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Let G be a pseudograph with vertex set V, edge set E, and incidence mapping f. Let n be a positive integer. A path of length n between vertex v and vertex w in G is a sequence e 1, e 2, …, e n of edges in E for which f(e 1 ) = {v 0, v 1 }, f(e 2 ) = {v 1, v 2 }, …, f(e n ) = {v n-1,v n }, for some set of vertices v 0, v 1, v 2, …, v n-1, v n in V, with v 0 = v and v n = w. (For loops e i, we would have v i-1 = v i, and f(e i ) would be a singleton set) A circuit of length n is simply a path of length n which ends where it started. In the above definition we simply have v = w. A simple path or simple circuit is one in which there are no repeated edges. Some Definitions from 9.4

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Connectedness in Undirected Graphs We say that an undirected graph G is connected provided…

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9.7 Planar Graphs Definition Water, Electricity, Gas

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Examples:

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Euler’s Formula

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K 5 and K 3,3

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Kuratowski’s Theorem

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9.8 Graph Coloring Planar Dual Graph of a map

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Coloring Define a coloring of a graph Define the chromatic number of a graph

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The Four Color Theorem Appel and Haken, 1976 If there is a counterexample, then there is a minimal counterexample. A reducible configuration is a subgraph which cannot occur in a minimal counterexample. Appel and Haken proved that every planar graph contains one of 1936 reducible configurations. The proof was constructed by a computer program.

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Three colors is not enough Can you come up with a quick proof of that fact?

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Example:

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Other Facts Since Appel and Haken’s proof, an O(n 2 ) algorithm has been discovered for 4- coloring planar graphs. The problem of finding a 3-coloring of a planar graph or deciding such does not exist is NP-complete. The problem of finding a 4-coloring of a general (non-planar) graph or deciding such does not exist is NP-complete.

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Computing the Chromatic Number Can you come up with a simple algorithm for coloring a graph with a reasonably small number of colors?

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Applications Scheduling rooms, final exams, etc. Assigning roles in a play Frequency assignments for TV stations

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Example There are three meeting rooms in the lodge where the Royal Squid Captains hold their annual convention. Seven meetings are scheduled. There are four officers: The Exalted Octopus, the Revered Clam, the Mighty Sea Bass, and the Mystic Eel. The Exalted Octopus must be present for talks 1, 3, and 7. The Mystic Eel must attend talks 2, 4, and 1. The Revered Clam can’t afford to miss talk 2 or talk 5. Finally, the Mighty Sea Bass must be present for talks 1, 4, and 6. What is the minimum number of time slots needed in which to conduct the meetings, so that each officer will be able to attend all the meetings he must attend?

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10.1 Introduction to Trees Definition: A tree is a connected undirected graph with no simple circuits Theorem: An undirected graph is a tree if and only if any two vertices are joined by a unique simple path.

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Rooted Trees A rooted tree is a directed graph with all vertices except one having indegree one. The exception is the root, which has indegree zero. All other nodes are accessible from the root via a unique path Canonical tree drawing is…

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Tree Terms Node, parent, child, sibling, ancestor, descendant Internal vertex, leaf

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Tree Terms, Continued m-ary tree, binary tree Full m-ary tree

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Ordered Trees An ordered tree is like a rooted tree, except that an ordering is assigned to the children of every node, so that the terms first child, left child, right child, etc, make sense.

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Properties of Trees Theorem: A tree with n vertices has n-1 edges Theorem: A full m -ary tree with i internal vertices has n = mi+1 vertices

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Relationships Between i, n, and l Let i, n, and l be the number of internal vertices, the total number of vertices, and the number of leaves, respectively. Theorem: In a full m -ary tree, all of the following formulae apply: a)i = (n – 1)/m and l = ((m – 1)n + 1)/m b)n = mi + 1 and l = (m – 1)i + 1 c)n = (m l – 1) / (m – 1) and i = ( l – 1)/(m – 1) In other words, with m fixed, any two of the attributes i, n, and l of a full m -ary tree can be computed given the remaining attribute

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Example: Suppose that someone starts a chain letter. Each person who receives the letter is asked to send it on to four other people. Some people do this, but others do not send any letters. How many people have seen the letter, including the first person, if no one receives more than one letter and if the chain letter ends after there have been 100 people who read it but did not send it out? How many people sent out the letter?

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Levels The level of a node is its distance from the root. The root is at level 0, its children are at level 1, their children are at level 2, etc. The height of a tree is the maximum of all the levels of its nodes A tree of height h is balanced provided all its leaves are either at height h or height h – 1.

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Examples:

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Leaves in an m -ary Tree Theorem: There are at most m h leaves in an m -ary tree of height h. Corollary: If an m -ary tree of height h has l leaves, then. If the tree is full and balanced, then

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10.2 Applications of Trees

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Binary Search Trees Maude Louise Ken Isaac George Zack Mary

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Decision Trees

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The Complexity of “Compare and Swap” Sorting Algorithms Theorem: A sorting algorithm based on binary comparisons requires at least __________ comparisons. Corollary: The number of comparisons used by a sorting algorithm to sort n elements based on binary comparisons is ________________.

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Prefix Codes and Huffman Encoding Binary code assigns a bit string to each character. Variable- length code can be used to compress a document- shorter codes for more frequent characters. One example is a prefix code where no code appears as the prefix of another.

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'a'.12 'c'.02 'd'.08 'o'.14 'p'.03 'r'.11 's'.20 't'.30 Example of Huffman Coding:

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Game Trees

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