Develop Reference

Speed up processing 32 bit numbers in combinations (k from n)

I have a list of 128 32 bit numbers, and I want to know, is there any combination of 12 numbers, so that all numbers XORed give the 32 bit number with all bits set to 1.
So I have started with naive approach and took combinations generator like that:
private static IEnumerable<int[]> Combinations(int k, int n)
{
var state = new int[k];
var stack = new Stack<int>();
stack.Push(0);
while (stack.Count > 0)
{
var index = stack.Count - 1;
var value = stack.Pop();
while (value < n)
{
state[index++] = value++;
if (value < n)
{
stack.Push(value);
}
if (index == k)
{
yield return state;
break;
}
}
}
}
and used it like that (data32 is an array of given 32bit numbers)
foreach (var probe in Combinations(12, 128))
{
int p = 0;
foreach (var index in probe)
{
p = p ^ data32[index];
}
if (p == -1)
{
//print out found combination
}
}
Of course it takes forever to check all 23726045489546400 combinations...
So my question(s) are - am I missing something in options how to speedup the check process?
Even if I do the calculation of combinations in partitions (e.g. I could start like 8 threads each will check combination started with numbers 0..8), or speed up the XORing by storing the perviously calculated combination - it is still slow.
P.S. I'd like it to run in reasonable time - minutes, hours not years.
Adding a list of numbers as was requested in one of the comments:
1571089837
2107702069
466053875
226802789
506212087
484103496
1826565655
944897655
1370004928
748118360
1000006005
952591039
2072497930
2115635395
966264796
1229014633
827262231
1276114545
1480412665
2041893083
512565106
1737382276
1045554806
172937528
1746275907
1376570954
1122801782
2013209036
1650561071
1595622894
425898265
770953281
422056706
477352958
1295095933
1783223223
842809023
1939751129
1444043041
1560819338
1810926532
353960897
1128003064
1933682525
1979092040
1987208467
1523445101
174223141
79066913
985640026
798869234
151300097
770795939
1489060367
823126463
1240588773
490645418
832012849
188524191
1034384571
1802169877
150139833
1762370591
1425112310
2121257460
205136626
706737928
265841960
517939268
2070634717
1703052170
1536225470
1511643524
1220003866
714424500
49991283
688093717
1815765740
41049469
529293552
1432086255
1001031015
1792304327
1533146564
399287468
1520421007
153855202
1969342940
742525121
1326187406
1268489176
729430821
1785462100
1180954683
422085275
1578687761
2096405952
1267903266
2105330329
471048135
764314242
459028205
1313062337
1995689086
1786352917
2072560816
282249055
1711434199
1463257872
1497178274
472287065
246628231
1928555152
1908869676
1629894534
885445498
1710706530
1250732374
107768432
524848610
2791827620
1607140095
1820646148
774737399
1808462165
194589252
1051374116
1802033814

