Major Themes. Trees. Types of Graphs. Which of the following are trees? G 3 G 1 G 2 G 4. Dr Papalaskari 1
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1 Major Themes Trees Villanova University Trees Binary trees Binary search trees Spanning trees Minimum spanning trees Villanova CSC Dr Papalaskari 2 Types of Graphs graph mulhgraph digraph: directed graph/mulhgraph weighted graphs/digraphs Special graphs Cycle C x Wheel W x Hypercube Q x Complete graph K x A Complete BiparHte K x,y Connected graph Tree: a connected graph with no cycles Forest: a graph with no cycles Review Which of the following are trees? G 1 G 3 G 2 G 4 Villanova CSC Dr Papalaskari 4 Villanova CSC Dr Papalaskari 5 Dr Papalaskari 1
2 Tree: a connected graph with no cycles A tree with n verhces has n - 1 edges A connected graph with n verhces and n - 1 edges is a tree. Every tree with at least 2 verhces has 2 or more leaves A leaf is a vertex of degree 1. Rooted Tree A rooted tree has a dishnguished vertex called the root Rooted trees are usually drawn with the root at the top and all other verhces connected by edges growing downward Each edge in a rooted tree connects a parent vertex with a child vertex below it. Villanova CSC Dr Papalaskari 6 Villanova CSC Dr Papalaskari 7 Binary Tree A rooted tree Each vertex has at most 2 children: le. and right child. the k th level (or rank) of a binary tree is the collechon of verhces at distance k from the root In a complete binary tree every vertex has exactly two children, except for the leaves in the bo\om level. More about binary trees: How many leaves in a complete binary tree with n levels? If a binary tree has 6 levels and 50 verhces, is it possible for it to be: a binary tree? a complete binary tree? Villanova CSC Dr Papalaskari 9 Villanova CSC Dr Papalaskari 10 Dr Papalaskari 2
3 Binary Search Trees A binary search tree has a datum associated with each node such that in the ordering of the data set, the datum occurs earlier than any of the data downward and to the right and occurs later than any of the data downward and to the le^. Exercise Place baa at the root and draw the binary search tree for aaa, ab, baa, baba Villanova CSC Dr Papalaskari 12 Villanova CSC Dr Papalaskari 13 A Spanning Trees Depth-first/Breadth-first Search Trees 5 5 Spanning Tree: A subgraph that is a tree and contains all the verhces of a given graph. Every connected graph has at least 1 spanning tree. Every weighted connected graph has a minimum spanning tree Villanova CSC Dr Papalaskari 14 Villanova CSC Dr Papalaskari Dr Papalaskari 3
4 Kruskal s Minimum Spanning Tree Algorithm Order the edges of G, by increasing weight G starts out as the graph with all the verhces of G and none of the edges Keep adding edges to G, at each iterahon adding the lowest weighted edge that does not introduce a cycle Stop when the graph becomes connected. G is now the minimum spanning tree. Prim s Minimum Spanning Tree Algorithm G starts out as the graph with just one vertex of graph G and no edges. Select the lowest weighted edge that connects a vertex in G to any of the remaining verhces of G; add edge and vertex to G and repeat unhl all the verhces of G are added to G G is now the minimum spanning tree. Villanova CSC Dr Papalaskari 16 Villanova CSC Dr Papalaskari 17 Kruskal s Algorithm Prim s Algorithm Villanova CSC Dr Papalaskari 18 Villanova CSC Dr Papalaskari 19 Dr Papalaskari 4
5 Fake coin puzzle You have eight gold coins and a balance scale. One of the coins is a fake and weighs less (gold is HEAVY). How can you determine which coin is fake with the least number of weighings? Villanova CSC Dr Papalaskari 22 Dr Papalaskari 5
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