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What modification should I apply to the dynamic programming implementation of two partition problem to solve the following task:

You are given an array of positive integers as input, denote it C. The program should decide if it is possible to partition the array into two equal sum subsequences. You are allowed to remove some elements from the array, but not all, in order to make such partition feasible.

Example:

Suppose the input is 4 5 11 17 9. Two partition is possible if we remove 11 and 17. My question is what adjustments to my two partition implementation I should make to determine if two partition is possible (may or may not require to remove some elements) or output that two partition is impossible even if some elements are removed. The program should run in O(sum^2 * C) time.

Here is my two partition implementation in Python:

def two_partition(C):
 n = len(C)
 s = sum(C)

 if s % 2 != 0: return False

 T = [[False for _ in range(n + 1)] for _ in range(s//2 + 1)]
 for i in range(n + 1): T[0][i] = True

 for i in range(1, s//2 + 1):
 for j in range(1, n + 1):
 T[i][j] = T[i][j-1]
 if i >= C[j-1]:
 T[i][j] = T[i][j] or T[i-C[j-1]][j-1]

 return T[s // 2][n]

Create a 3 dimensional array indexed by sum of 1st partition, sum of 2nd partition and number of elements.
T[i][j][k] if only true if it's possible to have two disjoint subsets with sum i & j respectively within the first k elements.

To calculate it, you need to consider three possibilities for each element. Either it's present in first set, or second set, or it's removed entirely.
Doing this in a loop for each combination of sum possible generates the required array in O(sum ^ 2 * C).

To find the answer to your question, all you need to check is that there is some sum i such that T[i][i][n] is true. This implies that there are two distinct subsets both of which sum to i, as required by the question.

If you need to find the actual subsets, doing so is easy using a simple backtracking function. Just check which of the three possibilities are possible in the back_track functions and recurse.

Here's a sample implementation:

def back_track(T, C, s1, s2, i):
 if s1 == 0 and s2 == 0: return , 
 if T[s1][s2][i-1]:
 return back_track(T, C, s1, s2, i-1)
 elif s1 >= C[i-1] and T[s1 - C[i-1]][s2][i-1]:
 a, b = back_track(T, C, s1 - C[i-1], s2, i-1)
 return ([C[i-1]] + a, b)
 else:
 a, b = back_track(T, C, s1, s2 - C[i-1], i-1)
 return (a, [C[i-1]] + b)

def two_partition(C):
 n = len(C)
 s = sum(C)

 T = [[[False for _ in range(n + 1)] for _ in range(s//2 + 1)] for _ in range(s // 2 + 1)]
 for i in range(n + 1): T[0][0][i] = True

 for s1 in range(0, s//2 + 1):
 for s2 in range(0, s//2 + 1):
 for j in range(1, n + 1):
 T[s1][s2][j] = T[s1][s2][j-1]
 if s1 >= C[j-1]:
 T[s1][s2][j] = T[s1][s2][j] or T[s1-C[j-1]][s2][j-1]
 if s2 >= C[j-1]:
 T[s1][s2][j] = T[s1][s2][j] or T[s1][s2-C[j-1]][j-1]
 for i in range(1, s//2 + 1):
 if T[i][i][n]:
 return back_track(T, C, i, i, n)
 return False

print(two_partition([4, 5, 11, 9]))
print(two_partition([2, 3, 1]))
print(two_partition([2, 3, 7]))

To determine if it's possible, keep a set of unique differences between the two parts. For each element, iterate over the differences seen so far; subtract and add the element. We're looking for the difference 0.

