functional programming – Proving injectivity for an algorithm computing a function between sets of different types of partitions

I am attempting to solve the following problem:

Let $A$ be the set of partitions of $n$ with elements $(a_1, dots, a_s)$ such that $a_i > a_{i+1}+a_{i+2}$ for all $i < s,$ taking $a_{s+1} = 0.$ Define $g_n = F_{n+2}-1$ and let $B$ be the set of partitions of $n$ as $b_1 ge dots ge b_s$ such that $b_i in {g_n}$ for all $i,$ and if $b_1 = g_k$ for some $k,$ then $g_1, dots, g_k$ all appear as some $b_i.$ Prove $|A|=|B|.$

Attempt: Let $e_i$ be the vector with $1$ at position $i$ and $0$ elsewhere. If $b_1 = g_k,$ let $c=(c_k, dots, c_1)$ count how many times $g_i$ appears. We calculate $f: B to A$ as follows:

Let $c=(c_k,dots,c_1), a=(0,dots,0).$ While $c ne 0,$ let $d_1 > dots > d_k$ be the indices such that $c_{d_i} ne 0.$ Replace $c, a$ with $c-(e_{d_1}+dots+e_{d_k}), a+(g_{d_1} e_1 + dots + g_{d_k} e_k)$ respectively. After the while loop ends, let $f(b)=a.$

Let $sum a, sum b, sum c$ be the sum of the components of $a, b, c$ respectively. Since $sum c$ decreases after every loop, the algorithm terminates and $f(b)$ is well-defined. Since $c_k g_k + dots + c_1 g_1 + sum a$ does not change after every iteration, is $sum b$ at the start and $sum a$ at the end, we have $sum f(b) = sum b = n,$ so $f(b)$ is also a partition of $n.$ Now $a = (g_k, dots, g_1)$ after the first loop, which satisfies the condition $g_i > g_{i-1}+g_{i-2}$ since $g_i = F_{n+2}-1 = (F_{n+1}-1)+(F_n-1)+1 > g_{i-1}+g_{i-2}.$ Furthermore, after every iteration of the loop, the difference $a_i – (a_{i-1}+a_{i-2})$ changes by $0, g_{d_j}-g_{d_{j-1}} > 0,$ or $g_{d_j}-(g_{d_{j-1}}+g_{d_{j-2}}) > 0,$ so we have $a_i > a_{i-1} + a_{i-2}$ at the end and hence $f(b) in A.$ Thus, $f: B to A$ is well-defined.

In order to prove the injectivity of $f,$ it suffices to prove each loop iteration as a mapping $(c,a) to (c’,a’)$ is injective, which would imply the mapping $(c,0) to (0,a)$ that the while loop creates is injective. Indeed, if $f(b_1) = f(b_2) = a$ with $(c_1, 0), (c_2, 0)$ being sent to $(0, f(b_1)) = (0,a), (0, f(b_2)) = (0,a)$ respectively, then we have $(c_1, 0) = (c_2, 0) Rightarrow c_1 = c_2 Rightarrow b_1 = b_2.$

Suppose $d_1 > dots > d_i, f_1 > dots > f_j$ are the non-zero indices of $c_1, c_2$ respectively and $c_1 – (e_{d_1}+dots+e_{d_i}) = c_2 – (e_{f_1}+dots+e_{f_j}), a_1+g_{d_1}e_1 + dots+ g_{d_i} e_i = a_2 + g_{f_1} e_1 + dots + g_{f_j} e_j.$ If $x ge 2$ is an entry of $c_1,$ it decreases by $1,$ so the corresponding entry in $c_2$ after $c_2$ is modified is also $x-1,$ which means it must’ve been $(x-1)+1 = x$ before since $x-1>0.$ Thus, if the values of two positions of $c_1, c_2$ differ, one is $1$ and the other is $0.$ However, if $c_1 = (1,0), a_1 = (3,1), c_2 = (0,1), a_2 = (4,1),$ then $(a_1, c_1), (a_2, c_2)$ both get sent to $((5,1), (0,0)).$ I can rule out this specific example by arguing that one of the pairs is illegal and could not have come from any choice of initial $c,$ but I have no idea on how to do this in general.

What should I do next in order to show $f$ is injective? Furthermore, since the problem I’m trying to prove is correct, injectivity would imply $f$ is secretly a bijection. But I have no clue on how to even start on the surjectivity of $f,$ so I just constructed a similar algorithm for $g: A to B$ in the hopes of proving $g$ is injective too. If I can show $f$ is injective I will probably know how to show $g$ is.