I don't know C#, I did something in Python, maybe interesting anyway. Takes about 0.8 seconds to find a solution for your sample set:
solution = {422056706, 2791827620, 506212087, 1571089837, 827262231, 1650561071, 1595622894, 512565106, 205136626, 944897655, 966264796, 477352958}
len(solution) = 12
solution.issubset(nums) = True
hex(xor(solution)) = '0xffffffff'
There are 128C12 combinations, that's 5.5 million times as many as the 232 possible XOR values. So I tried being optimistic and only tried a subset of the possible combinations. I split the 128 numbers into two blocks of 28 and 100 numbers and try combinations with six numbers from each of the two blocks. I put all possible XORs of the first block into a hash set A, then go through all XORs of the second block to find one whose bitwise inversion is in that set. Then I reconstruct the individual numbers.
This way I cover (28C6)2 × (100C6)2 = 4.5e14 combinations, still over 100000 times as many as there are possible XOR values. So probably still a very good chance to find a valid combination.
Code (Try it online!):
from itertools import combinations
from functools import reduce
from operator import xor as xor_
nums = list(map(int, '1571089837 2107702069 466053875 226802789 506212087 484103496 1826565655 944897655 1370004928 748118360 1000006005 952591039 2072497930 2115635395 966264796 1229014633 827262231 1276114545 1480412665 2041893083 512565106 1737382276 1045554806 172937528 1746275907 1376570954 1122801782 2013209036 1650561071 1595622894 425898265 770953281 422056706 477352958 1295095933 1783223223 842809023 1939751129 1444043041 1560819338 1810926532 353960897 1128003064 1933682525 1979092040 1987208467 1523445101 174223141 79066913 985640026 798869234 151300097 770795939 1489060367 823126463 1240588773 490645418 832012849 188524191 1034384571 1802169877 150139833 1762370591 1425112310 2121257460 205136626 706737928 265841960 517939268 2070634717 1703052170 1536225470 1511643524 1220003866 714424500 49991283 688093717 1815765740 41049469 529293552 1432086255 1001031015 1792304327 1533146564 399287468 1520421007 153855202 1969342940 742525121 1326187406 1268489176 729430821 1785462100 1180954683 422085275 1578687761 2096405952 1267903266 2105330329 471048135 764314242 459028205 1313062337 1995689086 1786352917 2072560816 282249055 1711434199 1463257872 1497178274 472287065 246628231 1928555152 1908869676 1629894534 885445498 1710706530 1250732374 107768432 524848610 2791827620 1607140095 1820646148 774737399 1808462165 194589252 1051374116 1802033814'.split()))
def xor(vals):
return reduce(xor_, vals)
A = {xor(a)^0xffffffff: a
for a in combinations(nums[:28], 6)}
for b in combinations(nums[28:], 6):
if a := A.get(xor(b)):
break
solution = {*a, *b}
print(f'{solution = }')
print(f'{len(solution) = }')
print(f'{solution.issubset(nums) = }')
print(f'{hex(xor(solution)) = }')

Arrange your numbers into buckets based on the position of the first 1 bit.
To set the first bit to 1, you will have to use an odd number of the items in the corresponding bucket....
As you recurse, try to maintain an invariant that the number of leading 1 bits is increasing and then select the bucket that will change the next 0 to a 1, this will greatly reduce the number of combinations that you have to try.

I have found a possible solution, which could work for my particular task.
As main issue to straitforward approach I see a number of 2E16 combinations.
But, if I want to check if combination of 12 elements equal to 0xFFFFFFFF, I could check if 2 different combinations of 6 elements with opposit values exists.
That will reduce number of combinations to "just" 5E9, which is achievable.
On first attempt I think to store all combinations and then find opposites in the big list. But, in .NET I could not find quick way of storing more then Int32.MaxValue elements.
Taking in account idea with bits from comments and answer, I decided to store at first only xor sums with leftmost bit set to 1, and then by definition I need to check only sums with leftmost bit set to 0 => reducing storage by 2.
In the end it appears that many collisions could appear, so there are many combinations with the same xor sum.
Current version which could find such combinations, need to be compiled in x64 mode and use (any impovements welcomed):
static uint print32(int[] comb, uint[] data)
{
uint p = 0;
for (int i = 0; i < comb.Length; i++)
{
Console.Write("{0} ", comb[i]);
p = p ^ data[comb[i]];
}
Console.WriteLine(" #[{0:X}]", p);
return p;
}
static uint[] data32;
static void Main(string[] args)
{
int n = 128;
int k = 6;
uint p = 0;
uint inv = 0;
long t = 0;
//load n numbers from a file
init(n);
var lookup1x = new Dictionary<uint, List<byte>>();
var lookup0x = new Dictionary<uint, List<byte>>();
Stopwatch watch = new Stopwatch();
watch.Start();
//do not use IEnumerable generator, use function directly to reuse xor value
var hash = new uint[k];
var comb = new int[k];
var stack = new Stack<int>();
stack.Push(0);
while (stack.Count > 0)
{
var index = stack.Count - 1;
var value = stack.Pop();
if (index == 0)
{
p = 0;
Console.WriteLine("Start {0} sequence, combinations found: {1}",value,t);
}
else
{
//restore previous xor value
p = hash[index - 1];
}
while (value < n)
{
//xor and store
p = p ^ data32[value];
hash[index] = p;
//remember current state (combination)
comb[index++] = value++;
if (value < n)
{
stack.Push(value);
}
//combination filled to end
if (index == k)
{
//if xor have MSB set, put it to lookup table 1x
if ((p & 0x8000000) == 0x8000000)
{
lookup1x[p] = comb.Select(i => (byte)i).ToList();
inv = p ^ 0xFFFFFFFF;
if (lookup0x.ContainsKey(inv))
{
var full = lookup0x[inv].Union(lookup1x[p]).OrderBy(x=>x).ToArray();
if (full.Length == 12)
{
print32(full, data32);
}
}
}
else
{
//otherwise put it to lookup table 2, but skip all combinations which are started with 0
if (comb[0] != 0)
{
lookup0x[p] = comb.Select(i => (byte)i).ToList();
inv = p ^ 0xFFFFFFFF;
if (lookup1x.ContainsKey(inv))
{
var full = lookup0x[p].Union(lookup1x[inv]).OrderBy(x=>x).ToArray();
if (full.Length == 12)
{
print32(full, data32);
}
}
}
}
t++;
break;
}
}
}
Console.WriteLine("Check was done in {0} ms ", watch.ElapsedMilliseconds);
//end
}

