Working with Binary numbers BASE(..., 2, 7) and bit operations BITXOR, BITAND
I recently came across an Excel challenge from Chinmay Amte on LinkedIn and I plan to post my solution here because it contains some 'off the beaten track' techniques that might be of some interest. ...
Looks like a nice approach, I'd like to make time to check this out in more detail. It does makes me think a lambda function for generating combinations, like the one in Python, would make for a useful utility function and worth looking at as a future challenge.
On the subject of more obscure functions, I have seen specialised financial functions such as DOLLARFR and even ODDFYIELD employed in the context of treasury yield computations. Bitwise functions are also fairly niche in my experience but become more useful in lambda development where one is looking to optimise algorithms for example replacing array comparisons with bitwise operations. One application I have found myself reusing on multiple occasions is in generating (pseudo-)random data but removing volatility e.g. for benchmarking or speeding up recalc
ooooh pseudo-random number. I totally thing Excel should add [seed] to RAND() and all of its 'sister' functions. That said I created my own pseudo-rand number generator (very primitive but fine for most non-scientific work) and will have to compare it to the option you give here. I have seen many good options for pseudo-rnd numbers here on the forum when people ask to create 'random' lists for work schedules or baseball call sheets or tournement lists, etc... Having a SEED value based on the date, or week, or whatever will give them that random feel but not change every time they re-open the sheet. thx for sharing.
Maybe we need an archive folder to store the definitive recommendations and methods thought to be of long-term value? I think a pseudo-random number generator would qualify.
m_tarlerYes, having a repeatable random function can come in handy, I've quite often divided by 2^32 to get a non-volatile RAND alternative
PeterBartholomew1 Indeed, another possible candidate might be a 'combinations' function,
=combinations({"a";"b";"c";"d"},2)
which would return {"a","b";"a","c";"b","c";"a","d";"b","d";"c","d"}. I had been thinking about this off and on for a while but like SergeiBaklan couldn't see any reasonable route. I revisited again in light of the bisection method you provided recently based on a recursion over an array version of,
COMBIN(n,k) = COMBIN(n-1,k-1) + COMBIN(n-1,k)
For example with the following lambda definition =combinations(sequence(20),10) returns COMBIN(20,10)*10=1847560 elements.
combinations
=LAMBDA(arr, k,
IF(
k <= ROWS(arr),
IF(
k = 1,
arr,
DROP(
VSTACK(
combinations(DROP(arr, -1), k),
EXPAND(combinations(DROP(arr, -1), k - 1), , k, @TAKE(arr, -1))
),
k = ROWS(arr)
)
)
)
)
That was quite an achievement! I am not sure that my mind is correctly wired for recursion. I did get there but it was far from easy.
I picked out one step in your recursive calculation and, at least, it explained why the manner in which the combinations are ordered is a little unusual.
The other thing I noticed was that various nested calculations are performed more than once. I suppose that goes for the recursive formula for the count of combinations as well
Something else that had me puzzled for a moment was the type conversion
@TAKE(arr, -1)
used to pad the expanded array.
