Week 12 - Wednesday

 What did we talk about last time?  Matching  Stable marriage  Started Euler paths

 By simplifying the problem into a graph, the important features are clear  To arrive as many times as you leave, the degrees of each node must be even (except for the starting and ending points) Center Island North Shore East Island South Shore

 There is actually an way to find such a path on a graph where one exists:  Start with a node of odd degree if there is one  Every time we move across an edge, delete it  If you have a choice, always pick an edge whose deletion will not disconnect the graph  At the end of the algorithm, you will either have an Eulerian path or an Eulerian cycle, depending on the graph

 We can label the nodes to make a more interesting problem  Now each piece of land is associated with the Blue Prince, the Red Prince, the Bishop, or the Tavern

 The Blue Prince wants to build an 8th bridge so that he can walk starting at his castle, cross every bridge once, and end at the Tavern to brag

 Put the bridge from the Bishop’s land to the Red Prince’s land

 Furious, the Red Prince wants to build his own bridge so that only he can start at his own castle, cross all the bridges once and then end at the Tavern

 Put the bridge from the Red Prince’s land to the Blue Prince’s land

 Upset by this pettiness, the Bishop decides to build a 10th bridge which allows all citizens to cross all bridges and return to their starting point

 Put the bridge from the Center Island to the Red Prince’s land, making all pieces of land have even degree

Student Lecture

 Binary trees are great  However, only two splits means that you have a height of log 2 n when you want to store n things  If n = 1,000,000, log 2 n = 20  What if depth was expensive? Could we have say, 10 splits?  If n = 1,000,000, log 10 n = 6

 Answer: When the tree is in secondary storage  Each read of a block from disk storage is slow  We want to get a whole node at once  Each node will give us information about lots of child nodes  We don’t have to make many decisions to get to the node we want

 A B-tree of order m has the following properties: 1. The root has at least two subtrees unless it is a leaf 2. Each nonroot and each nonleaf node holds k keys and k + 1 pointers to subtrees where m/2 ≤ k ≤ m 3. Each leaf node holds k keys where m/2 ≤ k ≤ m 4. All leaves are on the same level

Insert

Insert

 Insert the following numbers: 

 When the list of keys drops below half m, we have to redistribute keys  In the worst case, we have to delete a level

 Instead of requiring every non-root node to be half full, every non-root node must be at least 2/3 full  Key redistribution becomes more complex  However, the tree is fuller

 Essentially, make a B-tree such that all the leaves are tied together in a linked list  It is also necessary that all keys in a B-tree appear as leaves  Some other variations are possible, but we’ll end the list here

Are you ready to get Hamiltonian with it?

 The Eulerian Cycle problem asked if you could start at some node, cross every edge once, and return to the start  The Hamiltonian Cycle problem asks if you can start at some node, visit every node only once, and return to the start  In other words, find a tour  Sounds easy, right?

 On this graph: E E A A C C B B D D E A C B D

 Now, on this graph:  There isn’t one! E E A A C C B B D D F F

 Now, on this graph: Hmmm…

 Given a graph, find the shortest tour that visits every node and comes back to the starting point  Like a lazy traveling salesman who wants to visit all possible clients and then return to his home

 Find the shortest TSP tour:  Starts anywhere  Visits all nodes  Has the shortest length of such a tour E E A A D D B B F F C C

 Strategies:  Always add the nearest neighbor  Randomized approaches  Try every possible combination  Why are we only listing strategies?

 So far we haven’t given a foolproof solution to TSP  Let’s look at two possibilities:  Greedy Solution: Pick the closest neighbor  Brute Force: Try all possibilities

 Finish NP-completeness  Elementary sorts  Merge sort

 Start on Project 4  Form teams by Friday  Start on Assignment 6  Read sections 2.1 and 2.2