Random Integer, no repeat, within specific range

Detailed scenario:
There are 300 possible numbers that fit into the discription.
That is, the number must between 1123 and 5543 inclusive
The number is unique and not repeated
The number only has the integers 1,2,3,4 and 5.
What I am trying to achieve is a program that can display all those numbers at once in ascending order.
My current code:
var chars = "12345";
var stringChars = new char[4];
var random = new Random();
for (int i = 0; i < stringChars.Length; i++)
{
stringChars[i] = chars[random.Next(chars.Length)];
}
var finalString = new String(stringChars);
Console.WriteLine(finalString)
This code works fine, but I have 2 additional requirements.
Loop 300 times
Display all results in ascending order
EDIT:
Sorry for all confusion. What this code produces is a 4 digit number like I want. But I want to it to do so 300 times. I guess I can use a Loop, but this is my first time using C# (I've only used vb.net in the past). What I mean by random and in ascending order is: Produce a random number, and when all numbers are generated, order them in ascending order before displaying them all.
However, if ordering them will be too complicated, then I'm fine without that.

It's not entirely clear what you're asking, but here is some code that will print the 611 (not 300) numbers matching the description you gave in ascending order.
If I've understood your question correctly, you will want to sample 300 distinct elements from this larger set, sort them, and them print them out. You might look at using a Fisher-Yates shuffle to do this. Shuffle the list, take the first 300, and sort them.
public static void Main (string[] args) {
var digits = new [] { 1, 1, 2, 3 };
for (var num = DigitsToInt(digits); num <= 5543; num = DigitsToInt(digits)) {
Console.WriteLine(num);
for (int i = 3; i >= 0; i--) {
digits[i]++;
if (digits[i] < 6) {
break;
} else {
digits[i] = 1;
}
}
}
}
private static int DigitsToInt(int[] digits) {
return 1000 * digits[0] + 100 * digits[1] + 10 * digits[2] + digits[3];
}

How can I fast count how often a string appears in a given part of a sting list?

I have a list of strings where I need to count the number of list entries that have an occurances of a specific string inside of them (and the whole thing only for a subset of the list not the whole list).
The code below works quite well BUT its performance is.....sadly not in an acceptable niveau as I need to parse through 500k to 900k list entries.For these entries I need to run the code below about 10k times (as I have 10k parts of the list I need to analyse). For that it takes 177 seconds and even more. So my question is how can I do this...fast?
private int ExtraktNumbers(List<string> myList, int start, int end)
{
return myList.Where((x, index) => index >= start && index <= end
&& x.Contains("MYNUMBER:")).Count();
}

Well now we know you are calling the method 10,00 times here is my suggestion. I assume as you have hardcoded "Number:" that it means you are doing different ranges with each call? So if that's the case...
First, run an 'indexing' method and create a list of which indices are a match. Then you can easily count up the matches for the ranges you need.
NOTE: This is something quick, and you may even be able to further optimize this too:
List<int> matchIndex = new List<int>();
void RunIndex(List<string> myList)
{
for(int i = 0; i < myList.Count; i++)
{
if(myList[i].Contains("MYNUMBER:"))
{
matchIndex.Add(i);
}
}
}
int CountForRange(int start, int end)
{
return matchIndex.Count(x => x >= start && x <= end);
}
Then you can use like this, for example:
RunIndex(myList);
// I don't know what code you have here, this is just basic example.
for(int i = 0; i <= 10,000; i++)
{
int count = CountForRange(startOfRange, endOfRange);
// Do something with count.
}
In addition, if you have a lot of duplication in the ranges you check then you could consider caching range counts in a dictionary, but at this stage it's hard to tell if that will be worth doing anyway.