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If you’re looking for technical support, please visit Microsoft Support Community.","avatar":null,"profileSettings":{"__typename":"ProfileSettings","language":null},"parent":{"__ref":"Category:category:microsoft365"},"ancestors":{"__typename":"CoreNodeConnection","edges":[{"__typename":"CoreNodeEdge","node":{"__ref":"Community:community:gxcuf89792"}},{"__typename":"CoreNodeEdge","node":{"__ref":"Category:category:communities"}},{"__typename":"CoreNodeEdge","node":{"__ref":"Category:category:products-services"}},{"__typename":"CoreNodeEdge","node":{"__ref":"Category:category:microsoft365"}}]},"userContext":{"__typename":"NodeUserContext","canAddAttachments":false,"canUpdateNode":false,"canPostMessages":false,"isSubscribed":false},"boardPolicies":{"__typename":"BoardPolicies","canPublishArticleOnCreate":{"__typename":"PolicyResult","failureReason":{"__typename":"FailureReason","message":"error.lithium.policies.forums.policy_can_publish_on_create_workflow_action.accessDenied","key":"error.lithium.policies.forums.policy_can_publish_on_create_workflow_action.accessDenied","args":[]}}},"shortTitle":"Excel","repliesProperties":{"__typename":"RepliesProperties","sortOrder":"REVERSE_PUBLISH_TIME","repliesFormat":"threaded"},"tagProperties":{"__typename":"TagNodeProperties","tagsEnabled":{"__typename":"PolicyResult","failureReason":null}},"requireTags":true,"tagType":"PRESET_ONLY","eventPath":"category:microsoft365/category:products-services/category:communities/community:gxcuf89792board:ExcelGeneral/"},"Rank:rank:32":{"__typename":"Rank","id":"rank:32","position":13,"name":"Silver Contributor","color":"333333","icon":null,"rankStyle":"TEXT"},"User:user:214174":{"__typename":"User","id":"user:214174","uid":214174,"login":"PeterBartholomew1","deleted":false,"avatar":{"__typename":"UserAvatar","url":"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/dS0yMTQxNzQtMzY2MzA1aUI3OUMwMDM2QUQ1QkJEOUM"},"rank":{"__ref":"Rank:rank:32"},"email":"","messagesCount":3639,"biography":null,"topicsCount":29,"kudosReceivedCount":1143,"kudosGivenCount":410,"kudosWeight":1,"registrationData":{"__typename":"RegistrationData","status":null,"registrationTime":"2018-09-26T09:46:14.179-07:00","confirmEmailStatus":null},"followersCount":null,"solutionsCount":197,"entityType":"USER","eventPath":"community:gxcuf89792/user:214174"},"ForumTopicMessage:message:4047559":{"__typename":"ForumTopicMessage","uid":4047559,"subject":"Working with Binary numbers BASE(..., 2, 7) and bit operations BITXOR, BITAND","id":"message:4047559","revisionNum":2,"repliesCount":24,"author":{"__ref":"User:user:214174"},"depth":0,"hasGivenKudo":false,"board":{"__ref":"Forum:board:ExcelGeneral"},"conversation":{"__ref":"Conversation:conversation:4047559"},"readOnly":false,"editFrozen":false,"moderationData":{"__ref":"ModerationData:moderation_data:4047559"},"body@stripHtml({\"truncateLength\":200})":" I recently came across an Excel challenge from Chinmay Amte on LinkedIn and I plan to post my solution here because it contains some 'off the beaten track' techniques that might be of some interest. ...","body@stringLength":"4952","rawBody":"
I recently came across an Excel challenge from Chinmay Amte on LinkedIn and I plan to post my solution here because it contains some 'off the beaten track' techniques that might be of some interest.
Post | Feed | LinkedIn \"Yesterday, Indian state of Bihar was in news due to change in structure of coalition government. Assume you are an Analyst at Governor's office. He has asked you to evaluate ALL possible combinations of coalition between the political parties to form a government.\"
Please note, minimum vote of 122 is required to form government. The correct answer for the number of possible combinations is 61 (done manually)!
There are 127 combinations so I started with a decimal index but, within the Lambda functions SeatCountλ and ListNamesλ, this is first turned into a binary number (with 1 indicating that the corresponding party is part of the coalition.
MID is then used to split the binary number (a string) into separate cells to that matrix multiplication can be used to evaluate the total seats for each coalition.
I observed that the coalitions identified include cases that have more parties involved than are strictly necessary to form a government. Therefore I set out to discount any coalition that included a viable governing coalition as a subset. That is where the bit operations came into the equation.
Subsetλ\n\"Determines whether the first parameter is a subset of the second \nwhere set membership is determined by digits of a binary number\"\n= LAMBDA(set₁, \n LET(\n diff, BITXOR(set₁, set₂), \n IF(set₁ <> set₂, BITAND(set₁, diff) = 0)\n )\n )
Despite the fact that the concept involves binary numbers, the arguments to BITXOR and BITAND are actually their decimal equivalents. Exclusive BITXOR returns 0 where the corresponding bits match and 1 for mismatches. The bitwise BITAND examines one of the sets to see whether there is any non-zeros where the sets differ. That is does that coalition include any parties that are not in the second, in which case both combinations must be retained.
