associative arrays in awk challenging memory limits - memory

This is related to my recent post in Awk code with associative arrays -- array doesn't seem populated, but no error and also to optimizing loop, passing parameters from external file, naming array arguments within awk
My basic problem here is simply to compute from detailed ancient archival financial market data, daily aggregates of #transactions, #shares, value, BY DATE, FIRM-ID, EXCHANGE, etc. Learnt to use associative arrays in awk for this, and was thrilled to be able to process 129+ million lines in clock time of under 11 minutes. Literally before I finished my coffee.
Became a little more ambitious, and moved from 2 array subscripts to 4, and now I am unable to process more than 6500 lines at a time.
Get error messages of the form:
K:\User Folders\KRISHNANM\PAPERS\FII_Transaction_Data>zcat
RAW_DATA\2003_1.zip | gawk -f CODE\FII_daily_aggregates_v2.awk >
OUTPUT\2003_1.txt&
gawk: CODE\FII_daily_aggregates_v2.awk:33: (FILENAME=- FNR=49300)
fatal: more_no des: nextfree: can't allocate memory (Not enough space)
On some runs the machine has told me it lacks as little as 52 KB of memory. I have what I think of a std configuration with Win-7 and 8MB RAM.
(Economist by training, not computer scientist.) I realize that going from 2 to 4 arrays makes the problem computationally much more complex for the computer, but is there something one can do to improve memory management at least a little bit. I have tried closing everything else I am doing. The error always has to do only with memory, never with disk space or anything else.
Sample INPUT:
49290,C198962542782200306,6/30/2003,433581,F5811773991200306,S5405611832200306,B5086397478200306,NESTLE INDIA LTD.,INE239A01016,6/27/2003,1,E9035083824200306,REG_DL_STLD_02,591.13,5655,3342840.15,REG_DL_INSTR_EQ,REG_DL_DLAY_P,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
49291,C198962542782200306,6/30/2003,433563,F6292896459200306,S6344227311200306,B6110521493200306,GRASIM INDUSTRIES LTD.,INE047A01013,6/27/2003,1,E9035083824200306,REG_DL_STLD_02,495.33,3700,1832721,REG_DL_INSTR_EQ,REG_DL_DLAY_P,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
49292,C198962542782200306,6/30/2003,433681,F6513202607200306,S1724027402200306,B6372023178200306,HDFC BANK LTD,INE040A01018,6/26/2003,1,E745964372424200306,REG_DL_STLD_02,242,2600,629200,REG_DL_INSTR_EQ,REG_DL_DLAY_D,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
49293,C7885768925200306,6/30/2003,48128,F4406661052200306,S7376401565200306,B4576522576200306,Maruti Udyog Limited,INE585B01010,6/28/2003,3,E912851176274200306,REG_DL_STLD_04,125,44600,5575000,REG_DL_INSTR_EQ,REG_DL_DLAY_P,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
49294,C7885768925200306,6/30/2003,48129,F4500260787200306,S1312094035200306,B4576522576200306,Maruti Udyog Limited,INE585B01010,6/28/2003,4,E912851176274200306,REG_DL_STLD_04,125,445600,55700000,REG_DL_INSTR_EQ,REG_DL_DLAY_P,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
49295,C7885768925200306,6/30/2003,48130,F6425024637200306,S2872499118200306,B4576522576200306,Maruti Udyog Limited,INE585B01010,6/28/2003,3,E912851176274200306,REG_DL_STLD_04,125,48000,6000000,REG_DL_INSTR_EU,REG_DL_DLAY_P,DL_RPT_TYPE_N,DL_AMDMNT_DEL_00
Code
BEGIN { FS = "," }
# For each array subscript variable -- DATE ($10), firm_ISIN ($9), EXCHANGE ($12), and FII_ID ($5), after checking for type = EQ, set up counts for each value, and number of unique values.