I am pretty sure a simple iterative solution will perform better:
private int ExtractNumbers(List<string> myList, int start, int end)
{
int count = 0;
for (int i = start; i <= end; i++)
{
if (myList[i].Contains("MYNUMBER:"))
{
count++;
}
}
return count;
}

Well for my test stand for 10 millions (10 times more than you have) lines
var data = Enumerable
.Range(1, 10000000)
.Select(item => "123456789 bla-bla-bla " + "MYNUMBER:" + item.ToString())
.ToList();
Stopwatch sw = new Stopwatch();
sw.Start();
int result = ExtraktNumbers(data, 0, 10000000);
sw.Stop();
I've got these results:
2.78 seconds - your initial implementtation
Naive loop (2.60 seconds):
private int ExtraktNumbers(List<string> myList, int start, int end) {
int result = 0;
for (int i = start; i < end; ++i)
if (myList[i].Contains("MYNUMBER:"))
result += 1;
return result;
}
PLinq (1.72 seconds):
private int ExtraktNumbers(List<string> myList, int start, int end) {
return myList
.AsParallel() // <- Do it in parallel
.Skip(start - 1)
.Take(end - start)
.Where(x => x.Contains("MYNUMBER:"))
.Count();
}
Explicit parallel implementation (1.66 seconds):
private int ExtraktNumbers(List<string> myList, int start, int end) {
long result = 0;
Parallel.For(start, end, (i) => {
if (myList[i].Contains("MYNUMBER:"))
Interlocked.Increment(ref result);
});
return (int) result;
}
I just cannot reproduce your 177 seconds

If you know from the beginning the intervals you want to consider, it's probably a good idea to loop the list once, as Dmytro and musefan proposed above, so I won't repeat the same idea again.
However I have a different suggestion for performance improvement. How do you create your list? Do you know the number of items in advance? Because for such a big list, you may gest a significant performance boost by using the List<T> constructor that takes the initial capacity.

How to: Assign a unique number to every entry in a list?

What I want is to make tiles. These tiles (about 30 of them) should have a fixed position in the game, but each time I load the game they should have random numbers that should affect their graphical appearance.
I know how to use the Random method to give a single tile a number to change its appearance, but I'm clueless on how I would use the Random method if I were to make a list storing the position of multiple tiles. How can you assign each entry in a list a unique random number?
I need this for my game where you're in a flat 2D map, generated with random types of rooms (treasure rooms, arena rooms etc.) that you are to explore.

Take a look at the Fisher-Yates shuffle. It's super easy to use and should work well for you, if I read your question right.

Make an array of 30 consecutive numbers, mirroring your array of tiles. Then pick an array-shuffling solution you like from, say, here for instance:
http://forums.asp.net/t/1778021.aspx/1
Then tile[23]'s number will be numberArray[23].