It is not often I get to use BITXORetc. Do others have examples of their use?
Just as a final point which may cause reaction from Excel users that hate 'black boxes', my final worksheet formula was no more than
= PotentialCoalitionsλ(party,seats,122)
which gave rise to
I am happy to accept any feedback. It is, after all, well away from looking like a traditional spreadsheet.
","kudosSumWeight":3,"postTime":"2024-02-02T15:50:07.479-08:00","images":{"__typename":"AssociatedImageConnection","edges":[{"__typename":"AssociatedImageEdge","cursor":"MjUuMXwyLjF8b3wyNXxfTlZffDE","node":{"__ref":"AssociatedImage:{\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2MmlFRTc0QkE4QkM4N0YyOTg2?revision=2\"}"}},{"__typename":"AssociatedImageEdge","cursor":"MjUuMXwyLjF8b3wyNXxfTlZffDI","node":{"__ref":"AssociatedImage:{\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2OWk4RUM3QzExMTdFMzU4N0Q1?revision=2\"}"}}],"totalCount":2,"pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null}},"attachments":{"__typename":"AttachmentConnection","pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null},"edges":[{"__typename":"AttachmentEdge","cursor":"MjUuMXwyLjF8b3w1fF9OVl98MQ","node":{"__ref":"Attachment:{\"id\":\"attachment:message4047559AttachmentNumber2\",\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/attachments/gxcuf89792/ExcelGeneral/219504/2/Possible%20Coalitions.xlsx\"}"}}]},"tags":{"__typename":"TagConnection","pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null},"edges":[{"__typename":"TagEdge","cursor":"MjUuMXwyLjF8b3wxMHxfTlZffDE","node":{"__typename":"Tag","id":"tag:excel","text":"excel","time":"2016-06-21T14:08:51.129-07:00","lastActivityTime":null,"messagesCount":null,"followersCount":null}},{"__typename":"TagEdge","cursor":"MjUuMXwyLjF8b3wxMHxfTlZffDI","node":{"__typename":"Tag","id":"tag:Formulas and Functions","text":"Formulas and Functions","time":"2016-06-24T14:00:56.974-07:00","lastActivityTime":null,"messagesCount":null,"followersCount":null}},{"__typename":"TagEdge","cursor":"MjUuMXwyLjF8b3wxMHxfTlZffDM","node":{"__typename":"Tag","id":"tag:office 365","text":"office 365","time":"2016-06-27T01:27:02.990-07:00","lastActivityTime":null,"messagesCount":null,"followersCount":null}}]},"timeToRead":2,"currentRevision":{"__ref":"Revision:revision:4047559_2"},"latestVersion":null,"metrics":{"__typename":"MessageMetrics","views":6430},"visibilityScope":"PUBLIC","canonicalUrl":null,"seoTitle":null,"seoDescription":null,"isEscalated":null,"placeholder":false,"originalMessageForPlaceholder":null,"messagePolicies":{"__typename":"MessagePolicies","canModerateSpamMessage":{"__typename":"PolicyResult","failureReason":{"__typename":"FailureReason","message":"error.lithium.policies.feature.moderation_spam.action.moderate_entity.allowed.accessDenied","key":"error.lithium.policies.feature.moderation_spam.action.moderate_entity.allowed.accessDenied","args":[]}}},"archivalData":null,"customFields":[],"body":"
I recently came across an Excel challenge from Chinmay Amte on LinkedIn and I plan to post my solution here because it contains some 'off the beaten track' techniques that might be of some interest.