4 5 11 17 9

0 (empty parts)

|0 ± 4| = 4

set now has 4 and empty-parts-0

|0 ± 5| = 5
|4 - 5| = 1
|4 + 5| = 9

set now has 4,5,1,9 and empty-parts-0

|0 ± 11| = 11
|4 - 11| = 7
|4 + 11| = 15
|5 - 11| = 6
|5 + 11| = 16
|1 - 11| = 10
|1 + 11| = 12
|9 - 11| = 2
|9 + 11| = 20

... (iteration with 17)

|0 ± 9| = 9
|4 - 9| = 5
|4 + 9| = 13
|5 - 9| = 4
|5 + 9| = 14
|1 - 9| = 8
|1 + 9| = 10
|9 - 9| = 0

Bingo!

Python code:

def f(C):
 diffs = set()

 for n in C:
 new_diffs = [n]

 for d in diffs:
 if d - n == 0:
 return True
 new_diffs.extend([abs(d - n), abs(d + n)])

 diffs = diffs.union(new_diffs)

 return False

Output:

> f([2, 3, 7, 2])

=> True

> f([2, 3, 7])

=> False

> f([7, 1000007, 1000000])

=> True

I quickly adapted code for searching of three equal-sums subsets to given problem.

Algorithm tries to put every item A[idx] in the first bag, or in the second bag (both are real bags) or in the third (fake) bag (ignored items). Initial values (available space) in the real bags are half of overall sum. This approach as-is has exponential complexity (decision tree with 3^N leaves)

But there is a lot of repeating distributions, so we can remember some state and ignore branches with no chance, so a kind of DP - memoization is used. Here mentioned state is set of available space in real bags when we use items from the last index to idx inclusively.

Possible size of state storage might reach N * sum/2 * sum/2

Working Delphi code (is not thoroughly tested, seems has a bug with ignored items output)

function Solve2(A: TArray<Integer>): string;
var
 Map: TDictionary<string, boolean>;
 Lists: array of TStringList;
 found: Boolean;
 s2: integer;

function CheckSubsetsWithItem(Subs: TArray<Word>; idx: Int16): boolean;
var
 key: string;
 i: Integer;
begin
 if (Subs[0] = Subs[1]) and (Subs[0] <> s2) then begin
 found:= True;
 Exit(True);
 end;

 if idx < 0 then
 Exit(False);

 //debug map contains current rests of sums in explicit representation

 key := Format('%d_%d_%d', [subs[0], subs[1], idx]);

 if Map.ContainsKey(key) then
 //memoisation
 Result := Map.Items[key]
 else begin
 Result := false;
 //try to put A[idx] into the first, second bag or ignore it
 for i := 0 to 2 do begin
 if Subs[i] >= A[idx] then begin
 Subs[i] := Subs[i] - A[idx];
 Result := CheckSubsetsWithItem(Subs, idx - 1);
 if Result then begin
 //retrieve subsets themselves at recursion unwindning
 if found then
 Lists[i].Add(A[idx].ToString);
 break;
 end
 else
 //reset sums before the next try
 Subs[i] := Subs[i] + A[idx];
 end;
 end;
 //remember result - memoization
 Map.add(key, Result);
 end;
end;


var
 n, sum: Integer;
 Subs: TArray<Word>;
begin
 n := Length(A);
 sum := SumInt(A);
 s2 := sum div 2;
 found := False;

 Map := TDictionary<string, boolean>.Create;
 SetLength(Lists, 3);
 Lists[0] := TStringList.Create;
 Lists[1] := TStringList.Create;
 Lists[2] := TStringList.Create;

 if CheckSubsetsWithItem([s2, s2, sum], n - 1) then begin
 Result := '[' + Lists[0].CommaText + '], ' +
 '[' + Lists[1].CommaText + '], ' +
 ' ignored: [' + Lists[2].CommaText + ']';
 end else
 Result := 'No luck :(';
end;



begin
 Memo1.Lines.Add(Solve2([1, 5, 4, 3, 2, 16,21,44, 19]));
 Memo1.Lines.Add(Solve2([1, 3, 9, 27, 81, 243, 729, 6561]));
end;

[16,21,19], [1,5,4,2,44], ignored: [3]

No luck :(