if you have something like this:
public class Tile
{
public int Number {get;set;}
...
}
you can do it like this:
var numbers = Enumerable
.Range(1, tilesList.Count) // generates list of sequential numbers
.OrderBy(x => Guid.NewGuid()) // shuffles the list
.ToList();
for (int i = 0; i < tiles.Count; i++)
{
tile[i].Number = numbers[i];
}
I know, that Guid is not a Random alternative, but it should fit this scenario.
Update: As long as answer was downvoted, I've wrote simple test, to check if Guids are not usable for shuffling an array:
var larger = 0;
var smaller = 0;
var start = DateTime.Now;
var guid = Guid.NewGuid();
for (int i = 0; i < 10000000; i++)
{
var nextGuid = Guid.NewGuid();
if (nextGuid.CompareTo(guid) < 0)
{
larger++;
}
else
{
smaller++;
}
guid = nextGuid;
}
Console.WriteLine("larger: {0}", larger);
Console.WriteLine("smaller: {0}", smaller);
Console.WriteLine("took seconds: {0}", DateTime.Now - start);
Console.ReadKey();
What it does, it counts how many times next guid is smaller than current and how many times is larger. In perfect case, there should be equal number of larger and smaller next guids, which would indicate, that those two events (current guid and next guid) are independent. Also measured time, just to make sure, that it is not too slow.
And got following result (with 10 million guids):
larger: 5000168
smaller: 4999832
took seconds: 00:00:01.1980686
Another test is direct compare of Fisher-Yates and Guid shuffling:
static void Main(string[] args)
{
var numbers = Enumerable.Range(1, 7).ToArray();
var originalNumbers = numbers.OrderBy(x => Guid.NewGuid()).ToList();
var foundAfterListUsingGuid = new List<int>();
var foundAfterListUsingShuffle = new List<int>();
for (int i = 0; i < 100; i++)
{
var foundAfter = 0;
while (!originalNumbers.SequenceEqual(numbers.OrderBy(x => Guid.NewGuid())))
{
foundAfter++;
}
foundAfterListUsingGuid.Add(foundAfter);
foundAfter = 0;
var shuffledNumbers = Enumerable.Range(1, 7).ToArray();
while (!originalNumbers.SequenceEqual(shuffledNumbers))
{
foundAfter++;
Shuffle(shuffledNumbers);
}
foundAfterListUsingShuffle.Add(foundAfter);
}
Console.WriteLine("Average matching order (Guid): {0}", foundAfterListUsingGuid.Average());
Console.WriteLine("Average matching order (Shuffle): {0}", foundAfterListUsingShuffle.Average());
Console.ReadKey();
}
static Random _random = new Random();
public static void Shuffle<T>(T[] array)
{
var random = _random;
for (int i = array.Length; i > 1; i--)
{
// Pick random element to swap.
int j = random.Next(i); // 0 <= j <= i-1
// Swap.
T tmp = array[j];
array[j] = array[i - 1];
array[i - 1] = tmp;
}
}
By "direct compare" I mean, that I'm producing shuffled sequence and try to shuffle again to get same sequence, and assume, that the more tries I need to produce same sequence, the better random is (which is not necessary mathematically correct assumption, I think it is oversimplification).
So results for small set with 1000 iterations to reduce error, was:
Average matching order (Guid): 5015.097
Average matching order (Shuffle): 4969.424
So, Guid performed event better, if my metric is correct :)
with 10000 iterations they came closer:
Average matching order (Guid): 5079.9283
Average matching order (Shuffle): 4940.749
So in my opinion, for current usage (shuffle room number in game), guids are suitable solution.

Data Aggregations over time, when time is variable

I am in the process of developing an application which calculates the shared acquired in a product over a specified time period (Term).
After the calculations have been performed, it is necessary for me to aggregate the data into groups based on a predefined review period (for example if the time required to gain 100% ownership of the product is 25 years, and the review period value is 5 years, I would have 5 sets of data aggregations for the agreement).
I perform the aggregations as shown by looping through my calculation result set:
if (Year% ReviewPeriod == 0)
{
// Perform Aggregations
}
This works fine in most scenarios.
However I do have a number of scenarios where the product reaches 100% ownership before the end of term.
What I need to be able to do is aggregate the calculations performed based on the ReviewPeriod variable, but if the final number of values in the calculations is not equal to the review period, aggregate the items based on the number of items remaining.
For example, given a 22 year term, data would be aggregated based on the Review Period variable, however if there is a remainder, then the remainder should be aggregated based on the value of the remainder.
Worked Example
Year 0 - 5 = 5 Aggregations
Year 6 - 10 = 5 Aggregations
Year 11 - 15 = 5 Aggregations
Year 16 - 20 = 5 Aggregations
Year 21 - 22 = 2 Aggregations
Could anyone help me with the logic to aggregate the data as I have described.

Probably the simplest way would be something like:
for ( int year = 0; year <= max_year; year++ ) {
if ( year % reviewPeriod == 0 ) {
// start a new aggregation
}
// add year to current aggregation
}
You could keep a list of aggregations and add a new one at the start of each period.
Here is a working example that just groups years in lists:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace Aggregations
{
class Program
{
static void Main(string[] args)
{
int maxYear = 22;
int period = 5;
int year = 1985;
List<List<int>> aggregations = new List<List<int>>();
int i = -1;
for (int y = 0; y <= maxYear; y++)
{
if (y % period == 0)
{
aggregations.Add(new List<int>());
i++;
}
aggregations.ElementAt(i).Add(year);
year++;
}
foreach ( List<int> l in aggregations )
{
foreach (int yy in l)
{
Console.Write(yy + " ");
}
Console.WriteLine();
}
}
}
}

