# Calculate minimum common differences? **URL:** \ **Category:** Questions\ **Tags:** algebra, calculus, equation-solving\ **Created:** [August 6, 2017, 8:48pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066 "2017-08-06T20:48:16Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![lfelipemassena1](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/lfelipemassena1/32/9167_2.png) [@lfelipemassena1](https://community.wolfram.com/u/lfelipemassena1)\ **Post date:** [August 6, 2017, 8:48pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/1 "2017-08-06T20:48:16Z")
![minimadiferençacomum0007.jpg](https://global.discourse-cdn.com/wolfram/original/2X/8/8755345abf8840b341cd7f48220fc398b7c42171.jpeg) Can someone program something todo this? Calculate the least or greatest most different numbers between a series of numbers? So far I came out with these lines, but nothing comes out… ``` Unprotect [Element]Unprotect [NotElement]Unprotect[NestWhile]Unprotect[a,a1,a2]Unprotect[Slot] f[a1_]:= a&;&["|"a 1"\[Rule]a 1],|<=>"a1 "\[Rule] a1 "]; ``` Block ``` Thread[f[{a},{a1,a2},List AllTrue]] e=Intersection [a,a1,a2] Last[%] Block[{$RecursionLimit=1000},f= a] RandomChoice[a,27] Thread[a=NestWhileList[a!= a1>a!=a2[Element[a,aa]& Element[a1,aa]& Element[aa,a],Element[a,aa]!=Element[a1,aa]!=Element[a2,aa]&],a,27]] a=Flatten[List[%]] Thread[a1=NestWhileList[a1!= a1>b!=a2[Element[a1,aa]& Element[a1,aa]& Element[aa,a1],Element[a1,aa]!=Element[a1,aa]!=Element[a1,aa]&],a1,27]] a1=Flatten[%] Block[{$RecursionLimit=1000},c1] c1={#&,a+b1-c1} b1={a,a1+a} Map[MatchQ[#,_Integer]&,b] Pick[c1,b1] Do[If[a1 a1 And Or >=a1,Print["a1 = ",a1],Throw[a]],{a1,d}] Catch[Do[If[a2!= a And Or #3&,Print["a1 = ",a1],Throw[a1]],{a,d}]] Directory[] Export["aa.txt","aa"] Import["aa.txt"] a=Flatten[a] b=Flatten[a1] c=Flatten[a2] d=Union[a,b,c] s=Subtract[a1,a2] Take[a1,"-s"] Solve[aa->aa!= aa!= #& ,aa] a={3,4,5,7,10,15,23,27} a1={4,6,7,8,9,10,12,15,25} a2={2,6,7,10,11,12,23,13,14,15,25} While a<=a1!=a2 Take[Last[a]] ``` ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [December 28, 2017, 6:36pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/2 "2017-12-28T18:36:14Z")
If your a2 is corrected (a 23 between 12 and 13 has been canceled) then ``` In[11]:= a0 = {3, 4, 5, 7, 10, 15, 23, 27}; a1 = {4, 6, 7, 8, 9, 10, 12, 15, 25}; a2 = {2, 6, 7, 10, 11, 12, 13, 14, 15, 25}; In[8]:= Plus @@ Most[a0] Out[8]= 67 In[9]:= Plus @@ Most[a1] Out[9]= 71 In[14]:= Plus @@ Most[a2] Out[14]= 90 In[16]:= 32/13 // N Out[16]= 2.46154 ``` A has another sum as hand-written and 32/13 is of course not bigger than 24 … then: in the last sequence `C`: the difference between 2 and 6 is 4, so 2 and 6 are more different than 7 and 10, why 7 and 10 have been selected? ---
**Author:** ![shujiec](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/shujiec/32/18914_2.png) [@shujiec](https://community.wolfram.com/u/shujiec)\ **Post date:** [December 30, 2017, 12:12am UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/3 "2017-12-30T00:12:10Z")
It’s an interesting algorithm problem, but I cannot understand your example ☹ Would you mind providing more sample cases? ---
**Author:** ![lfelipemassena1](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/lfelipemassena1/32/9167_2.png) [@lfelipemassena1](https://community.wolfram.com/u/lfelipemassena1)\ **Post date:** [December 31, 2017, 8:35pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/4 "2017-12-31T20:35:28Z")
The point i wanted to make was to catch all the numbers that were the smallest numbers of a serie but that were common to each serie presented in the three series above and the three smalest number that were different from each serie of numbers(minimum common difference ) and then make up a new seried of numbers with those characteristics.Forget my example because i started simulating numbers ( elements ) to try to catch the least common number different from each serie but the programing ( lines of program ) came out completely odd from what i supposed, i think you have to start different from me. ---
