MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 13, Monday, September 29

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MATH 310, FALL 2003(Combinatorial Problem
Solving)Lecture 13, Monday, September 29
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Example 3 Pitcher Pouring Puzzle

• We are given three pitchers of water, of sizes 10
  quarts, 7 quarts and 4 quarts.
• Initially, the 10-quart is full and the other two
  pitchers are empty.
• Is there a way to pour among pitchers to obtain
  exactly 2 quarts in one of the pitchers?

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The State Space

• The state space contains the following
• S set of states
• s0 initial state
• F final or goal states (could be only one)
• P - set of rules
• D Admissibility function DP ? S ?
  True,False
• U Rule application UP ? S ? S.

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Example 3 Pitcher Pouring Puzzle Exercise 16(a)

• We are given three pitchers of water, of sizes 8
  quarts, 5 quarts and 3 quarts.
• Initially, the 8-quart is full and the other two
  pitchers are empty.
• Is there a way to pour among pitchers to obtain
  exactly 4 quarts in one of the pitchers?

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Example 3 Pitcher Pouring Puzzle Exercise 16(a)

• We are given three pitchers of water, of sizes 8
  quarts, 5 quarts and 3 quarts.
• Initially, the 8-quart is full and the other two
  pitchers are empty.
• Is there a way to pour among pitchers to obtain
  exactly 4 quarts in one of the pitchers?

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State Space

• Here is a part of the state space for Exercise
  16(a).
• What is the initial state?
• What are the final states?
• What are the rules?
• Are there any states missing?
• Are there any edges missing?
• Find DFS and BFS trees.

800
503
350
053
530
323
620
602
152
233
251
440
143
701
710
413
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Example 4 Jealous Wives Puzzle.

• Three jealous wives and three husbands come to a
  river.
• The boat is for two people.
• Find a sequence of boat trips that will get 6
  people across the river without ever letting any
  husband to be without his wife in the presence of
  another wife.

A,a,B,b,C,c
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Example 4 Jealous Wives Puzzle.

• Three jealous wives and three husbands come to a
  river.
• The boat is for two people.
• Find a sequence of boat trips that will get 6
  people across the river without ever letting any
  husband to be without his wife in the presence of
  another wife.

A,a,B,b,C,c
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A Center of a Tree T

• Let T be an arbitrary tree and let r be any of
  its vertices. Let Tr be the tree rooted at r.
  Vertex r is called a center if the correspondig
  rooted tree Tr has minimal height. Compare
  Exercise 18, p. 103.

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2
3
4
8
1
6
5
root 1 2 3 4 5 6 7 8
height 3 4 3 4 4 3 4 2
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4.1. Shortest Paths

• Homework (MATH 3105M)
• Read 4.2.
• Do Exercises 4.1 1,2,4,6,8,10,12
• Volunteers
• ____________
• ____________
• Problem 1.
• No Section of Chapter 4 will be on Test 1.

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Dijkstras Algorithm

• At each step of the algorithm the (green) edge e
  between a labeled vertex p and unlabeled vertex q
  is selected in such a way, that
• d(p) k(e)
• is minimal.