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Complete the following inductive proof:
Theorem: any set S of natural numbers has a largest
element.
Basis case: n=________ 13 ________. Then S = {x} and x
≤ x. Thus x is the largest element of {x}=S.
• If y>x, then y is the largest element of S.
• If x>y, then x is the largest element of S.
Inductive step: Assume true for n. Choose S with |S|=n+1.
Choose x
S. Let T=S ________ 14 ________ {x}. Then
|T|=________ 15 ________ so the inductive hypothesis
applies and ________ 16 ________ has a largest element y.
Then there are two cases,
Thus in either case, there is a largest element of S. and the
inductive hypothesis is true with |S|=n+1.
Thus the whole theorem is true by induction.
Pasted from
Quiz 1 quandary
Monday, October 04, 2010 12:52 PM Docsity.com
We have developed a precise language for discussing
computation
We have developed some skill in a powerful
computer language.
So far,
It is time to do some real computer science.
It is time to engage in the grand dance of time, space,
and complexity.
Now,
Complexity
Saturday, October 02, 2010 5:40 PM Docsity.com
or die "can't read the dictionary: $!"; open(DICT, ") { chomp; push(@words, $_); } print "Enter words to look up: "; while () { chomp; my @line = split(/\s+/,$_); foreach my $word (@line) { my $found = &FALSE; for (my $i=0; $i<@words; $i++) { if ($words[$i] eq $word) { $found = &TRUE; last; } } if ($found) { print "'$word' is in the dictionary\n"; } else { print "'$word' is not in the dictionary\n"; } } print "Enter words to look up: "; } Pasted from
Linear search
Saturday, October 02, 2010 5:48 PM Docsity.com
or die "can't read the dictionary: $!"; open(DICT, ") { chomp; push(@words, $_); } @words = sort (@words); print "Enter words to look up: "; while () { chomp; my @line = split(/\s+/,$_); foreach my $word (@line) { my $found = &FALSE; my $low = 0; my $high = @words-1; while ($high-$low>1) { my $mid = ($high + $low)/2; if ($word gt $words[$mid]) { $low=$mid; # rule out #$words[$low]-$words[$mid-1] } elsif ($word lt $words[$mid]) { $high=$mid; # rule out #words[$mid+1]-$words[$high] } else { # found it: end $found=&TRUE; last; } } if ($found) { print "'$word' is in the dictionary\n"; } else { print "'$word' is not in the dictionary\n"; } } print "Enter words to look up: "; } Pasted from
Binary search
Saturday, October 02, 2010 6:28 PM Docsity.com
Not a trivial question.
Time to do things (time complexity)
Space to store things (space complexity)
Each approach requires
How can we measure these?
Which one is better?
Which one of these should we utilize?
Which broomstick?
Saturday, October 02, 2010 6:35 PM Docsity.com
Hashing is optimally fast, but takes more space (for
complex reasons). This is an example of trading space
for time.
Storing something in an array is space-efficient, but
takes a lot more time. This is an example of trading
time for space.
Sorting an array takes time, but then, searching is
faster. This is an example of the value of preparation.
There is more preparation than search time.
These involve space/time tradeoffs:
Trading space and time
Saturday, October 02, 2010 6:37 PM Docsity.com
read my own memory usage from the proc table
It is a line of the form "VmSize: size" in status
sub readmemory { open(PROC, ") { chomp; my ($key, $value)=split(/[\s:]+/); close PROC; if ($key eq 'VmSize') { return $value } } return undef; } $mem1 = &readmemory; for ($i=0; $i<100000; $i++) { $a[$i]=$i; } $mem2 = &readmemory; $diff = $mem2 - $mem1; print "mem1=$mem1 mem2=$mem2 diff=$diff kbytes\n"; See and run http://www.cs.tufts.edu/comp/14/examples/Complexity/space.perl Caution : not subtracting $mem1 would count the size of perl itself!
Measuring space used
Sunday, October 03, 2010 2:49 PM Docsity.com
Worst-case time (or space) : an upper bound on how
much time (or space) is taken to compute something.
How slow can it be?
Best-case time : a lower bound. How fast can it run?
Average-case time : on average, how long (or how much
memory) does it take? Depends upon input!
Concepts of time and space
Sunday, October 03, 2010 2:58 PM Docsity.com
How can we characterize the time spent in a
program?
How can we distinguish between different
approaches to solving the same problem?
Concentrate on worst-case behavior.
For today, our goal is to concentrate upon time.
For today
Sunday, October 03, 2010 3:06 PM Docsity.com
We define some concept of a "program step". Then we "count the steps". How long does a program take? If we define a step as a program statement, then for ($i=0; $i<100; $i++) { $sum+=$i; } takes 302 steps. Example: for ($i=0; $i<100; $i++) { $sum += $i; } Reasoning: is equivalent with $i=0; # 1 step while ($i<100) { # 1 step, TRUE 100 times, then FALSE $sum+=$i; # 1 step $i++; # 1 step } So the total is 1+(1+1+1)*100 +1 =302 total steps.
"Steps"
Saturday, October 02, 2010 7:05 PM Docsity.com
The concept of a step can differ greatly depending
upon who defines it.
Our general strategy: arrange things so the definition
of the size of a step does not matter.
This is painful!
Concentrate on the relationship between size of
input and computation time.
Discuss bounds rather than actual times.
Two parts to our strategy:
Wow, this is confusing
Saturday, October 02, 2010 7:11 PM Docsity.com
We start by defining some size of input, e.g., n things.
We don't care what units one uses as long as they're
consistent: bytes, records, etc.
Size of input
Saturday, October 02, 2010 7:12 PM Docsity.com
Let n be some measure of input size. C is a multiplicative constant that represents a change in units N is a boundary condition that determines how large n has to be for the change in units to work. We say that a program takes O(n) time (or space) if there are constants C and N such that if n>N, the time (or space) required < Cn.
Order notation
Saturday, October 02, 2010 7:14 PM Docsity.com
A picture of linear time
Sunday, October 03, 2010 4:14 PM Docsity.com
