Euler circuit is start at 1 then abc.

27 Three simple networks 1 a 2 d b 3 The middle one does NOT have an Euler circuit. You can try all cases 3 places to start and the only choice is a or b at node 1.

28 Three simple networks 1 a 2 d b 3 The right hand one is more difficult. There are three places to start, and more choices but not So many that you can t try them all

29 d Start at 2: acb nope bca nope bcd nope bda nope bdc nope ADempts at Euler paths: Start at 1: abcd nope abdc nope cbad nope cdab nope dbac nope dcab nope Start at 3: bacd nope badc nope cabd nope dabc nope cdba nope dcba nope NO POSSIBLE PATH IS AN EULER CIRCUIT tried them all Proof by exhausbon.

30 What did we learn? Even simple networks can have lots of paths! But what keeps a path from becoming an Euler circuit? In all the cases we tried, we either ran out of edges before we got back to the starbng node or we got stuck that is, we got to a node where there were no unused edges to leave the node on (but the circuit wasn t completed).

31 How can we use this observabon? Say it a different way. If a network HAS an Euler circuit, then when you walk along that Euler circuit, every Bme you reach a node there is an unused edge available to leave that node, unbl you get to the end of the circuit and are back where you started.

32 So If you have a network that has an Euler circuit, then at every node that isn t the start=end node of the circuit, every Bme you arrive at the node via an edge, you can leave by another (not yet used) edge. So At every node the arriving edges match up with the leaving edges.

33 Since every edge is used by the Euler circuit, for each node, the edges touching are either arriving edges or leaving edges (never both) and the number of arriving edges must equal the number of leaving edges.

34 New idea (counbng the edges that touch a node), so we need a new word. The degree of a node is the number of edges that start or end at that node

35 But then If a network has an Euler circuit, a node that is not the start=end node has Degree =arriving edges + leaving edges =arriving edges + arriving edges = 2 (arriving edges). We call an integer that is 2 Bmes another integer an even number.

36 So saying this more efficiently If a network has an Euler circuit then every node that is not the start=end node must have even degree. Wow What about the start=end node? Well, the first edge in the Euler circuit is a leaving edge, but the last edge is an arriving edge. Every other Bme you arrive at this node you must leave. So even at the start=end node, the number of arriving edges matches the number of leaving edges so the degree is even.

37 StarBng Node What about the starbng node? If you are standing there, your friend starts the Euler circuit by leaving (on a leaving edge). Each Bme they return (on an arriving edge), the either leave again (on a leaving edge) like at other nodes OR the circuit is finished.

38 So, we can pair the arriving and leaving edges as before, by pairing the first leaving edge with the last arriving edge. So the degree of the starbng node is even also!! Conjecture: If a network has an Euler circuit, then the degree of every node of the network must be an even number.

39 Conjecture: If a network has an Euler circuit, then the degree of every node of the network must be an even number. What this says (and what it doesn t say!). IF you have a network that (somehow) you already know has an Euler Circuit, THEN the degree of every node is even. (IF I have a network with all even degree nodes, this conjecture says NO COMMENT only says something one way.)

40 Proof: Remember, a proof is an explanabon of why a statement MUST be true. It isn t evidence that it is true, OR an example where it is true. Proofs are forever

41 Since a proof is just an explanabon why a conjecture (or theory) must be true, we don t need special language or special symbols. We just need to write carefully, making sure each statement follows from the ones that come before. For our conjecture on Euler circuits, we need only repeat carefully our thinking above that lead to the conjecture.

42 Proof of our conjecture: Suppose you have any network that has an Euler circuit (I m not telling you ANYTHING else about the network other than it has an Euler circuit). At any node (that isn t the starbng node of the circuit), every edge touching the node is an arriving edge or a leaving edge of the circuit, and each arriving edge can be paired with the leaving edge that follows it in the circuit.

43 Does this help? Does this answer say anything about Konigsberg? If Konigsberg had an Euler circuit, then the degree of every node would have to be even But

44 Hey there are nodes of odd degree In fact, every node has odd degree!! So this graph can NOT have an Euler circuit!! There was no path around Konigsberg that was an Euler circuit!. VICTORY FAME $$$