**Author:** ![hdolhaine](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/hdolhaine/32/35924_2.png) [@hdolhaine](https://community.wolfram.com/u/hdolhaine)\ **Post date:** [January 4, 2018, 12:08pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/5 "2018-01-04T12:08:04Z")
I am not quite sure, but perhaps you mean this: your numbers (hand written from above) ``` aA = {3, 4, 5, 7, 10, 15}; bB = {4, 6, 7, 8, 9, 10, 12, 15}; cC = {2, 6, 7, 10, 11, 12, 13, 14, 15}; ``` Find common elements ``` IS = Intersection[aA, bB, cC] {7, 10, 15} ``` Remove them from your lists ``` cp = Complement[#, IS] & /@ {aA, bB, cC} ``` {{3, 4, 5}, {4, 6, 8, 9, 12}, {2, 6, 11, 12, 13, 14}} Now again find elements in cp common to all members in cp comparing all subsets ``` pp = Table[Intersection[cp[[i]], cp[[j]]], {i, Length[cp] - 1}, {j, i + 1, Length[cp]}] // Flatten {4, 6, 12} ``` Remove them from the elements of cp and take the first elements of the results. These should be the smallest numbers different in each series ``` First /@ (Complement[#, pp] & /@ cp) {3, 8, 2} ``` ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [January 6, 2018, 12:45pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/6 "2018-01-06T12:45:33Z")
This can be made into a two-liner, generical counting for subsets ``` In[98]:= (* for the WC *) a0 = {3, 4, 5, 7, 10, 15, 23, 27}; a1 = {4, 6, 7, 8, 9, 10, 12, 15, 25}; a2 = {2, 6, 7, 10, 11, 12, 13, 14, 15, 25}; In[101]:= Clear[cp] cp = Complement[#, Intersection[a0, a1, a2]] & /@ {a0, a1, a2} Out[102]= {{3, 4, 5, 23, 27}, {4, 6, 8, 9, 12, 25}, {2, 6, 11, 12, 13, 14, 25}} In[103]:= First /@ (Complement[#, Flatten[Intersection @@@ (Part[cp, #] & /@ Subsets[Range[Length[cp]], {2, Length[cp] - 1}])]] & /@ cp) Out[103]= {3, 8, 2} ``` but from the (insinuated) point of view of the problem owner it might consist a problem, that not all subset intersections are non-empty ``` In[105]:= Intersection @@@ (Part[cp, #] & /@ Subsets[Range[Length[cp]], {2, Length[cp] - 1}]) Out[105]= {{4}, {}, {6, 12, 25}} ``` which has been _flattened_ away, ---
**Author:** ![hdolhaine](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/hdolhaine/32/35924_2.png) [@hdolhaine](https://community.wolfram.com/u/hdolhaine)\ **Post date:** [January 6, 2018, 4:35pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/7 "2018-01-06T16:35:38Z")
Hi Udo, great, albeit somewhat complex. I wanted to show the single steps. Moreover I think it suffices to look at all pairs of subsets of cp, and it is not neccesary to look at all subsets of cp with more elements than 2. This would save time and memory. So I think one could write as well ``` First /@ (Complement[#, Flatten[Intersection @@@ (Part[cp, #] & /@ Subsets[Range[Length[cp]], {2}])]] & /@ cp) ``` The empty sets could indeed be an issue indicating that there are elements which don’t occur in some of the original sets after removing the elements common to all sets. Perhaps you could include an error-variable ( perhaps Catch, Throw ?) in your 2-liner? ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [January 6, 2018, 8:15pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/8 "2018-01-06T20:15:25Z")
Well, if one considers test input a0, a1, a2 as _series_ and some substructure in a given series as _succession_ and defines a _ **succession** _ as a list of consecutive elements with constant difference between sucession members, the following arises: ``` (* given series *) a0 = {3, 4, 5, 7, 10, 15, 23, 27}; a1 = {4, 6, 7, 8, 9, 10, 12, 15, 25}; a2 = {2, 6, 7, 10, 11, 12, 13, 14, 15, 25}; (* helpers *) Clear[taker, successor] taker[{l_List, o_Integer}] := First[l] /; Length[l] == 1 && o == 1 taker[{l_List, o_Integer}] := Drop[First[l], 1] /; Length[l] == 1 && o != 1 taker[{l_List, o_Integer}] := Union @@ l /; Length[l] != 1 && o == 1 taker[{l_List, o_Integer}] := Union @@ Drop[l, 1] /; Length[l] != 1 && o != 1 successor[l_List?VectorQ] := taker /@ Transpose[{#, Range[Length[#]]}] &[Split[Partition[l, 2, 1], (#1.{1, -1} == #2.