( $17~/_EQ\>/ ) { if (date[$10]++ == 0) date_list[d++] = $10;
if (isin[$9]++ == 0) isin_list[i++] = $9;
if (exch[$12]++ == 0) exch_list[e++] = $12;
if (fii[$5]++ == 0) fii_list[f++] = $5;
}
# For cash-in, buy (B), or cash-out, sell (S) count NR = no of records, SH = no of shares, RV = rupee-value.
(( $17~/_EQ\>/ ) && ( $11~/1|2|3|5|9|1[24]/ )) {{ ++BNR[$10,$9,$12,$5]} {BSH[$10,$9,$12,$5] += $15} {BRV[$10,$9,$12,$5] += $16} }
(( $17~/_EQ\>/ ) && ( $11~/4|1[13]/ )) {{ ++SNR[$10,$9,$12,$5]} {SSH[$10,$9,$12,$5] += $15} {SRV[$10,$9,$12,$5] += $16} }
END {
{ print NR, "records processed."}
{ print " " }
{ printf("%-11s\t%-13s\t%-20s\t%-19s\t%-7s\t%-7s\t%-14s\t%-14s\t%-18s\t%-18s\n", \
"DATE", "ISIN", "EXCH", "FII", "BNR", "SNR", "BSH", "SSH", "BRV", "SRV") }
{ for (u = 0; u < d; u++)
{
for (v = 0; v < i; v++)
{
for (w = 0; w < e; w++)
{
for (x = 0; x < f; x++)
#check first below for records with zeroes, don't print them
{ if (BNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]] + SNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]] > 0)
{ BR = BNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SR = SNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
BS = BSH[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
BV = BRV[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SS = SSH[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SV = SRV[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
{ printf("%-11s\t%13s\t%20s\t%19s\t%7d\t%7d\t%14d\t%14d\t%18.2f\t%18.2f\n", \
date_list[u], isin_list[v], exch_list[w], fii_list[x], BR, SR, BS, SS, BV, SV) } }
}
}
}
}
}
}
Expected output
6 records processed.
DATE ISIN EXCH FII BNR SNR BSH SSH BRV SRV
6/27/2003 INE239A01016 E9035083824200306 F5811773991200306 1 0 5655 0 3342840.15 0.00
6/27/2003 INE047A01013 E9035083824200306 F6292896459200306 1 0 3700 0 1832721.00 0.00
6/26/2003 INE040A01018 E745964372424200306 F6513202607200306 1 0 2600 0 629200.00 0.00
6/28/2003 INE585B01010 E912851176274200306 F4406661052200306 1 0 44600 0 5575000.00 0.00
6/28/2003 INE585B01010 E912851176274200306 F4500260787200306 0 1 0 445600 0.00 55700000.00
It is in this case that as the number of input records exceeds 6500, I end up having memory problems. Have about 7 million records in all.
For a 2 array subscript problem, albeit on a different data set, where 129+ million lines were processed in clock time of 11 minutes using the same GNU-AWK on the same machine, see optimizing loop, passing parameters from external file, naming array arguments within awk
Question: is it the case that awk is not very smart with memory management, but that some other more modern tools (say, SQL) would accomplish this task with the same memory resources? Or is this simply a characteristic of associative arrays, which I found magical in enabling me to avoid many passes over the data, many loops and SORT procedures, but which maybe work well up to 2 array subscripts, and then face exponential memory resource costs after that?
Afterword: the super-detailed almost-idiot-proof tutorial along with the code provided by Ed Morton in comments below makes a dramatic difference, especially his GAWK script tst.awk. He taught me about (a) using SUBSEP intelligently (b) tackling needless looping, which is crucial in this problem which tends to have very sparse arrays, with various AWK constructs. Compared to performance with my old code (only up to 6500 lines of input accepted on one machine, another couldn't even get that far), the performance of Ed Morton's tst.awk can be seen from the table below:
**filename start end min in ln out lines
2008_1 12:08:40 AM 12:27:18 AM 0:18 391438 301160
2008_2 12:27:18 AM 12:52:04 AM 0:24 402016 314177
2009_1 12:52:05 AM 1:05:15 AM 0:13 302081 238204
2009_2 1:05:15 AM 1:22:15 AM 0:17 360072 276768
2010_1 "slept" 507496 397533
2010_2 3:10:26 AM 3:10:50 AM 0:00 76200 58228
2010_3 3:10:50 AM 3:11:18 AM 0:00 80988 61725
2010_4 3:11:18 AM 3:11:47 AM 0:00 86923 65885
2010_5 3:11:47 AM 3:12:15 AM 0:00 80670 63059**
Times were obtained simply from using %time% on lines before and after tst.awk was executed, all put in a simple batch script, "min" is the clock time taken (per whatever rounding EXCEL does by default), "in ln" and "out lines" are lines of input and output, respectively. From processing the entire data that we have, from Jan 2003 to Jan 2014, we find the theoretical max number of output records = #dates*#ISINs*#Exchanges*#FIIs = 2992*2955*567*82268, while the actual number of total output lines is only 5,261,942, which is only 1.275*10^(-8) of the theoretical max -- very sparse indeed. That there was sparseness, we did guess earlier, but that the arrays could be SO sparse -- which matters a lot for memory management -- we had no way of telling till something actually completed, for a real data set. Time taken seems to increase exponentially in input size, but within limits that pose no practical difficulty. Thanks a ton, Ed.