Post | Feed | LinkedIn \"Yesterday, Indian state of Bihar was in news due to change in structure of coalition government. Assume you are an Analyst at Governor's office. He has asked you to evaluate ALL possible combinations of coalition between the political parties to form a government.\"
Please note, minimum vote of 122 is required to form government. The correct answer for the number of possible combinations is 61 (done manually)!
There are 127 combinations so I started with a decimal index but, within the Lambda functions SeatCountλ and ListNamesλ, this is first turned into a binary number (with 1 indicating that the corresponding party is part of the coalition.
MID is then used to split the binary number (a string) into separate cells to that matrix multiplication can be used to evaluate the total seats for each coalition.
I observed that the coalitions identified include cases that have more parties involved than are strictly necessary to form a government. Therefore I set out to discount any coalition that included a viable governing coalition as a subset. That is where the bit operations came into the equation.
Subsetλ\n\"Determines whether the first parameter is a subset of the second \nwhere set membership is determined by digits of a binary number\"\n= LAMBDA(set₁, \n LET(\n diff, BITXOR(set₁, set₂), \n IF(set₁ <> set₂, BITAND(set₁, diff) = 0)\n )\n )
Despite the fact that the concept involves binary numbers, the arguments to BITXOR and BITAND are actually their decimal equivalents. Exclusive BITXOR returns 0 where the corresponding bits match and 1 for mismatches. The bitwise BITAND examines one of the sets to see whether there is any non-zeros where the sets differ. That is does that coalition include any parties that are not in the second, in which case both combinations must be retained.
It is not often I get to use BITXORetc. Do others have examples of their use?
Just as a final point which may cause reaction from Excel users that hate 'black boxes', my final worksheet formula was no more than
= PotentialCoalitionsλ(party,seats,122)
which gave rise to
I am happy to accept any feedback. It is, after all, well away from looking like a traditional spreadsheet.
"},"Conversation:conversation:4047559":{"__typename":"Conversation","id":"conversation:4047559","solved":false,"topic":{"__ref":"ForumTopicMessage:message:4047559"},"lastPostingActivityTime":"2024-06-17T12:20:58.998-07:00","lastPostTime":"2024-06-17T12:20:58.998-07:00","unreadReplyCount":24,"isSubscribed":false},"ModerationData:moderation_data:4047559":{"__typename":"ModerationData","id":"moderation_data:4047559","status":"APPROVED","rejectReason":null,"isReportedAbuse":false,"rejectUser":null,"rejectTime":null,"rejectActorType":null},"AssociatedImage:{\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2MmlFRTc0QkE4QkM4N0YyOTg2?revision=2\"}":{"__typename":"AssociatedImage","url":"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2MmlFRTc0QkE4QkM4N0YyOTg2?revision=2","title":"image.png","associationType":"BODY","width":214,"height":316,"altText":null},"AssociatedImage:{\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2OWk4RUM3QzExMTdFMzU4N0Q1?revision=2\"}":{"__typename":"AssociatedImage","url":"https://techcommunity.microsoft.com/t5/s/gxcuf89792/images/bS00MDQ3NTU5LTU0ODU2OWk4RUM3QzExMTdFMzU4N0Q1?revision=2","title":"image.png","associationType":"BODY","width":401,"height":316,"altText":null},"Attachment:{\"id\":\"attachment:message4047559AttachmentNumber2\",\"url\":\"https://techcommunity.microsoft.com/t5/s/gxcuf89792/attachments/gxcuf89792/ExcelGeneral/219504/2/Possible%20Coalitions.xlsx\"}":{"__typename":"Attachment","id":"attachment:message4047559AttachmentNumber2","filename":"Possible