Here is an example of $f, g$ in practice:

Let $n = 41, b = (12, 7, 7, 4, 4, 2, 2, 2, 1) Rightarrow c = (1, 2, 2, 3, 1).$

$$((1, 2, 2, 3, 1), (0,0,0,0,0)) to ((0, 1, 1, 2, 0), (12, 7, 4, 2, 1)) to ((0, 0, 0, 1, 0), (19,11,6,2,1)) to ((21,11,6,2,1),(0,0,0,0,0)),$$ so $f(b) = (21,11,6,2,1).$

Let $a = (21, 11, 6, 2, 1).$

$$((21,11,6,2,1),(0,0,0,0,0)) to ((9,4,2,0,0), (1,1,1,1,1)) to ((2,0,0,0,0),(1,2,2,2,1)) to ((0,0,0,0,0),(1,2,2,3,1)),$$ so $g(a) = (12, 7, 7, 4, 4, 2, 2, 2, 1).$

algorithm – Is the number of swap function calls and the number of swaps done while selection sort the same thing?

I know that for n elements, in selection sort:

Best cases: 1 swap done.
Worst cases: n-1 swaps done.
Average cases: (n-1)/2 swaps done.

So, if i were to say that the number of times the swap function has been called is same as the number of swaps done in 3 different cases, would I be correct?

nonlinear optimization – Gradient of the function in the BFGS Quasi-Newton Algorithm

I have a function $f=sumlimits_{k=1}^{K}|R_{k}^{dl}- R_{k}^{ul}|$ that I want to minimize using the BFGS Quasi-Newton algorithm.

If $R_{k}^{dl} = y_k times A times B times C$.

$y_k$ and $R_k^{ul}~ forall~ k in {1, 2, dots, K}$ are given.

Also, $A$ is given, $B$ is not given, and $C$ is a function of $B$ only.

If I want to calculate to the gradient of the function ($nabla f$). Should I calculate it with respect to all variables or with respect to $B$ only?

Which algorithm to use ? Master Theorem

I have two algorithms for the same problem. Given an input of size n, first algorithm solves the problem by dividing it into five subproblems of size n/2, recursively solving each subproblem, and then combining the solutions in linear time. Second algorithm solves the problem by dividing the problem into nine subproblems of size n/3, recursively solving each subproblem, and then combining the solutions in O(n2) time. Express the number of operations performed by each algorithm using a recurrence relation. Solve the recurrence relations using Master Theorem. Which algorithm would you prefer ?

data structures – Algorithm for splitting an array into k sub-arrays

We want implement a data structure that have the following methods: Init(A,k)- Gets an array A with n different values and initialize our data structure that so it will divide A into k equally sized sub-arrays (+-1), that each value in the i’th sub-array will be larger than any value in the (i-1)’th sub-array, and smaller than any value in the (i+1)’th sub-array. This method need to be applied in complexity of O(n log k).

Insert(x)- Gets a value x which isn’t in our data structure, and adds it. This method need to be applied in complexity of O(k logn).

I did the init method using Medians- ofMedians QuickSelect, by dividing the array into k’ sub arrays when k’ equals to the closest power of 2 for k, and then I adjusted my pointers to the dividers by using Select on the smaller arrays which added me only O(n).

With the Insert part I’m having some trouble and would appreciate any help, Thanks:)

public key – Is there any documentation for Ripple XRP signing algorithm?

I’m recently digging into the cryptocurrency Ripple (XRP). However, I found XRP leaks cryptographic explanation for how to sign a transaction.

I could get the cryptographic explanation of address encoding and the public&private key in xrpl.org. But for how to sign a transaction with private key in XRP, I’ve searched online for days but found no such reference. They are just telling me to call API to sign, and I feel hard to understand the API source code.

I want to sign an offline transaction manually, but I got stuck at what to sign. What should be signed with the private key and attached to the transaction? Is there any reference?

Is there a polynomial time algorithm for this decision problem?

Is there a factor in $M$ that is $>$ $1$, but $<$ $M$ that is NOT a factor of $N$?

False Result Example

$N$ = 8

$M$ = 16

1, 2, 4, 8, 16

There is no integer that is NOT a factor of $N$ that is $>$ $1$ but < $M$

True Result Example

$N$ = 2

$M$ = 26

1, 2, 13, 26

There is an integer $13$ which is NOT a factor of $N$ that is > 1 but < $M$

Question

I have found a pseudo-polynomial solution, but is there a polynomial solution for this problem in the length of input?