There is no problem with associative arrays in general. In awk (except gawk for true 2D arrays) an associative array with 4 subscripts is identical to one with 2 subscripts since in reality it only has one subscript which is the concatenation of each of the pseudo-subscripts separated by SUBSEP.
Given you say I am unable to process more than 6500 lines at a time. the problem is far more likely to be in the way you wrote your code than any fundamental awk issue so if you'd like more help, post a small script with sample input and expected output that demonstrates your problem and attempted solution to see if we have suggestions on way to improve it's memory usage.
Given your posted script, I expect the problem is with those nested loops in your END section When you do:
for (i=1; i<=maxI; i++) {
for (j=1; j<=maxJ; j++) {
if ( arr[i,j] != 0 ) {
print arr[i,j]
}
}
}
you are CREATING arr[i,j] for every possible combination of i and j that didn't exist prior to the loop just by testing for arr[i,j] != 0. If you instead wrote:
for (i=1; i<=maxI; i++) {
for (j=1; j<=maxJ; j++) {
if ( (i,j) in arr ) {
print arr[i,j]
}
}
}
then the loop itself would not create new entries in arr[].
So change this block:
if (BNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]] + SNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]] > 0)
{
BR = BNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SR = SNR[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
BS = BSH[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
BV = BRV[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SS = SSH[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
SV = SRV[date_list[u],isin_list[v],exch_list[w],fii_list[x]]
which is probably unnecessarily turning each of BNR, SNR, BSH, BRV, SSH, and SRV into huge but highly sparse arrays, to something like this:
idx = date_list[u] SUBSEP isin_list[v] SUBSEP exch_list[w] SUBSEP fii_list[x]
BR = (idx in BNR ? BNR[idx] : 0)
SR = (idx in SNR ? SNR[idx] : 0)
if ( (BR + SR) > 0 )
{
BS = (idx in BSH ? BSH[idx] : 0)
BV = (idx in BRV ? BRV[idx] : 0)
SS = (idx in SSH ? SSH[idx] : 0)
SV = (idx in SRV ? SRV[idx] : 0)
and let us know if that helps. Also check your code for other places where you might be doing the same.
The reason you have this problem with 4 subscripts when you didn't with 2 is simply that you have 4 levels of nesting in the loops now creating much larger and more sparse arrays when when you just had 2.
Finally - you have some weird syntax in your script, some of which #MarkSetchell pointed out in a comment, and your script isn't as efficient as it could be since you're not using else statements and so testing for multiple conditions that can't possibly all be true and you're testing the same condition repeatedly, and it's not robust as you aren't anchoring your REs (e.g you test /4|1[13]/ instead of /^(4|1[13])$/ so for example your 4 would match on 14 or 41 etc. instead of just 4 on its own) so change your whole script to this:
$ cat tst.awk
BEGIN { FS = "," }
# For each array subscript variable -- DATE ($10), firm_ISIN ($9), EXCHANGE ($12), and FII_ID ($5), after checking for type = EQ, set up counts for each value, and number of unique values.
