       Re:Challenge! ....RESULTS..., (Corrected)

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1069] Re:[mg950] Challenge! ....RESULTS..., [mg986] (Corrected)
• From: Allan Hayes <hay at haystack.demon.co.uk>
• Date: Fri, 12 May 1995 16:50:58 -0400

```I have corrected some typing errors in my previous posting. The
corrected version (CORRECTED COPY) follows the first section.
I have also, got closer to seeing why codes involving Module and
With can be inefficient. I suggest some ways of using them more
efficiently. Since this is of more general consequence I will
explain it at once.

 USING MODULE AND WITH

*************************
(*6*)
Module[{a=A},B] and With[{a=A},B]
can be slow when A or B are big.

Some Suggestions:

Name A, preferaby inside Block,: eg
Block[{a = A}, With[{a=a},B]];

Minimize B by placing Module and With as deep inside the
expression as possible; consistent with its working as
required. Try naming all or parts of B (eg.
Block[{b = B}, With[{a=A},b]]) but don't remove symbols
that you wish With to replace.

In the test below it turns out that we need to deal with both A and
B to get a significant improvement.

***********************
Examples: For brevity I deal only with With.
Please note that thes are not realstic examples: they are chosen to
investigate the behaviour of Module and With. The fastest form of
he code is simply test[L_] :=  {5 len, L[[1,1]]}.

data =Array[#,{300}]&/@(Range);

First the direct application of With
testWith[L_] :=

With[ {len = Length[L]}, {5 len, L[[1,1]]}  ]
testWith[data]//Timing
{1.48333 Second, {500, 1}}

Now to try the suggestions individually and together

testWith2[L_] :=
Block[{len = Length[L]}, 	   (*name A *)
With[ {len = len}, {5 len, L[[1,1]]}  ]
];
testWith2[data]//Timing
{1.46667 Second, {500, 1}}

testWith3[L_] :=
{With[{len = Length[L]},5 len],  L[[1,1]]};(*With inside*)
testWith3[data]//Timing
{1.41667 Second, {500, 1}}

testWith4[L_] :=
Block[{len = Length[L]},		      (*name A*)
{ With[ {len = len},5 len ],  L[[1,1]] } (*With inside*)
]
testWith4[data]//Timing
{0. Second, {500, 1}}

testWith5[L_] :=
Block[{len = Length[L], l = L},        (*name A and B*)
With[ {len = len}, {5 len, l[[1,1]]}  ]
];
testWith5[data]//Timing
{0. Second, {500, 1}}

Why do we need to deal with both A and B before any significant
improvement is seen?

There is another way: use replacement directly like this

testReplace[L_] :=
Block[{len},
{5 len, L[[1,1]]} /. len -> Length[L]
];
testReplace[data]//Timing
{0. Second, {500, 1}}

For repeated use we have the following timings

Do[testWith4[data],{1000}]//Timing
Do[testReplace[data],{1000}]//Timing
{1.46667 Second, Null}
{1.61667 Second, Null}

So using With seems better. Unfortunately this speed up does not
seem to carry through to the interleaving code.

(*previous code using replacement*)
interleave3Replace[L:{s_,___}] :=
Block[{len},
Transpose[
L[[(Random[Integer,{1,len}]&/.len-> Length[L])/@s ]], (*4*)
{1,1}
]
];
Do[interleave3Replace[data],{30}]//Timing//First
3.55 Second

(*new code using With*)
interleave3With[L:{s_,___}] :=
Block[{len = Length[L]},
Transpose[
L[[With[{len=len},Random[Integer,{1,len}]&]/@s ]], (*4*)
{1,1}
]
];
Do[interleave3With[data],{30}]//Timing//First
3.51667 Second

One final point: we can avoid the need to take With inside as follows

interleave3WithA[L_] :=
Block[{len = Length[L]L},
With[{len=len},
Transpose[
L[[Random[Integer,{1,len}]&/@First[LL] ]],
{1,1}
]
]];

Do[interleave3WithA[data],{30}]//Timing//First
Do[interleave3WithA[{list1,list2}],{1000}]//Timing//First