{1, -1}) &]] /; OrderedQ[l] ``` finding successions within the series ``` In[75]:= successor /@ {a0, a1, a2} Out[75]= {{{3, 4, 5}, {7}, {10}, {15}, {23}, {27}}, {{4, 6}, {7, 8, 9, 10}, {12}, {15}, {25}}, {{2, 6}, {7}, {10}, {11, 12, 13, 14, 15}, {25}}} ``` the smallest elements of successions ``` Map[First, #] & /@ (successor /@ {a0, a1, a2}) {{3, 7, 10, 15, 23, 27}, {4, 7, 12, 15, 25}, {2, 7, 10, 11, 25}} ``` the smallest elements of successions common to all series ``` In[99]:= Clear[smac] smac = Intersection[Union @@ (Map[First, #] & /@ (successor /@ {a0, a1, a2})), a0, a1, a2] Out[100]= {7, 10, 15} ``` get new series excluding the common smallest succession elements ``` In[102]:= {b0, b1, b2} = Complement[#, smac] & /@ {a0, a1, a2} Out[102]= {{3, 4, 5, 23, 27}, {4, 6, 8, 9, 12, 25}, {2, 6, 11, 12, 13, 14, 25}} ``` finding again successions in the new series b0, b1, and b2; Attention: 8 is **not** smallest in any succession ``` In[103]:= successor /@ {b0, b1, b2} Out[103]= {{{3, 4, 5}, {23}, {27}}, {{4, 6, 8}, {9}, {12}, {25}}, {{2, 6}, {11}, {12, 13, 14}, {25}}} ``` the smallest elements of these successions ``` In[104]:= Map[First, #] & /@ (successor /@ {b0, b1, b2}) Out[104]= {{3, 23, 27}, {4, 9, 12, 25}, {2, 11, 12, 25}} ``` getting smallest succession elements not in the non-home series; the X trick shouldn’t be necessary ``` In[152]:= cp = Complement @@@ (Thread[ X[List /@ (Map[First, #] & /@ (successor /@ {b0, b1, b2})), RotateLeft[Partition[{b0, b1, b2}, 2, 1, 1]]]] //. X -> Join) Out[152]= {{3, 23, 27}, {9}, {2, 11}} ``` take the smallest of them ``` In[153]:= First /@ cp Out[153]= {3, 9, 2} ``` Finally, discuss it - no list came empty - result is `{3,9,2}`, not `{3,8,2}` - not sure whether the multi-valued usage of the term _series_ in problem owner’s description has been demangled correctly - problem owner is free the decide that and to convert the whole thing into something useful - notebook attached for convenience - why the trick with the `X` ruling to `Join` after threading `X` was necessary? ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [January 7, 2018, 12:19pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/9 "2018-01-07T12:19:51Z")
> Moreover I think it suffices to look at all pairs of subsets of cp This means that for let’s say 4 non-empty sets A,B,C,D the sets A\_2 = A\cap B \vee A\cap C \vee A\cap D \vee B\cap C \vee B\cap D \vee C\cap D and A\_{23} = A\cap B \vee A\cap C \vee A\cap D \vee B\cap C \vee B\cap D \vee C\cap D \vee A\cap B\cap C \vee A\cap B\cap D \vee A\cap C\cap D \vee B\cap C\cap D are identical A\_2 \equiv A\_{23}, which is clear because of the \vee (logical OR) and the fact, that intersections of three sets are not more permissive as intersections of two sets. Actually I was driven by your other remark > Now again find elements in cp common to all members in cp comparing > **all** subsets to deliver a generic code sample which could deliver all subsets. It was restricted to the subsets with 2,3,...,n-1 members and should be restricted even more in the given context, as you say. ---
**Author:** ![hdolhaine](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/hdolhaine/32/35924_2.png) [@hdolhaine](https://community.wolfram.com/u/hdolhaine)\ **Post date:** [January 8, 2018, 12:18pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/10 "2018-01-08T12:18:59Z")
Very interesting. But what does the OR between two sets (here intersections) mean? OR is a logical function and I don’t know how it works concerning sets: ``` In[1]:= m = {a, b, c, d}; sm = Subsets[m, {2, 3}] Out[2]= {{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}} In[3]:= (Or @@ (Intersection @@@ sm)) // LogicalExpand During evaluation of In[3]:= Intersection::normal: Nonatomic expression expected at position 1 in a\[Intersection]b. >> During evaluation of In[3]:= Intersection::normal: Nonatomic expression expected at position 1 in a\[Intersection]c. >> During evaluation of In[3]:= Intersection::normal: Nonatomic expression expected at position 1 in a\[Intersection]d. >> During evaluation of In[3]:= General::stop: Further output of Intersection::normal will be suppressed during this calculation. >> Out[3]= a \[Intersection] b || a \[Intersection] c || a \[Intersection] d || b \[Intersection] c || b \[Intersection] d || c \[Intersection] d || a \[Intersection] b \[Intersection] c || a \[Intersection] b \[Intersection] d || a \[Intersection] c \[Intersection] d || b \[Intersection] c \[Intersection] d ``` And unfortunately this is not your A2 ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [January 14, 2018, 4:19pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/11 "2018-01-14T16:19:15Z")