$17 ~ /_EQ\>/ {
if (!seenDate[$10]++) date_list[++d] = $10
if (!seenIsin[$9]++) isin_list[++i] = $9
if (!seenExch[$12]++) exch_list[++e] = $12
if (!seenFii[$5]++) fii_list[++f] = $5
# For cash-in, buy (B), or cash-out, sell (S) count NR = no of records, SH = no of shares, RV = rupee-value.
idx = $10 SUBSEP $9 SUBSEP $12 SUBSEP $5
if ( $11 ~ /^([12359]|1[24])$/ ) {
++BNR[idx]; BSH[idx] += $15; BRV[idx] += $16
}
else if ( $11 ~ /^(4|1[13])$/ ) {
++SNR[idx]; SSH[idx] += $15; SRV[idx] += $16
}
}
END {
print NR, "records processed."
print " "
printf "%-11s\t%-13s\t%-20s\t%-19s\t%-7s\t%-7s\t%-14s\t%-14s\t%-18s\t%-18s\n",
"DATE", "ISIN", "EXCH", "FII", "BNR", "SNR", "BSH", "SSH", "BRV", "SRV"
for (u = 1; u <= d; u++)
{
for (v = 1; v <= i; v++)
{
for (w = 1; w <= e; w++)
{
for (x = 1; x <= f; x++)
{
#check first below for records with zeroes, don't print them
idx = date_list[u] SUBSEP isin_list[v] SUBSEP exch_list[w] SUBSEP fii_list[x]
BR = (idx in BNR ? BNR[idx] : 0)
SR = (idx in SNR ? SNR[idx] : 0)
if ( (BR + SR) > 0 )
{
BS = (idx in BSH ? BSH[idx] : 0)
BV = (idx in BRV ? BRV[idx] : 0)
SS = (idx in SSH ? SSH[idx] : 0)
SV = (idx in SRV ? SRV[idx] : 0)
printf "%-11s\t%13s\t%20s\t%19s\t%7d\t%7d\t%14d\t%14d\t%18.2f\t%18.2f\n",
date_list[u], isin_list[v], exch_list[w], fii_list[x], BR, SR, BS, SS, BV, SV
}
}
}
}
}
}
I added seen in front of 4 array names just because by convention arrays testing for the pre-existence of a value are typically named seen. Also, when populating the SNR[] etc arrays I created an idx variable first instead of repeatedly using the field numbers every time for both ease of changing it in future and mostly because string concatenation is relatively slow in awk and that's whats happening when you use multiple indices in an array so best to just do the string concatenation once explicitly. And I changed your date_list[] etc arrays to start at 1 instead of zero because all awk-generated arrays, strings and field numbers start at 1. You CAN create an array manually that starts at 0 or -357 or whatever number you want but it'll save shooting yourself in the foot some day if you always start them at 1.
I expect it could be made more efficient still by restricting the nested loops to only values that could exist for the enclosing loop index combinations (e.g. not every value of u+v+w is possible so there will be times when you shouldn't bother looping on x). For example:
$ cat tst.awk
BEGIN { FS = "," }
# For each array subscript variable -- DATE ($10), firm_ISIN ($9), EXCHANGE ($12), and FII_ID ($5), after checking for type = EQ, set up counts for each value, and number of unique values.
$17 ~ /_EQ\>/ {
if (!seenDate[$10]++) date_list[++d] = $10
if (!seenIsin[$9]++) isin_list[++i] = $9
if (!seenExch[$12]++) exch_list[++e] = $12
if (!seenFii[$5]++) fii_list[++f] = $5
# For cash-in, buy (B), or cash-out, sell (S) count NR = no of records, SH = no of shares, RV = rupee-value.
idx = $10 SUBSEP $9 SUBSEP $12 SUBSEP $5
if ( $11 ~ /^([12359]|1[24])$/ ) {
seen[$10,$9]
seen[$10,$9,$12]
++BNR[idx]; BSH[idx] += $15; BRV[idx] += $16
}
else if ( $11 ~ /^(4|1[13])$/ ) {
seen[$10,$9]
seen[$10,$9,$12]
++SNR[idx]; SSH[idx] += $15; SRV[idx] += $16
}
}
END {
printf "d = %d\n", d | "cat>&2"
printf "i = %d\n", i | "cat>&2"
printf "e = %d\n", e | "cat>&2"
printf "f = %d\n", f | "cat>&2"
print NR, "records processed."
