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in reply to Re^8: Re-orderable keyed access structure?
in thread Re-orderable keyed access structure?
Sorry, I'm not the one who seems to only have read headlines. A heap does not somehow entail a bubble sort. But let's leave the ad hominem out and look at facts.
Yes, you inspect log N item and move one, (steps 1, 2 and 3) below).
A single swap requires inspecting exactly two elements, not log n. You need at most log n swaps total at any time.
But then you are not finished. You still need to swap items 1 and 2.
Why? The heap condition is not violated at any point after your step 3 (which is really step 2, and swapping step 1). $a[0] > $a[1] and $a[0] > $a[2] is fulfilled, so the root and its children satisfy the condition. Likewise $a[1] > $a[3] and $a[1] > $a[4], so the left child of the root and its children satisfy the condition as well. $a[2] has no children, so it automatically satisfies the condition as well. Your step 4 is not required in a heap.
Want me to demonstrate on a larger heap? Sure.
X) 0 [ 13 ] 0) 0 [ 13 ] 1) 0 [ 13 ] 2) 0 [ 13 ] 3) 0 * 16 ]
1 [ 12 ] 1 [ 12 ] 1 [ 12 ] 1 [ 12 ] 1 [ 12 ]
2 [ 11 ] 2 [ 11 ] 2 [ 11 ] 2 * 16 ] 2 * 13 ]
3 [ 10 ] 3 [ 10 ] 3 [ 10 ] 3 [ 10 ] 3 [ 10 ]
4 [ 9 ] 4 [ 9 ] 4 [ 9 ] 4 [ 9 ] 4 [ 9 ]
5 [ 8 ] 5 [ 8 ] 5 * 16 ] 5 * 11 ] 5 [ 11 ]
6 [ 7 ] 6 [ 7 ] 6 [ 7 ] 6 [ 7 ] 6 [ 7 ]
7 [ 6 ] 7 [ 6 ] 7 [ 6 ] 7 [ 6 ] 7 [ 6 ]
8 [ 5 ] 8 [ 5 ] 8 [ 5 ] 8 [ 5 ] 8 [ 5 ]
9 [ 4 ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 ]
10 [ 3 ] 10 [ 3 ] 10 [ 3 ] 10 [ 3 ] 10 [ 3 ]
11 [ 2 ] 11 * 16 ] 11 * 8 ] 11 [ 8 ] 11 [ 8 ]
12 [ 1 ] 12 [ 1 ] 12 [ 1 ] 12 [ 1 ] 12 [ 1 ]
That's it. 3 swaps among a segment of 12 elements.
In a heap with 100 elements, you need at most 7 swaps to get an item from the bottom of the heap to the top without violating the heap condition. I am doubtful of whether splice would win.
In a heap with 1,000 elements, you need at most 10 swaps. How much money will you bet on splice?
Makeshifts last the longest.
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Re^10: Re-orderable keyed access structure?
by BrowserUk (Patriarch) on Aug 15, 2004 at 20:47 UTC
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But let's leave the ad hominem out...
Please do pick up a book or two on algorithms and data structures; this is stuff anyone who is serious about programming should know.
Yes. Let's do that.
In a heap with 1,000 elements, you need at most 10 swaps. How much money will you bet on splice?
Quite a lot, were I a betting man! :)
From where you left off. A new item not currently in cache is called for, it is read from disk, the lowest item* (currently index 12) is replaced by the new item in the array** and the new item given a weight of 17.
a) 0 [ 16 ] b) 0 [ 16 ] c) 0 [ 16 ] d) 0 [ 16 ] e) 0 * 17 ]
1 [ 12 ] 1 [ 12 ] 1 [ 12 ] 1 * 17 ] 1 * 16 ]
2 [ 13 ] 2 [ 13 ] 2 [ 13 ] 2 [ 13 ] 2 [ 13 ]
3 [ 10 ] 3 [ 10 ] 3 * 17 ] 3 * 12 ] 3 [ 12 ]
4 [ 9 ] 4 [ 9 ] 4 [ 9 ] 4 [ 9 ] 4 [ 9 ]
5 [ 11 ] 5 [ 11 ] 5 [ 11 ] 5 [ 11 ] 5 [ 11 ]
6 [ 7 ] 6 * 17 ] 6 * 10 ] 6 [ 10 ] 6 [ 10 ]
7 [ 6 ] 7 [ 6 ] 7 [ 6 ] 7 [ 6 ] 7 [ 6 ]
8 [ 5 ] 8 [ 5 ] 8 [ 5 ] 8 [ 5 ] 8 [ 5 ]
9 [ 4 ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 ]
10 [ 3 ] 10 [ 3 ] 10 [ 3 ] 10 [ 3 ] 10 [ 3 ]
11 [ 8 ] 11 [ 8 ] 11 [ 8 ] 11 [ 8 ] 11 [ 8 ]
12 * 17 ] 12 * 7 ] 12 [ 7 ] 12 [ 7 ] 12 [ 7 ]
Now, another new item is called for, so I need to locate the lowest weighted item in the array. *How do I do this?
And another problem, when I need to locate one of these items that are moving around in this heap via it's key.
**How do I locate it?
Actually, it's just the original one. That of maintaining the linkage between the items in the array(heap) and their keys. No matter how long I "look at the pictures"--or read the text--at heaps, I do not see the mechanism by which the lowest weighted item in the heap is located (other than a linear search).