> But what does the OR between two sets (here intersections) mean? What do you think does it mean? That A\_2 consists of elements from A \cap B (inclusively) or from A \cap C … but this must be correctly written as A\_2 = (A \cap B) \cup (A \cap C) \cup \dots which means with respect to A\_2 and A\_{23} that A\_2 = A\_{23} because A\cap B \cap C \subseteq (A \cap B) \cup (A \cap C) and A\cap B \cap C \subseteq (B \cap C) \cup (B \cap A) and A\cap B \cap C \subseteq (C \cap A) \cup (C \cap B) and so on and so on. ---
**Author:** ![hdolhaine](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/hdolhaine/32/35924_2.png) [@hdolhaine](https://community.wolfram.com/u/hdolhaine)\ **Post date:** [January 15, 2018, 6:22pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/12 "2018-01-15T18:22:11Z")
Hello Udo, I see what you mean: ``` Off[Intersection::"normal", Union::"normal"] m = {a, b, c, d, e, f}; sm2 = Subsets[m, {2}] sm23 = Subsets[m, {2, 3}] Intersection @@@ sm2 Intersection @@@ sm23 ``` and everything is clear now ``` Union @@ Intersection @@@ sm2 Union @@ Intersection @@@ sm23 ``` Btw, i tried that with labeld variables and found that Intersection and Union behave in this case somewhat strange… ``` Subsets[Table[a[i], {i, 6}], {2}] Intersection @@@ Subsets[Table[a[i], {i, 6}], {2}] Union @@ Subsets[Table[a[i], {i, 6}], {2}] Union @@ Table[a[i], {i, 6}] ``` ---
**Author:** ![chaiselongue](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/chaiselongue/32/25455_2.png) [@chaiselongue](https://community.wolfram.com/u/chaiselongue)\ **Post date:** [January 21, 2018, 7:46am UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/13 "2018-01-21T07:46:32Z")
[Intersection](http://reference.wolfram.com/language/ref/Intersection.html) has to be applied usually to a sequence of lists, but as mostly often > Intersection works with any head, not just List: So, if something isn’t an atom ``` In[8]:= AtomQ[a[1]] Out[8]= False ``` it has a head and this head remains unchanged, the content is intersected - as it is done with `List[]`: ``` In[8]:= AtomQ[a[1]] Out[8]= False In[9]:= a[1] \[Intersection] a[2] Out[9]= a[] In[10]:= {1} \[Intersection] {2} Out[10]= {} ``` If the heads do not fit, you get it to know ``` In[11]:= a[1, 2, 3, 4] \[Intersection] a[2, 3, 4, 5] \[Intersection] b[1, 2, a[1, 2, 3, 4]] During evaluation of In[11]:= Intersection::heads: Heads b and a at positions 3 and 1 are expected to be the same. >> Out[11]= a[1, 2, 3, 4] \[Intersection] a[2, 3, 4, 5] \[Intersection] b[1, 2, a[1, 2, 3, 4]] ``` it’s once again a reflection of the deep insight that heads are not as much different from each other as they appear in education for childs and students: ``` In[13]:= a[1, 2, 3, 4] \[Intersection] a[2, 3, 4, 5] \[Intersection] a[1, 2, b[1, 2, 3, 4]] Out[13]= a[2] ``` ---
**Author:** ![lfelipemassena1](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/lfelipemassena1/32/9167_2.png) [@lfelipemassena1](https://community.wolfram.com/u/lfelipemassena1)\ **Post date:** [March 1, 2019, 2:30pm UTC](https://community.wolfram.com/t/calculate-minimum-common-differences/11066/14 "2019-03-01T14:30:36Z")
I just would like to thank you all for coming out with a solution for the puzzle I presented , i think you are all top mathmaticians and programmers. Now i am using it to take the pixels that are least common and and least ifferent form 2 pictures. I think all you know how important it is to take out the sequences a it applies to many areas of imaging. Unfortunately i do not know how to apply it tomodifying image pixels so that i get an image from imagelevels or image data.