Great Theoretical Ideas In Computer Science John Lafferty Lecture 21 CS 15 251 November 7 2006 Fall 2006 Carnegie Mellon University Graphs II Recap Theorem Let G be a graph with n nodes and e edges The following are equivalent 1 G is a tree connected acyclic 2 Every two nodes of G are joined by a unique path 3 G is connected and n e 1 4 G is acyclic and n e 1 5 G is acyclic and if any two nonadjacent points are joined by a line the resulting graph has exactly one cycle Cayley s formula The number of labeled trees on n nodes is n n 2 A graph is planar if it can be drawn in the plane without crossing edges A plane graph is any such drawing which breaks up the plane into a number f of faces or regions Euler s Formula If G is a connected plane graph with n vertices e edges and f faces then n e f 2 Euler s Formula If G is a connected plane graph with n vertices e edges and f faces then n e f 2 The beauty of Euler s formula is that it yields a numeric property from a purely topological property Graph Coloring A coloring of a graph is an assignment of a color to each vertex such that no neighboring vertices have the same color Finding Optimal Trees Trees have many nice properties uniqueness of paths no cycles etc May want to compute the best tree approximation to a graph If all we care about is communication then a tree may be enough Want tree with smallest communication link costs Finding Optimal Trees Problem Find a minimum spanning tree that is a tree that has a node for every node in the graph such that the sum of the edge weights is minimum Tree Approximations 7 8 4 5 9 7 8 6 11 9 Finding an MST Kruskal s Algorithm Create a forest where each node is a separate tree Make a sorted list of edges S While S is non empty Remove an edge with minimal weight If it connects two different trees add the edge Otherwise discard it Applying the Algorithm 7 4 1 5 9 9 10 3 8 7 Analyzing the Algorithm The algorithm outputs a spanning tree T Suppose that it s not minimal For simplicity assume all edge weights in graph are distinct Let M be a minimum spanning tree Let e be the first edge chosen by the algorithm that is not in M If we add e to M it creates a cycle Since this cycle isn t fully contained in T it has an edge f not in T N M e f is another spanning tree Analyzing the Algorithm 7 4 1 5 9 f 10 3 8 7 e Analyzing the Algorithm N M e f is another spanning tree Claim e f and therefore N M Suppose not e f Then f would have been visited before e by the algorithm but not added because adding it would have formed a cycle But all of these cycle edges are also edges of M since e was the first edge not in M This contradicts the assumption M is a tree Greed is Good In this case The greedy algorithm by adding the least costly edges in each stage succeeds in finding an MST But in math and life if pushed too far the greedy approach can lead to bad results The Greedy Traveling Salesman Tours from Trees We can use an MST to derive a tour that is no more expensive than twice the optimal tour Idea walk around the MST and take shortcuts if a node has already been visited We assume that all pairs of nodes are connected and edge weights satisfy the triangle inequality d x y d x z d z y Tours from Trees Shortcuts only decrease the cost so Cost Greedy Tour 2 Cost MST 2 Cost Optimal Tour This is a 2 competitive algorithm Dancing Partners A group of 100 boys and girls attend a dance Every boy knows 5 girls and every girl knows 5 boys Can they be matched into dance partners so that each pair knows each other Dancing Partners Perfect Matchings Theorem If every node in a bipartite graph has the same degree d 1 then the graph has a perfect matching Note if degrees are the same then A B where A is the set of nodes on the left and B is the set of nodes on the right A Matter of Degree Claim If degrees are the same then A B The Marriage Theorem or What s Love Got to Do With it Each woman would happily marry some subset of the men and any man would be happy to marry any woman who would be happy with him Is it possible to match the men and women into pairs of happy couples The Marriage Theorem Theorem A bipartite graph has a perfect matching if and only if A B and for any subset of say k nodes of A there are at least k nodes of B that are connected to at least one of them The Marriage Theorem The condition fails for this graph The Feeling is Mutual The condition of the theorem still holds if we swap the roles of A and B If we pick any k nodes in B they are connected to at least k nodes in A At least k k At most n k n k Proof of Marriage Theorem Call a bipartite graph matchable if it has the same number of nodes on left and right and any k nodes on the left are connected to at least k on the right Strategy Break up the graph into two matchable parts and recursively partition each of these into two matchable parts etc until each part has only two nodes Proof of Marriage Theorem Select two nodes a A and b B connected by an edge Idea Take G1 a b and G2 everything else Problem G2 need not be matchable There could be a set of k nodes that has only k 1 neighbors Proof of Marriage Theorem a b k 1 k The only way this could fail is if one of the missing nodes is b Add this in to form G1 and take G2 to be everything else This is a matchable partition Generalized Marriage Hall s Theorem Let S S1 S2 be a set of finite subsets that satisfies For any subset T Ti of S UTi T Thus any k subsets contain at least k elements Then we can choose an element xi Si from each Si so that x1 x2 are all distinct Example Suppose that a standard deck of cards is dealt into 13 piles of 4 cards each Then it is possible to select a card from each pile so that the 13 chosen cards contain exactly one card of each rank Graph Spectra Finally we ll discuss a different representation of graphs that is …
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