print " "
printf "%-11s\t%-13s\t%-20s\t%-19s\t%-7s\t%-7s\t%-14s\t%-14s\t%-18s\t%-18s\n",
"DATE", "ISIN", "EXCH", "FII", "BNR", "SNR", "BSH", "SSH", "BRV", "SRV"
for (u = 1; u <= d; u++)
{
date = date_list[u]
for (v = 1; v <= i; v++)
{
isin = isin_list[v]
if ( (date,isin) in seen )
{
for (w = 1; w <= e; w++)
{
exch = exch_list[w]
if ( (date,isin,exch) in seen )
{
for (x = 1; x <= f; x++)
{
fii = fii_list[x]
#check first below for records with zeroes, don't print them
idx = date SUBSEP isin SUBSEP exch SUBSEP fii
if ( (idx in BNR) || (idx in SNR) )
{
if (idx in BNR)
{
bnr = BNR[idx]
bsh = BSH[idx]
brv = BRV[idx]
}
else
{
bnr = bsh = brv = 0
}
if (idx in SNR)
{
snr = SNR[idx]
ssh = SSH[idx]
srv = SRV[idx]
}
else
{
snr = ssh = srv = 0
}
printf "%-11s\t%13s\t%20s\t%19s\t%7d\t%7d\t%14d\t%14d\t%18.2f\t%18.2f\n",
date, isin, exch, fii, bnr, snr, bsh, ssh, brv, srv
}
}
}
}
}
}
}
}

Related

Is there a more succinct way of performing the same task below? Rounding down to nearest 1 or 5mod6

var n = 27;
void main() {
while (n>=5)
if ( (n) % 3==0 && (n) % 2==0 ){
print(n-1);
break;
}
else if ( (n) % 3==0 && (n) % 2!=0 ){
print(n-2);
break;
}
else if ((n) % 3!=0 && (n) % 2==0){
print((n/6).floor()*6 +(1));
break;
}
else { n=n+1;}
}
I was looking for a way to have an input reduced to the nearest 1 or 5mod6 (for n>=5) and came up with the code above in Dart (project is in flutter). Does anyone have any better way of doing the same thing?
The result (rounded down input) will then be passed to another function.
For the current n value of 27 the console will print 25... try other values 24 maps to 23, 23 to itself, 22,21,20 to 19, 19 to itself, 18 to 17, 17 to itself, 16,15,14 to 13.....and I hope you get the idea.
I would get rid of the loop by finding the next lower multiple of 6 and then adding 1 or 5. The code ends up being shorter, and I think it expresses your intent much more clearly than code involving % 3 and % 2.
int lowerMultiple(int n, int multipleOf) =>
((n - 1) / multipleOf).floor() * multipleOf;
void main() {
var n = 27;
var m = lowerMultiple(n, 6);
print((n - m) >= 5 ? (m + 5) : (m + 1));
}
The above code should work for integers less than 5 as well, including non-positive ones.

Why doesn't this Fibonacci Number function work in O(log N)?

So the Fibonacci number for log (N) — without matrices.
Ni // i-th Fibonacci number
= Ni-1 + Ni-2 // by definition
= (Ni-2 + Ni-3) + Ni-2 // unwrap Ni-1
= 2*Ni-2 + Ni-3 // reduce the equation
= 2*(Ni-3 + Ni-4) + Ni-3 //unwrap Ni-2
// And so on
= 3*Ni-3 + 2*Ni-4
= 5*Ni-4 + 3*Ni-5
= 8*Ni-5 + 5*Ni-6
= Nk*Ni-k + Nk-1*Ni-k-1
Now we write a recursive function, where at each step we take k~=I/2.
static long N(long i)
{
if (i < 2) return 1;
long k=i/2;
return N(k) * N(i - k) + N(k - 1) * N(i - k - 1);
}
Where is the fault?
You get a recursion formula for the effort: T(n) = 4T(n/2) + O(1). (disregarding the fact that the numbers get bigger, so the O(1) does not even hold). It's clear from this that T(n) is not in O(log(n)). Instead one gets by the master theorem T(n) is in O(n^2).
Btw, this is even slower than the trivial algorithm to calculate all Fibonacci numbers up to n.