You've not really given enough of your code to go on. Hopefully you should be able to use this however your loop is currently set up. It "leaks" the mod value to the outside of the loop; after the loop is over, you can check the final mod value to see how many aggregations are left.
int modValue = 0;
for //foreach/while/... - your loop here
{
...
modValue = Year % ReviewPeriod;
if (modValue == 0)
{
// Perform Aggregations
}
...
} // end of your loop
if (modValue != 0)
{
// Perform final aggregation. There are modValue items to aggregate.
}

I think my suggestion is not worth 300rep bounty, and either I misunderstood your problem, or you've overshot the bounty..
Do your existing code that calculates the final aggregations works well? If so, then to determine the ranges yo umay just use modulo (%) and simple math:
int minYear = ...the first year // inclusive, i.e. 1970
int maxYear = ...the last year // inclusive, i.e. 2012
int span = maxYear - minYear + 1; // 1970..2012->43, 2001..2006->6
int fullFives = span / 5; // 1970..2012->8, 2001..2006->1
int remainder = span % 5; // 2001..2006->3, 2001..2006->1
for(int i=0; i<fullFives; ++i)
{
int yearFrom = minYear + 5*i
int yearTo = minYear + 5*(i+1) - 1
// 1970..2012 -> 1970-1974, 1975-1979,1980-1984,1985-1989,1990-1994,1995-1999,2000-2004,2005-2009
// 2001..2006 -> 2001-2005
aggregate(yearFrom, yearTo);
}
if(remainder > 0)
{
int yearFrom = minYear + 5*fullFives
int yearTo = minYear + maxYear
// 1970..2012 -> 2010-2012
// 2001..2006 -> 2006-2006
aggregate(yearFrom, yearTo);
}
This is written "out of thin air", I've not checked/compiled it - it is just to sketch the idea.
Note: you've said that everything works but sometimes "a number of scenarios where the product reaches 100% ownership before the end of term." - that would suggest that you rather have an error in the calculations, not in the looping. If the error were in the loop or year boundary detection, then probably almost all would be off. It's hard to say without more of the calculating code is revealed.

The code sample will fire on years 0, 5, 10 etc rather than for every year.
If you just need the number of years to aggregate when that code fires, and the term can be set in advance when a product reaches 100% ownership early, I think this would work:
int term = 22;
int reviewperiod = 5;
for (int year = 0; year < term; year++)
{
if (year % reviewperiod == 0)
{
var endyear = Math.Min(year + reviewperiod, term);
Console.WriteLine("Aggregate years {0} to {1}, {2} Aggregations ", year, endyear, endyear - year);
}
}

Do you think of something like
private int reviewPeriod = 5;
public void Aggregate(int term)
{
Enumerable.Range(0, term)
.ToList()
.Foreach(this.AggregateYear);
}
when this.AggregateYear is defined as follows
public void AggregateYear(int year)
{
var currentRemainder = year % this.reviewPeriod;
var aggregatePeriod = (currentRemainder == 0)
? this.reviewPeriod
: currentRemainder;
this.PerformAggregation(aggregatePeriod);
}
and this.PerformAggregation is defined as follows
private void PerformAggregation(int aggregatePeriod)
{
//...
}

Assuming this data is in memory (since you have not specified otherwise), then you can just use the GroupBy function from Linq:
struct YearValue
{
public int Year, Value;
}
static void Main()
{
// Create some data, hopefully representative of what you are dealing with...
Random r = new Random();
YearValue[] dataValues = new YearValue[22];
for (int i = 0; i < dataValues.Length; i++)
dataValues[i] = new YearValue {Year = i, Value = r.Next(200)};
// Average of values across 'ReviewPeriod' of five:
foreach (var item in dataValues.AsEnumerable().GroupBy(i => i.Year / 5))
{
YearValue[] items = item.ToArray();
Console.WriteLine("Group {0} had {1} item(s) averaging {2}",
item.Key,
items.Length,
items.Average(i => i.Value)
);
}
}
This program then outputs the following text:
Group 0 had 5 item(s) averaging 143.6
Group 1 had 5 item(s) averaging 120.4
Group 2 had 5 item(s) averaging 83
Group 3 had 5 item(s) averaging 145.2
Group 4 had 2 item(s) averaging 98.5