To re-state the requirements. I need to be able to:
- Locate the highest weighted item.
This is required to allow promotion of the lastest accessed item to the top in the classic LRU algorithm.
- Locate the lowest weighted item.
Also an LRU requirement(or any variation), as this is the one that will be discarded when the cache is full and a new element must be added.
- Locate an item in the cache via it's key.
As the items get moved around, that linkage *must* be maintained.
Embedding the key within the item would require a linear search to locate it. The purpose of the exercise was to avoid a linear search.
Examine what is said, not who speaks.
"Efficiency is intelligent laziness." -David Dunham
"Think for yourself!" - Abigail
"Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon
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I was going to address this at some point, but to be honest, I've lost interest. So I decided to post a closing note instead of letting this silently slip into oblivion.
The point I was going to make is that your heap code is not exactly efficient. The algorithm description is formulated recursively, but you don't need to implement it that way. Since it's just tail recursion, you can trivially write it iteratively, which would gain a lot of ground.
Still, that would almost certainly only accelerate the code by a constant factor, which does not make it worth the effort for the smallish data sets you're working with.
I confess my surprise to find Perl is that slow. I am no stranger to optimizing Perl, but I've never come across such a stellar disparity between a builtin and explicit code before.
Makeshifts last the longest.
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By all means rewrite the recursive routines iteratively and re-run the benchmark, but to be realistic, you would need to run this one.
However, even if the iterative approach was 3 orders of magnitude faster than the recursive (which I strongly doubt), it still leaves the original problem of how to retain keyed access to the data stored in the array after it has been re-ordered.
Examine what is said, not who speaks.
"Efficiency is intelligent laziness." -David Dunham
"Think for yourself!" - Abigail
"Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon
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Re^10: Re-orderable keyed access structure?
by BrowserUk (Patriarch) on Aug 16, 2004 at 05:00 UTC
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I wanted to see if how heaps could be made to work for this.
As well as O(N) -v- O(log N) for any given part of an algorithm not telling the whole story, you also have to consider the cost of all parts of the algorithm.
This benchmarks not only the promotion, but also building the list, promoting from any given position and removing (lowest weighted) items, until empty.
Feel free to show me where I am using a bad implementation.
#! perl -slw
use strict;
use List::Util;
use Benchmark qw[ cmpthese ];
our $SIZE ||= 1000;
our $ITERS||= -5;
sub spliceIt{
my @a;
## Add $SIZE items
push @a, $_ for 0 .. $SIZE;
## Promote $SIZE items, 1 from each position.
push @a, splice @a, $_, 1 for 1 .. $SIZE;
## Remove $SIZE (lowest) items.
shift @a for 0 .. $SIZE;
}
sub heapIt {
my @a;
## Add $SIZE items
for( 0 .. $SIZE ) {
$a[ @a ] = ( $a[ 0 ] || 0 ) + 1;
moveUp( \@a, $#a );
}
## Promote $SIZE items, 1 from each position.
## !!Ass-uming I could locate the item that needs promoting!!
for( 0 .. $SIZE ) {
$a[ $_ ] = $a[ 0 ] + 1;
moveUp( \@a, $_ );
}
## Remove $SIZE (lowest) items.
for( 0 .. $SIZE ) {
## Find the lowest (linear search unless you know a better way
+?)
my $low = 0;
for( 1 .. $#a ) {
$a[ $_ ] < $a[ $low ] and $low = $_;
}
## If the lowest is the last
## remove and and move on.
$#a-- and next if $low == $#a;
## overwrite the lowest with the highest
$a[ $low ] = $a[ 0 ];
## Move the last to the highest
$a[ 0 ] = $a[ $#a ];
## Discard the last
$#a--;
## Now move the (moved) highest item up
moveUp( \@a, $low );
}
}
sub moveUp {
my( $ref, $l ) = @_;
my $p = int $l /2;
return if $p >= $l;
my $temp = $ref->[ $p ];
$ref->[ $p ] = $ref->[ $l ];
$ref->[ $l ] = $temp;
moveUp( $ref, $p );
}
print "Testing $SIZE items for $ITERS iterations";
cmpthese( $ITERS, {
splice => \&spliceIt,
heap => \&heapIt,
});
__END__
## After making the benchmark more realistic
## By benchmarking adding, promoting & removing (lowest) items.
P:\test>heaptest -ITERS=-5 -SIZE=100
Testing 100 items for -5 iterations
Rate heap splice
heap 156/s -- -98%
splice 8335/s 5235% --
P:\test>heaptest -ITERS=-5 -SIZE=100
Testing 100 items for -5 iterations
Rate heap splice
heap 157/s -- -98%
splice 8330/s 5221% --
P:\test>heaptest -ITERS=-5 -SIZE=1000
Testing 1000 items for -5 iterations
Rate heap splice
heap 4.21/s -- -99%
splice 662/s 15613% --
P:\test>heaptest -ITERS=-5 -SIZE=10000
Testing 10000 items for -5 iterations
(warning: too few iterations for a reliable count)
s/iter heap splice
heap 17.9 -- -100%
splice 5.18e-002 34393% --
Examine what is said, not who speaks.
"Efficiency is intelligent laziness." -David Dunham
"Think for yourself!" - Abigail
"Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon
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