Coalitions.xlsx","filesize":23380,"contentType":"application/vnd.openxmlformats-officedocument.spreadsheetml.sheet","url":"https://techcommunity.microsoft.com/t5/s/gxcuf89792/attachments/gxcuf89792/ExcelGeneral/219504/2/Possible%20Coalitions.xlsx"},"Revision:revision:4047559_2":{"__typename":"Revision","id":"revision:4047559_2","lastEditTime":"2024-02-03T13:34:34.529-08:00"},"ForumReplyMessage:message:4068047":{"__typename":"ForumReplyMessage","id":"message:4068047","conversation":{"__ref":"Conversation:conversation:4047559"},"author":{"__ref":"User:user:2171967"},"revisionNum":1,"uid":4068047,"depth":3,"hasGivenKudo":false,"subscribed":false,"board":{"__ref":"Forum:board:ExcelGeneral"},"subject":"Re: Working with Binary numbers BASE(..., 2, 7) and bit operations BITXOR, BITAND","readOnly":false,"editFrozen":false,"moderationData":{"__ref":"ModerationData:moderation_data:4068047"},"parent":{"__ref":"ForumReplyMessage:message:4048229"},"body":"ooooh pseudo-random number. I totally thing Excel should add [seed] to RAND() and all of its 'sister' functions. That said I created my own pseudo-rand number generator (very primitive but fine for most non-scientific work) and will have to compare it to the option you give here. I have seen many good options for pseudo-rnd numbers here on the forum when people ask to create 'random' lists for work schedules or baseball call sheets or tournement lists, etc... Having a SEED value based on the date, or week, or whatever will give them that random feel but not change every time they re-open the sheet. thx for sharing.","body@stringLength":"627","rawBody":"ooooh pseudo-random number. I totally thing Excel should add [seed] to RAND() and all of its 'sister' functions. That said I created my own pseudo-rand number generator (very primitive but fine for most non-scientific work) and will have to compare it to the option you give here. I have seen many good options for pseudo-rnd numbers here on the forum when people ask to create 'random' lists for work schedules or baseball call sheets or tournement lists, etc... Having a SEED value based on the date, or week, or whatever will give them that random feel but not change every time they re-open the sheet. thx for sharing.","kudosSumWeight":0,"repliesCount":10,"postTime":"2024-02-26T19:49:31.655-08:00","images":{"__typename":"AssociatedImageConnection","edges":[],"totalCount":0,"pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null}},"attachments":{"__typename":"AttachmentConnection","pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null},"edges":[]},"timeToRead":1,"currentRevision":{"__ref":"Revision:revision:4068047_1"},"latestVersion":null,"metrics":{"__typename":"MessageMetrics","views":5143},"visibilityScope":"PUBLIC","isEscalated":null,"placeholder":false,"originalMessageForPlaceholder":null,"messagePolicies":{"__typename":"MessagePolicies","canModerateSpamMessage":{"__typename":"PolicyResult","failureReason":{"__typename":"FailureReason","message":"error.lithium.policies.feature.moderation_spam.action.moderate_entity.allowed.accessDenied","key":"error.lithium.policies.feature.moderation_spam.action.moderate_entity.allowed.accessDenied","args":[]}}},"customFields":[],"replies":{"__typename":"MessageConnection","edges":[{"__typename":"MessageEdge","cursor":"MjUuMXwyLjF8aXwxMHwxMzI6MHxpbnQsNDA3NDE2MSw0MDc0MTYx","node":{"__ref":"ForumReplyMessage:message:4074161"}}],"pageInfo":{"__typename":"PageInfo","hasNextPage":false,"endCursor":null,"hasPreviousPage":false,"startCursor":null}}},"CachedAsset:theme:customTheme1-1745486991417":{"__typename":"CachedAsset","id":"theme:customTheme1-1745486991417","value":{"id":"customTheme1","animation":{"fast":"150ms","normal":"250ms","slow":"500ms","slowest":"750ms","function":"cubic-bezier(0.07, 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Confession.
I have changed the code that checks whether a given proposed coalition contains a smaller viable coalition. Originally I worked with arrays of the form {1,1,0,0,1,0,1} to check the subset condition and that required nested MAPs which were not pretty. Since BITXOR and BITAND work with the decimal representation (i.e. scalars rather than arrays) that part could be simplified.