The four N calls inside the function each have an argument of around i/2. So the length of the stack of N calls in total is roughly equal to log2N, but because each call generates four more, the bottom 'layer' of calls has 4^log2N = O(n2) Thus, the fault is that N calls itself four times. With only two calls, as in the conventional iterative method, it would be O(n). I don't know of any way to do this with only one call, which could be O(log n).
An O(n) version based on this formula would be:
static long N(long i) {
if (i<2) {
return 1;
}
long k = i/2;
long val1;
long val2;
val1 = N(k-1);
val2 = N(k);
if (i%2==0) {
return val2*val2+val1*val1;
}
return val2*(val2+val1)+val1*val2;
}
which makes 2 N calls per function, making it O(n).
public class fibonacci {
public static int count=0;
public static void main(String[] args) {
Scanner scan = new Scanner(System.in);
int i = scan.nextInt();
System.out.println("value of i ="+ i);
int result = fun(i);
System.out.println("final result is " +result);
}
public static int fun(int i) {
count++;
System.out.println("fun is called and count is "+count);
if(i < 2) {
System.out.println("function returned");
return 1;
}
int k = i/2;
int part1 = fun(k);
int part2 = fun(i-k);
int part3 = fun(k-1);
int part4 = fun(i-k-1);
return ((part1*part2) + (part3*part4)); /*RESULT WILL BE SAME FOR BOTH METHODS*/
//return ((fun(k)*fun(i-k))+(fun(k-1)*fun(i-k-1)));
}
}
I tried to code to problem defined by you in java. What i observed is that complexity of above code is not completely O(N^2) but less than that.But as per conventions and standards the worst case complexity is O(N^2) including some other factors like computation(division,multiplication) and comparison time analysis.
The output of above code gives me information about how many times the function
fun(int i) computes and is being called.
OUTPUT
So including the time taken for comparison and division, multiplication operations, the worst case time complexity is O(N^2) not O(LogN).
Ok if we use Analysis of the recursive Fibonacci program technique.Then we end up getting a simple equation
T(N) = 4* T(N/2) + O(1)
where O(1) is some constant time.
So let's apply Master's method on this equation.
According to Master's method
T(n) = aT(n/b) + f(n) where a >= 1 and b > 1
There are following three cases:
If f(n) = Θ(nc) where c < Logba then T(n) = Θ(nLogba)
If f(n) = Θ(nc) where c = Logba then T(n) = Θ(ncLog n)
If f(n) = Θ(nc) where c > Logba then T(n) = Θ(f(n))
And in our equation a=4 , b=2 & c=0.
As case 1 c < logba => 0 < 2 (which is log base 2 and equals to 2) is satisfied
hence T(n) = O(n^2).
For more information about how master's algorithm works please visit: Analysis of Algorithms
Your idea is correct, and it will perform in O(log n) provided you don't compute the same formula
over and over again. The whole point of having N(k) * N(i-k) is to have (k = i - k) so you only have to compute one instead of two. But if you only call recursively, you are performing the computation twice.
What you need is called memoization. That is, store every value that you already have computed, and
if it comes up again, then you get it in O(1).
Here's an example
const int MAX = 10000;
// memoization array
int f[MAX] = {0};
// Return nth fibonacci number using memoization
int fib(int n) {
// Base case
if (n == 0)
return 0;
if (n == 1 || n == 2)
return (f[n] = 1);
// If fib(n) is already computed
if (f[n]) return f[n];
// (n & 1) is 1 iff n is odd
int k = n/2;
// Applying your formula
f[n] = fib(k) * fib(n - k) + fib(k - 1) * fib(n - k - 1);
return f[n];
}

Dividing a value between non-equal rows in order to balance them

I have a spreadsheet that's structured like:
Section Total Incoming New Total
AK 56,445 2,655 59,100
AL 58,304 796 59,100
B 55,524 3,576 59,100
C 54,272 4,828 59,100
D 53,956 5,144 59,100
S 59,161 0 59,161
-
Generated Pts 16,999
I'm trying to automate the "Incoming" column. The goal of the sheet is to balance the Totals as closely as possible by distributing the Generated Pts between each row until no more points remain, ensuring that the lowest totals are always increased first so that higher values aren't increased while lower values exist.
Is this possible in a spreadsheet? Any suggestions on how this could be done?
