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The Sum Composition Problem.
TL;DR: A first algorithm is presented solving the exhaustive problem and then a second algorithm solving the existential problem of the "sum composition problem" between two lists of positive integers.
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Abstract: In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$ is the length of $B$ and the sum of each part $A_k$ is equal to the corresponding part of $B$. Then, we consider the following two problems: $i)$ the "exhaustive problem", consisting in the generation of all partitions of $A$ for which $B$ is sum composition of $A$, and $ii)$ the "existential problem", consisting in the verification of the existence of a partition of $A$ for which $B$ is sum composition of $A$. Starting from some general properties of the sum compositions, we present a first algorithm solving the exhaustive problem and then a second algorithm solving the existential problem. We also provide proofs of correctness and experimental analysis for assessing the quality of the proposed solutions along with a comparison with related works.
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Figures
![Figure 5: Experiments with function sumCompExist. (a) Execution times varying the length of An1,200 (3 ≤ n ≤ 32) and the length of B (2 ≤ `(B) ≤ 31); (b) number of enabling decompositions and execution times varying the length of An1,200 (3 ≤ n ≤ 32) and for `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 32 and varying the length of B (2 ≤ `(B) ≤ 31); execution times with `(A) = 32 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]).](/figures/figure5-1-3ke6mnqtggjc.png)
Figure 5: Experiments with function sumCompExist. (a) Execution times varying the length of An1,200 (3 ≤ n ≤ 32) and the length of B (2 ≤ `(B) ≤ 31); (b) number of enabling decompositions and execution times varying the length of An1,200 (3 ≤ n ≤ 32) and for `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 32 and varying the length of B (2 ≤ `(B) ≤ 31); execution times with `(A) = 32 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]). 
Figure 2: Trace of the execution of function sumCompAux 
Figure 3: Recursive calls of the sumComp Algorithm ![Figure 4: Experiments with function sumComp by applying rule R1. (a) Execution times varying the length of An1,200 and B (1 ≤ `(A), `(B) ≤ 22); (b) number of enabling decompositions and execution times varying the length of An1,200 (with 1 ≤ n ≤ 22) and fixing `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 23 and varying the length of B (1 ≤ `(B) ≤ 22); and, (d) execution times with `(A) = 23 and `(B) = 2 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]).](/figures/figure4-1-6ilsnpjcysep.png)
Figure 4: Experiments with function sumComp by applying rule R1. (a) Execution times varying the length of An1,200 and B (1 ≤ `(A), `(B) ≤ 22); (b) number of enabling decompositions and execution times varying the length of An1,200 (with 1 ≤ n ≤ 22) and fixing `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 23 and varying the length of B (1 ≤ `(B) ≤ 22); and, (d) execution times with `(A) = 23 and `(B) = 2 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]). 
Table 1: Number of AB-decompositions by varying the length of A and the range of values ![Figure 1: (a) Ferrers diagram ΦB of the integer partition B = [2, 2, 4, 7]. (b) Tiling of ΦB corresponding to the decomposition [[1, 1], [2], [2, 2], [1, 3, 3]] of the integer partition A = [1, 1, 1, 2, 2, 2, 3, 3].](/figures/figure1-1-5jzhscv5nw20.png)
Figure 1: (a) Ferrers diagram ΦB of the integer partition B = [2, 2, 4, 7]. (b) Tiling of ΦB corresponding to the decomposition [[1, 1], [2], [2, 2], [1, 3, 3]] of the integer partition A = [1, 1, 1, 2, 2, 2, 3, 3].
![Figure 5: Experiments with function sumCompExist. (a) Execution times varying the length of An1,200 (3 ≤ n ≤ 32) and the length of B (2 ≤ `(B) ≤ 31); (b) number of enabling decompositions and execution times varying the length of An1,200 (3 ≤ n ≤ 32) and for `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 32 and varying the length of B (2 ≤ `(B) ≤ 31); execution times with `(A) = 32 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]).](/figures/figure5-1-3ke6mnqtggjc.webp)
![Figure 4: Experiments with function sumComp by applying rule R1. (a) Execution times varying the length of An1,200 and B (1 ≤ `(A), `(B) ≤ 22); (b) number of enabling decompositions and execution times varying the length of An1,200 (with 1 ≤ n ≤ 22) and fixing `(B) = 4; (c) number of enabling decompositions and execution times for `(A) = 23 and varying the length of B (1 ≤ `(B) ≤ 22); and, (d) execution times with `(A) = 23 and `(B) = 2 varying the number of repetitions in A and the considered ranges ([1, 100], [1, 150], [1, 200]).](/figures/figure4-1-6ilsnpjcysep.webp)
![Figure 1: (a) Ferrers diagram ΦB of the integer partition B = [2, 2, 4, 7]. (b) Tiling of ΦB corresponding to the decomposition [[1, 1], [2], [2, 2], [1, 3, 3]] of the integer partition A = [1, 1, 1, 2, 2, 2, 3, 3].](/figures/figure1-1-5jzhscv5nw20.webp)
Citations
On the poset of partitions of an integer
TL;DR: A rotary pickles making device comprises a base body including spaced drive and guide rollers rotatably mounted in the body, and a pickling casing for containing pickling rice-bran paste with foodstuffs.
2
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