3.61667 Second
5.68333 Second

 CORRECTED COPY
(I have change the numbering: to [2a] and  to [2b])
*************************
Paul Howland <PEHOWLAND at taz.dra.hmg.gb> asked in [mg889] for the
most efficient function for generating a new list by randomly
interchanging the elements of two other lists.

I give in [2a] below a further speed up to my best code in my
response [mg986] and find  another speed up rule:

(*4*) Simple replacements can be very fast

But first a correction: in [mg986] I let my keenness to try out
larger tests take me away from the original challenge problem.The
statement that interleave3 is faster than Dave Wagner's code swap is
incorrect for the two lists in the original problem. However, the
new code, is faster for these lists.

In [2b] I compare this new code interleave3Replace with the results
of using straightforward Block, Module and With on interleave3.
Module and With come out badly for large data sets.
Hence, the rule:

(*5*) Try Block for speed.

[2a]

My previous best

interleave3[L:{s_,___}] :=
Transpose[
L[[Random[Integer,{1,Length[L]}]&/@s ]],
{1,1}
];

The new code

interleave3Replace[L:{s_,___}] :=
Block[{len},
Transpose[
L[[(Random[Integer,{1,len}]&/.len-> Length[L])/@s ]], (*4*)
{1,1}
]
];

TIMINGS

list1 = {a,b,c,d,e,f,g,h,i,j};
list2 = {A,B,C,D,E,F,G,H,I,J};

Timing[Do[interleave3[{list1,list2}],{1000}]][]

7.33333 Second

Timing[Do[interleave3Replace[{list1,list2}],{1000}]][]

5.6 Second

Dave Wagner's swap code is

swap[list1_List,list2_List] :=
#[[Random[Integer,{1,2}]]]& /@ Transpose[{list1,list2}]

and gives the timing

Timing[Do[swap[list1,list2],{1000}]][]

6.28333 Second

For bigger data interleave3Replace maintains its advantage over
interleave3.

test =Array[#,{300}]&/@(Range);

Do[interleave3[test],{30}]//Timing//First

4.58333 Second

Do[interleave3Replace[test],{30}]//Timing//First

3.6 Second

[2b]

Is the complication of interleave3Replace necessary? Let's try a
straightforward use of Block, Module and With

interleave3Block[L:{s_,___}] :=
Block[{len = Length[L]},
Transpose[
L[[Random[Integer,{1,len}]&/@s ]],
{1,1}
]
];

interleave3Module[L:{s_,___}] :=
Module[{len = Length[L]},
Transpose[
L[[Random[Integer,{1,len}]&/@s ]],
{1,1}
]
];

interleave3With[L:{s_,___}] :=
With[{len = Length[L]},
Transpose[
L[[Random[Integer,{1,len}]&/@s ]],
{1,1}
]
];

Timings:

Timing[Do[interleave3Replace[{list1,list2}],{1000}]][]
5.7 Second

Timing[Do[interleave3Block[{list1,list2}],{1000}]][]
5.93333 Second

Timing[Do[interleave3Module[{list1,list2}],{1000}]][]
8.3 Second

Timing[Do[interleave3With[{list1,list2}],{1000}]][]
6.08333 Second

But for bigger data interleave3Replace has more of an advantage
over interleave3Block, but Module and With fall dramatically behind.
This may be due to their being scoping constructs. For example
Module has to look up \$ModuleNumber and if it is n it then replaces
len with len\$n and sets up the assignment

len\$n = k  (* where k is the length of L*)

Block simply sets up the assignment

len = k.

But this seems quite inadequate to explain such large differneces.

Do[interleave3Replace[test],{30}]//Timing//First
3.66667 Second

Do[interleave3Block[test],{30}]//Timing//First
4.28333 Second

Do[interleave3Module[test],{30}]//Timing//First
48.9667 Second                                      (*!*)

Do[interleave3With[test],{30}]//Timing//First
48.5333 Second                                      (*!*)

Allan Hayes
hay at haystack.demon.co.uk

```

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