I made an attempt at a custom function. Two parameters are passed: the range corresponding with your Total column, and the cell containing the generated pts. Then the Incoming array is returned.
function distribute(range, value) {
var indexedRange = range.map(function (e, index) {return [e[0], e[0], index];});
indexedRange.sort(function (a, b) {return a[0] < b[0] ? -1 : a[0] > b[0] ? 1 : 0;});
var count = 0, i = 0, limit = indexedRange.length - 1;
while (count < value) {
indexedRange[i][0] ++;
i = i == limit || indexedRange[i][0] <= indexedRange[i + 1][0] ? 0 : i + 1;
count++;
}
indexedRange.sort(function (a, b) {return a[2] < b[2] ? -1 : a[2] > b[2] ? 1 : 0;});
return indexedRange.map(function (e) {return [e[0] - e[1]];});
}
It matches your expected results, but you might want to try it out on different data to check my logic is OK.

Huffman's encoding and decoding

I have to build a compressor based on the Huffman algorithm.
So far, I managed to create the tree with the frequencies of each character and generate a representation with a smaller number of bits for each character.
Is something like this, for the phrase "good this sugarplum":
'o' 000, '' 001, 't' 0100, 'r' 0101, 'p' 0110, 'm' 0111, 'l' 1000, 'i' 1001, 'h' 1010, 'd' 1011, 'a'1100, 'u' 1101, 'g' 1110, 's' 1111
The problem I'm having now is finding a way to save the tree in the archive, so I can rebuild it and then decompress the file.
Any suggestions?
I did some research but found it difficult to understand, so if you can explain in detail, I would appreciate it.
The code I used to read the frequencies from file is:
int main (int argc, char *argv[])
{
int i;
TipoSentinela *sentinela;
TipoLista *no = NULL;
Arv *arvore, *arvore2, *arvore3;
int *repete = (int *) calloc (256, sizeof(int));
if(argc == 2)
{
in = load_base(argv[1]);
le_dados_arquivo (repete); //read the frequencies from the file
sentinela = cria_lista (); //create a marker for the tree node list
for (i = 0; i < 256; i++)
{
if(repete[i] > 0 && i != 0)
{
arvore = arv_cria (Cria_info (i, repete[i])); //create a tree node with the character i and the frequence of it in the file
no = inicia_lista (arvore, no, sentinela); //create the list of tree nodes
}
}
Ordena (sentinela); //sort the tree nodes list by the frequencies
for(Seta_primeiro(sentinela); Tamanho_lista(sentinela) != 1; Move_marcador(sentinela))
{
Seta_primeiro(sentinela); //put the marker in the first element of the list
no = Retorna_marcador(sentinela);
arvore2 = Retorna_arvore (no); //return the tree represented by the list marker
Move_marcador(sentinela); //put the marker to the next element
arvore3 = Retorna_arvore (Retorna_marcador (sentinela)); //return the tree represented by the list marker
arvore = Cria_pai (arvore2, arvore3); //create a tree node that will contain the both arvore2 and arvore3
Insere_arvoreFinal (sentinela, arvore); //insert the node at the end of the list
Remove_arvore (sentinela); //remove the node arvore2 from the list
Remove_arvore (sentinela); //remove the node arvore3 from the lsit
Ordena (sentinela); //sort the list again
}
out = load_out(argv[1]); //open the output file
Codificacao (arvore); //generate the code from each node of the tree
rewind(in);
char c;
while(!feof(in))
{
c = fgetc(in);
if(c != EOF)
arvore2 = Procura_info (arvore, c); //search the character c in the tree
if(arvore2 != NULL)
imprimebit(Retorna_codigo(arvore2), out); //write the code in the file
}
fclose(in);
fclose(out);
free(repete);
arvore = arv_libera (arvore);
Libera_Lista(sentinela);
}
return 0;
}
//bit_counter and cur_byte are global variables
void write_bit (unsigned char bit, FILE *f)
{
static k = 0;
if(k != 0)
{
if(++bit_counter == 8)
{
fwrite(&cur_byte,1,1,f);
bit_counter = 0;
cur_byte = 0;
}
}
k = 1;
cur_byte <<= 1;
cur_byte |= ('0' != bit);
}
//aux is the code of a character in the tree
void imprimebit(char *aux, FILE *f)
{
int i, j;
if(aux == NULL)
return;
for(i = 0; i < strlen(aux); i++)
{
write_bit(aux[i], f); //write the bits of the code in the file
}
}
With this, I can write the code of all characters in the output file, but I can't see a way to store the tree too.
