Study of Decision Trees & Recursive Algorithms: Binary Search & Merge Sort, Lecture notes of Computer Science

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if f(x) is O(x) then f(x) is O(x

2

Homework problem:

Friday, October 15, 2010 6:26 PM

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Basis: a question and yes and no branches.

Inductive step: a new question has yes and no branches that are decision trees. Application: the classic game of "animals".

A decision tree

Decision trees Friday, October 15, 2010 5:53 PM

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Goal: traverse a decision tree to guess an animal.

A decision is a question with two consequences,

corresponding to "yes" and "no" answers.

a guess as to an animal.

another decision to make.

A consequence can be either

Form of the tree:

Animals

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a question.

a yes consequence.

a no consequence.

Each decision has three parts:

Representing a decision tree:

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["Is it a mammal?", "horse", "fish"]; Step 1: Distinguish between a horse and a fish Step 2: Distinguish between a horse and a pig. ["Is it rideable", "horse", "pig"] Step 3: Combine the above into one "tree": ["Is it a mammal?", ["Is it rideable", "horse", "pig"], "fish"] This is a recursively defined array structure. Step 4: distinguish between a fish and a lizard: ["Does it live in the desert?", "lizard", "fish"] Step 5: combine steps 3 and 4: ["Is it rideable?", "horse", "pig"] ["Does it live in the desert?", "lizard", "fish"] [ "Is it a mammal?", ] We can continue this as long as we want, refining lower levels. Building a decision tree Friday, October 15, 2010 5:59 PM

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A program can learn new decisions to correct its

mistakes!

Key is to modify the structure of the array to codify new

knowledge.

Just as we did in building the original tree.

Can learn from example: "supervised learning".

Point of decision trees

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Many common algorithms "divide" a problem into

smaller ones and then "conquer" the problem at a

small size.

Dividing and conquering

Binary search

Merge sort

Examples:

Dividing and conquering

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searching for an item among n sorted things => searching n/2 things. Binary search: Serial version: 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; } elsif ($word lt $words[$mid]) { $high=$mid; } else { $found=&TRUE; last; } } if ($found) { print "'$word' is in the dictionary\n"; } else { print "'$word' is not in the dictionary\n"; } Pasted from Example: binary search Friday, October 15, 2010 2:33 PM

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Start with a big array.

Cut it in half, repeatedly.

Until it has one element.

Recursive version:

# return 1 if a word is in a sorted list.

sub search {

my $words = shift; # a sorted list

my $word = shift; # word to look up

my $elts = @$words;

if ( $elts==1 ) { // inductive basis

if ($words->[0] eq $word) {

return 1;

} else {

return undef;

} else { // recursive step

my $mid = int($elts/2);

if ($word gt $words->[$mid]) {

return &search([@$words[$mid+1..$elts-1]],

$word);

} elsif ($word lt $words->[$mid]) {

return &search([@$words[0..$mid-1]],

$word);

} else {

return 1; # found it!

Pasted from

Recursive version of binary search

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sub sorter { my $thing = shift; my $length = @$thing; if ($length==0) { return []; } elsif ($length==1) { return [$thing->[0]]; } else { my $size = @$thing; my $mid = int($size/2); my $bot = [@$thing[0..$mid-1]]; my $top = [@$thing[$mid..$size-1]]; my $sbot = &sorter($bot); my $stop = &sorter($top); return &merge($sbot,$stop) ; } } my $out = &sorter([1,6,3,2,9,1,203,40,20]); Pasted from Example: merge sort Friday, October 08, 2010 1:49 PM

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if ($a->[0] <= $b->[0]) { # choose lowest

push(@$c, shift @$a);

} else {

push(@$c, shift @$b);

if (@$a && @$b) { # if both have elements

} elsif (@$a) { # a is all that's left

push(@$c,@$a); $a=[];

} else { # b is all that's left

push(@$c,@$b); $b=[];

return $c;

sub merge {

my $a = shift; $a = [@$a]; # make a copy

my $b = shift; $b = [@$b]; # make a copy

my $c = [];

while (@$a || @$b) { # while one has an element

Merging two (sorted) arrays Sunday, October 10, 2010 4:42 PM

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if ($a->[0] <= $b->[0]) { # choose lowest

push(@$c, shift @$a);

} else {

push(@$c, shift @$b);

Monday, October 18, 2010 3:26 PM

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