.\" Copyright (c) 1980 Regents of the University of California.
.\" All rights reserved.  The Berkeley software License Agreement
.\" specifies the terms and conditions for redistribution.
.\"
.\"	@(#)manCh5.rno	6.1 (Berkeley) 04/29/86
.\"
.NS 1 "Programming Examples"
.pp
We will start off by developing a larger FP program, \fImergeSort\fP.
We measure \fImergeSort\fP using the trace package, and then we
comment on the measurements.
Following \fImergeSort\fP
we show an actual session at the terminal.
.NS 2 "MergeSort"
.pp
The source code for
\fImergeSort\fP is:
.(b
.sp 4p
.hl
.sp 4p
.(c
# Use a divide and conquer strategy
.sp 3p
{mergeSort $^"\fB|\fP"^$ merge}
.sp 6p
{merge atEnd @ mergeHelp @ [[], fixLists]}
.sp 6p
# Must convert atomic arguments into sequences
# Atomic arguments occur at the leaves of the execution tree
.sp 3p
{fixLists &(atom -> [id] ; id)}
.sp 6p
# Merge until one or both input lists are empty
.sp 3p
{mergeHelp (while \kxand @ &(not@null) @ 2
\h'\nxu'(firstIsSmaller -> \kytakeFirst ;
\h'\nyu'takeSecond))}
.sp 6p
# Find the list with the smaller first element
.sp 3p
{firstIsSmaller < @ [1@1@2, 1@2@2]}
.sp 6p
# Take the first element of the first list
.sp 3p
{takeFirst [apndr@[1,1@1@2], [tl@1@2, 2@2]]}
.sp 6p
# Take the first element of the second list
.sp 3p
{takeSecond [apndr@[1,1@2@2], [1@2, tl@2@2]]}
.sp 6p
# If one list isn't null, then append it to the
# end of the merged list
.sp 3p
{atEnd (firstIsNull -> \kzconcat@[1,2@2] ;
\h'\nzu'concat@[1,1@2])}
.sp 6p
{firstIsNull null@1@2}
.)c
.sp 4p
.hl
.sp 4p
.)b
.pp
The merge sort algorithm uses a divide and conquer strategy;
it splits the input in half, recursively sorts each half,
and then merges the sorted lists.
Of course, all these sub-sorts can execute
in parallel, and the
tree-insert ($"\fB|\fP" $) functional form expresses this
concurrency.  \fIMerge\fP removes successively larger elements
from the heads of the
two lists (either \fItakeFirst\fP or \fItakeSecond\fP)
and appends these elements to the end of the merged sequence.
\fIMerge\fP terminates when
one sequence is empty, and then
\fIatEnd\fP appends
any remaining non-empty sequence
to the end of the merged one.
.CH "Dynamic Statistics"
.pp
On the next page we
give the trace of the function \fImerge\fP, which information
we can use to determine the structure of
\fImerge\fP's execution tree.
Since the tree is well-balanced,
many of the \fImerge\fP's
could be executed in parallel.
Using this trace we
can also calculate
the average length of the
arguments passed to \fImerge\fP, or a distribution of argument lengths.
This information is useful for
determining communication costs.
.(b
.nf
.sp 4p
.nf
.hl
.sp 4p
.(c
	)trace on merge

	mergeSort\ :\ <0\ 3\ -2\ 1\ 11\ 8\ -22\ -33>
|\ 3\ >Enter>\ merge\ [<0\ 3>]
|\ 3\ <EXIT<\ \ merge\ \ <0\ 3>
|\ 3\ >Enter>\ merge\ [<-2\ 1>]
|\ 3\ <EXIT<\ \ merge\ \ <-2\ 1>
|2\ >Enter>\ merge\ [<<0\ 3>\ <-2\ 1>>]
|2\ <EXIT<\ \ merge\ \ <-2\ 0\ 1\ 3>
|\ 3\ >Enter>\ merge\ [<11\ 8>]
|\ 3\ <EXIT<\ \ merge\ \ <8\ 11>
|\ 3\ >Enter>\ merge\ [<-22\ -33>]
|\ 3\ <EXIT<\ \ merge\ \ <-33\ -22>
|2\ >Enter>\ merge\ [<<8\ 11>\ <-33\ -22>>]
|2\ <EXIT<\ \ merge\ \ <-33\ -22\ 8\ 11>
1\ >Enter>\ merge\ [<<-2\ 0\ 1\ 3>\ <-33\ -22\ 8\ 11>>]
1\ <EXIT<\ \ merge\ \ <-33\ -22\ -2\ 0\ 1\ 3\ 8\ 11>

<-33\ -22\ -2\ 0\ 1\ 3\ 8\ 11>
.)c
.sp 4p
.hl
.fi
.)b
.bp
.NS 2 "FP Session"
.pp
User input is
\fBemboldened\fP, terminal output in Roman script.
.sp 0.5i
.nf
\fBfp\fP

FP, v. 4.1 11/31/82
\fB	)load ex_man\fP
{all_le}
{sort}
{abs_val}
{find}
{ip}
{mm}
{eq0}
{fact}
{sub1}
{alt_fnd}
{alt_fact}
\fB	)fns\fP

.TS
l l l l l l l.
abs_val	all_le	alt_fact	alt_fnd	eq0	fact	find
ip	mm	sort	sub1	\&	\&	\&
.TE

\fB	abs_val : 3\fP

3

\fB	abs_val : -3\fP

3

\fB	abs_val : 0\fP

0

\fB	abs_val : <-5 0 66>\fP

?