You don't need to send the tree. Just send the lengths. Then establish a consistent algorithm to convert the lengths to codes on both ends. The consistency is called a "canonical" Huffman code. You sort the codes by length, and within each length, sort by the symbol. Then assign codes starting at 0. So you would end up with (_ means space):
_ 000
o 001
a 0100
d 0101
g 0110
h 0111
i 1000
l 1001
m 1010
p 1011
r 1100
s 1101
t 1110
u 1111
I did found a way to store the code of each character.
For example:
I write the tree, starting by the root and going down to the left, then right.
So, if my tree was something like
0
/ \
0 1
/ \ / \
'a' 'b' 'c' 'd'
The header of my file would be someting like this:
001[8 bits from 'a']1[8 bits from b]01[8 bits from c]1[8 bits from d]
With this, I would be able to rebuild my tree.
My problem now is in read bit-by-bit of the header of file to know in wich direction I have to create a new node.

How can I do mod without a mod operator?

This scripting language doesn't have a % or Mod(). I do have a Fix() that chops off the decimal part of a number. I only need positive results, so don't get too robust.
Will
// mod = a % b
c = Fix(a / b)
mod = a - b * c
do? I'm assuming you can at least divide here. All bets are off on negative numbers.
a mod n = a - (n * Fix(a/n))
For posterity, BrightScript now has a modulo operator, it looks like this:
c = a mod b
If someone arrives later, here are some more actual algorithms (with errors...read carefully)
https://eprint.iacr.org/2014/755.pdf
There are actually two main kind of reduction formulae: Barett and Montgomery. The paper from eprint repeat both in different versions (algorithms 1-3) and give an "improved" version in algorithm 4.
Overview
I give now an overview of the 4. algorithm:
1.) Compute "A*B" and Store the whole product in "C" that C and the modulus $p$ is the input for that algorithm.
2.) Compute the bit-length of $p$, say: the function "Width(p)" returns exactly that value.
3.) Split the input $C$ into N "blocks" of size "Width(p)" and store each in G. Start in G[0] = lsb(p) and end in G[N-1] = msb(p). (The description is really faulty of the paper)
4.) Start the while loop:
Set N=N-1 (to reach the last element)
precompute $b:=2^{Width(p)} \bmod p$
while N>0 do:
T = G[N]
for(i=0; i<Width(p); i++) do: //Note: that counter doesn't matter, it limits the loop)
T = T << 1 //leftshift by 1 bit
while is_set( bit( T, Width(p) ) ) do // (N+1)-th bit of T is 1
unset( bit( T, Width(p) ) ) // unset the (N+1)-th bit of T (==0)
T += b
endwhile
endfor
G[N-1] += T
while is_set( bit( G[N-1], Width(p) ) ) do
unset( bit( G[N-1], Width(p) ) )
G[N-1] += b
endwhile
N -= 1
endwhile
That does alot. Not we only need to recursivly reduce G[0]:
while G[0] > p do
G[0] -= p
endwhile
return G[0]// = C mod p
The other three algorithms are well defined, but this lacks some information or present it really wrong. But it works for any size ;)
What language is it?
A basic algorithm might be:
hold the modulo in a variable (modulo);
hold the target number in a variable (target);
initialize modulus variable;
while (target > 0) {
if (target > modulo) {
target -= modulo;
}
else if(target < modulo) {
modulus = target;
break;
}
}
This may not work for you performance-wise, but:
while (num >= mod_limit)
num = num - mod_limit
In javascript:
function modulo(num1, num2) {
if (num2 === 0 || isNaN(num1) || isNaN(num2)) {
return NaN;
}
if (num1 === 0) {
return 0;
}
var remainderIsPositive = num1 >= 0;
num1 = Math.abs(num1);
num2 = Math.abs(num2);
while (num1 >= num2) {
num1 -= num2
}
return remainderIsPositive ? num1 : 0 - num1;
}

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