4 Combinatorics-F09-581-st

# 4 Combinatorics-F09-581-st - Combinatorics Animation...

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January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 1 Combinatorics Animation fill-in work-on answer only read only Melvin A. Breuer EE581, Fall 2009 Ming Hsieh Department of Electrical Engineering Viterbi School of Engineering University of Southern California

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January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 2 Some combinatoric problems Consider a Boolean switching function that has thousands of minterms and probably many thousands of prime implicants. What are some combinatoric problems associated with this situation? Counting : determine the number of prime implicants without necessary generating them all Enumeration : determine all the prime implicants Optimization : find one or all of the minimal cost covers Constrained optimization : find a minimal cost cover that uses the variable X as few times as possible
January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 3 Combinatorics is a branch of mathematics that studies __________ (usually finite) __________ that satisfy specified criteria. In particular, it is concerned with ___________ the objects in those collections (enumerative combinatorics), with _______ when the criteria can be met, with constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding "largest", "smallest", or ” ____________ " objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). Combinatorics http://en.wikipedia.org/wiki/Combinatorics fill-in

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January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 4 Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century. Much of combinatorics is about graphs, to whose study all types of combinatorics can contribute. An example of a combinatorial question is the following: What is the number of possible orderings of a deck of 52 playing cards? That number equals 52! (i.e., "fifty-two factorial"). It may seem surprising that this number, about 8.065817517094 X10 67 , is so large —a little bit more than 8 followed by 67 zeros! Comparing that number to some other large numbers. E.g., it is greater than the square of Avogadro's number*, 6.022X10 23 . (*Number of carbon-12 atoms in 12 grams of carbon-12.) Overview and History
January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 5 A hard problem An example of another kind of combinatoric problem is: Given n people, is it possible to assign them to sets so that each person is in at least one set, each pair of people are in exactly one set together, every pair of sets have exactly one person in common, and no set contains all or all but one of the people? The answer depends on n and is only partially known to this day.

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January 1, 2006; rev. 10/1/06, 9/09 Melvin A. Breuer (c) 6 Enumerative combinatorics came to prominence because counting configurations is essential to elementary probability, starting with the work of Pascal and others. Modern combinatorics began to develop in the late nineteenth century and became a
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4 Combinatorics-F09-581-st - Combinatorics Animation...

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