\fB	&abs_val : <-5 0 66>\fP

<5 0 66>

\fB	)pfn abs_val\fP

{abs_val ((> @ [id,%0]) -> id ; (- @ [%0,id]))}

\fB	[id,%0] : -3\fP

<-3 0>

\fB	[%0,id] : -3\fP

<0 -3>

\fB	- @ [%0,id] : -3\fP

3

\fB	all_le : <1 3 5 7>\fP

T

\fB	all_le : <1 0 5 7>\fP

F

\fB	)pfn all_le\fP

{all_le ! and @ &<= @ distl @ [1,tl]}

\fB	distl @ [1,tl] : <1 2 3 4>\fP

<<1 2> <1 3> <1 4>>

\fB	&<= @ distl @ [1,tl] : <1 2 3 4>\fP

<T T T>

\fB	! and : <F T T>\fP

F

\fB	! and : <T T T>\fP

T

\fB	sort : <3 1 2 4>\fP

<1 2 3 4>

\fB	sort : <1>\fP

<1>

\fB	sort : <>\fP

?

\fB	sort : 4\fP

?

\fB	)pfn sort\fP

{sort (null @ tl -> [1] ; (all_le -> apndl @ [1,sort@tl]; sort@rotl))}

\fB	fact : 3\fP

6

\fB	)pfn fact sub1 eq0\fP

{fact (eq0 -> %1 ; *@[id , fact@sub1])}

{sub1 -@[id,%1]}

{eq0 = @ [id,%0]}

\fB	&fact : <1 2 3 4 5>\fP

<1 2 6 24 120>

\fB	eq0 : 3\fP

F

\fB	eq0 : <>\fP

F

\fB	eq0 : 0\fP

T

\fB	sub1 : 3\fP

2

\fB	%1 : 3\fP

1

\fB	alt_fact : 3\fP

6

\fB	)pfn alt_fact\fP

{alt_fact !* @ iota}

\fB	iota : 3\fP

<1 2 3>

\fB	!* @ iota : 3\fP

6

\fB	!+ : <1 2 3>\fP

6

\fB	find : <3 <3 4 5>>\fP

T

\fB	find : <<> <3 4 <>>>\fP

T

\fB	find : <3 <4 5>>\fP

F

\fB	)pfn find\fP

{find (null@2 -> %F ; (=@[1,1@2] -> %T ; find@[1,tl@2]))}

\fB	[1,tl@2] : <3 <3 4 5>>\fP

<3 <4 5>>

\fB	[1,1@2] : <3 <3 4 5>>\fP

<3 3>

\fB	alt_fnd : <3 <3 4 5>>\fP

T

\fB	)pfn alt_fnd\fP

{alt_fnd ! or @ &eq @ distl }

\fB	distl : <3 <3 4 5>>\fP

<<3 3> <3 4> <3 5>>

\fB	&eq @ distl : <3 <3 4 5>>\fP

<T F F>

\fB	!or : <T F T>\fP

T

\fB	!or : <F F F>\fP

F

\fB	)delete alt_fnd\fP

\fB	)fns\fP

.TS
l l l l l l l.
abs_val	all_le	alt_fact	eq0	fact	find	ip
mm	sort	sub1	\&	\&	\&	\&
.TE

\fB	alt_fnd : <3 <3 4 5>>\fP

alt_fnd not defined

?
\fB	{g g}\fP

{g}
\fB	g : 3\fP

non-terminating

?

[Return to top level]

FP, v. 4.0 10/8/82
\fB	[+,*] : <3 4>\fP

<7 12>

.(b
\fB	[+,* : <3 4>\fP

syntax error:

[+,* \kx: <3 4>
\h'\nxu'^
.)b
\fB	ip : <<3 4 5> <5 6 7>>\fP

74

\fB	)pfn ip\fP

{ip !+ @ &* @ trans}

\fB	trans : <<3 4 5> <5 6 7>>\fP

<<3 5> <4 6> <5 7>>

\fB	&* @ trans : <<3 4 5> <5 6 7>>\fP

<15 24 35>

\fB	mm : <<<1 0> <0 1>> <<3 4> <5 6>>>\fP

<<3 4> <5 6>>

\fB	)pfn mm\fP

{mm &&ip @ &distl @ distr @[1,trans@2]}

\fB	[1,trans@2] : <<<1 0> <0 1>> <<3 4> <5 6>>>\fP

<<<1 0> <0 1>> <<3 4> <5 6>>>

\fB	distr : <<<1 0> <0 1>> <<3 4> <5 6>>>\fP

<<<1 0> <<3 4> <5 6>>> <<0 1> <<3 4> <5 6>>>>

\fB	&distl : <<<1 0> <<3 4> <5 6>>> <<0 1> <<3 4> <5 6>>>>\fP

<<<<1 0> <3 4>> <<1 0> <5 6>>> <<<0 1> <3 4>> <<0 1> <5 6>>>>

.(b
\fB	&ip @ &dist & distr @ [1,trans @ 2] : <<<1 0> <0 1>> <<3 4> <5 6>>>\fP

syntax error:

[+,* \kx: <3 4>
\h'\nxu'^
&ip @ &distl \kx& distr @ [1,trans @ 2] : <<<1 0> <0 1>> <<3 4> <5 6>>>
\h'\nxu'^
.)b
\fB	&ip @ &distl @ distr @ [1,trans@2] : <<<1 0> <0 1>> <<3 4> <5 6>>>\fP

?
.sp 2
.fi
.bp