HasSubsetλ\n\"Tests whether each of an array of sets is a superset of the others\"\n= LET(\n q, TRANSPOSE(p),\n s, SIGN(IF(p <> q, BITAND(BITXOR(p, q), q) = 0)),\n BYROW(s, LAMBDA(x, SUM(x))) > 0\n )
Sadly, settling for the first thing that works has its shortcomings.
","body@stringLength":"791","rawBody":"
Confession.
I have changed the code that checks whether a given proposed coalition contains a smaller viable coalition. Originally I worked with arrays of the form {1,1,0,0,1,0,1} to check the subset condition and that required nested MAPs which were not pretty. Since BITXOR and BITAND work with the decimal representation (i.e. scalars rather than arrays) that part could be simplified.
HasSubsetλ\n\"Tests whether each of an array of sets is a superset of the others\"\n= LET(\n q, TRANSPOSE(p),\n s, SIGN(IF(p <> q, BITAND(BITXOR(p, q), q) = 0)),\n BYROW(s, LAMBDA(x, SUM(x))) > 0\n )
Sadly, settling for the first thing that works has its shortcomings.
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Looks like a nice approach, I'd like to make time to check this out in more detail. It does makes me think a lambda function for generating combinations, like the one in Python, would make for a useful utility function and worth looking at as a future challenge.
On the subject of more obscure functions, I have seen specialised financial functions such as DOLLARFR and even ODDFYIELD employed in the context of treasury yield computations. Bitwise functions are also fairly niche in my experience but become more useful in lambda development where one is looking to optimise algorithms for example replacing array comparisons with bitwise operations. One application I have found myself reusing on multiple occasions is in generating (pseudo-)random data but removing volatility e.g. for benchmarking or speeding up recalc
Looks like a nice approach, I'd like to make time to check this out in more detail. It does makes me think a lambda function for generating combinations, like the one in Python, would make for a useful utility function and worth looking at as a future challenge.
On the subject of more obscure functions, I have seen specialised financial functions such as DOLLARFR and even ODDFYIELD employed in the context of treasury yield computations. Bitwise functions are also fairly niche in my experience but become more useful in lambda development where one is looking to optimise algorithms for example replacing array comparisons with bitwise operations. One application I have found myself reusing on multiple occasions is in generating (pseudo-)random data but removing volatility e.g. for benchmarking or speeding up recalc
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Maybe we need an archive folder to store the definitive recommendations and methods thought to be of long-term value? I think a pseudo-random number generator would qualify.
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m_tarlerYes, having a repeatable random function can come in handy, I've quite often divided by 2^32 to get a non-volatile RAND alternative
PeterBartholomew1 Indeed, another possible candidate might be a 'combinations' function,
=combinations({\"a\";\"b\";\"c\";\"d\"},2)
which would return {\"a\",\"b\";\"a\",\"c\";\"b\",\"c\";\"a\",\"d\";\"b\",\"d\";\"c\",\"d\"}. I had been thinking about this off and on for a while but like SergeiBaklan couldn't see any reasonable route. I revisited again in light of the bisection method you provided recently based on a recursion over an array version of,
COMBIN(n,k) = COMBIN(n-1,k-1) + COMBIN(n-1,k)
For example with the following lambda definition =combinations(sequence(20),10) returns COMBIN(20,10)*10=1847560 elements.
combinations\n=LAMBDA(arr, k,\n IF(\n k <= ROWS(arr),\n IF(\n k = 1,\n arr,\n DROP(\n VSTACK(\n combinations(DROP(arr, -1), k),\n EXPAND(combinations(DROP(arr, -1), k - 1), , k, @TAKE(arr, -1))\n ),\n k = ROWS(arr)\n )\n )\n )\n)
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That was quite an achievement! I am not sure that my mind is correctly wired for recursion. I did get there but it was far from easy.
I picked out one step in your recursive calculation and, at least, it explained why the manner in which the combinations are ordered is a little unusual.
The other thing I noticed was that various nested calculations are performed more than once. I suppose that goes for the recursive formula for the count of combinations as well
Something else that had me puzzled for a moment was the type conversion
@TAKE(arr, -1)
used to pad the expanded array.
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