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algorithms – Translating the in-order index of a node in a complete, balanced binary tree into the level-order index

Consider the topmost part of a complete, balanced binary tree of all 64-bit numbers, exemplified here.
As highlighted by the lack of a 7*2^64/8 term it is not necessarily full, but it is always complete. The nodes are stored in numerical order (in-order) in a list of which I know the size, and I have the index (call it the in-order index) of a node in that sorted list.

I’m trying to get the index (call it level-order index) of that same node if the array was stored according to the breadth-first traversal order (level-order) of the tree instead. In the example, the tree root would have level-order index 0 and in-order index 3.

What I’ve found already is that the level-order index consists of two parts:

  • A base part which is the amount of nodes contained in the full tree above the level in which the node resides.
  • An offset part which is the amount of nodes to the left of this node in the level in which it is contained.

Summed together, they produce the level-order index.

I’ve been able to solve this for the full tree case, as then the level at which a node is contained is the number of times its in-order index + 1 is divisible by 2. However, for the general case in which the tree is not necessarily full, how would that base part be calculated from the in-order index?

Of course, it is possible by iterating the tree programmatically, but since the exact layout and contents of the entire tree are known simply from the amount of nodes that it contains, I was wondering whether a mathematical relation between these two indices holds. Does anyone have an idea?

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powershell – can any one provide complete script to compare csv file and SharePoint list column(People) and extract records not exist in SharePoint list

I have one excel file and a SharePoint list(Employee master). I want to extract the user in csv file who is not in the SharePoint list(Employee Master).

If the employee not exists in SharePoint list we treat him as user terminated for internal tracking.

Please provide the PowerShell solution or if any other way.

virtual machines – Azure VM Scale Set: When exactly is a state change considered ‘complete’?

I’m asking because this is relevant for the autorepair grace period.
https://docs.microsoft.com/en-us/azure/virtual-machine-scale-sets/virtual-machine-scale-sets-automatic-instance-repairs#grace-period

When an instance goes through a state change operation because of a PUT, PATCH or POST action performed on the scale set (for example reimage, redeploy, update, etc.), then any repair action on that instance is performed only after waiting for the grace period. Grace period is the amount of time to allow the instance to return to healthy state. The grace period starts after the state change has completed.

We use a stock image, and then use the custom scripts extension to configure the machine. These scripts take a long time, think ~30 minutes.
I’ve seen that when the custom scripts throw an error, that the VM creation is then marked as a failure.

What’s not clear to me, is whether the run time of these custom scripts is included in the ‘state change’ or not.

Has anyone tested this, is there documentation of this somewhere?

complexity theory – How can I randomly sample from the set of NP Complete problems?

I’d like to create some program that can keep spitting out verification algorithms.

My verification algorithms take two inputs: problem instance, and solution (both encoded in binary), and output True or False. Every verification algorithm my program returns needs to satisfy the following criteria:

  • It terminates in polynomial time (in terms of size(problem instance)+size(solution))
  • The problem of: “Given a problem instance input, find a solution input that results in the verification algorithm returning True” is NP-Complete.

It seems like “sampling uniformly” from this space has the same problem of sampling uniformly from the set of integers, so I’m okay with a biased sampling method. However, I’d like every NP-Complete problem to have a non-zero chance of being sampled. Because verification programs are countable, this should be doable.

One idea would be to have some initial set of NP-Complete problems, and consider things like (problem input=(problem A instance, problem B instance, …) solution input=(problem A solution, problem B solution, …))) and then AND the individual verifiers. You could even go more general and consider boolean functions on the individual problem’s verifiers. This fails because I’m not aware of any “basis” of NP-Complete problems that by composition can give all NP-Complete problems, so we’d be missing some problems that can’t be formed as compositions of our problems.

Is this studied somewhere? Or how would I do this? I’m having trouble finding the words for it.

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Will BTC script be Turing complete in future?

I want to know if BTC script will have loops in it in the future?

Dúvida com desafio de JavaScript, complete o cálculo [fechada]

Os meses de Junho e Julho são tradicionalmente conhecidos por todo Brasil por suas festas típicas, e para o grupo de amigos da Marlene não é diferente, toda segunda quinzena do mês de Julho ela e seus amigos se reunem para tradicional mandiocada! Na festiva sempre se reúnem na casa da Marlene, o Chico, o Beto, o Bernardo, a Marina e a Iara. E como não poderia ser diferente o prato principal dessa reunião é a mandioca. Cada um deles come de uma a dez porções de mandioca e eles sempre avisam a Marlene com antecedência a respeito de quantas porções irão comer nesse dia. O tamanho da porção de cada um é diferente, mas sempre são os mesmos. As porções são as seguintes (em gramas):

O Chico come 300

O Bento come 1500

O Bernardo come 600

A Marina 1000

A Iara come 150

Marlene por sua vez sempre come 225 gramas de mandioca. Cansada de todo ano ter que calcular quanta mandioca preparar ela te desafiou para escrever um programa que informe quanta mandioca deve ser preparada em gramas.

Entrada
A entrada consiste de 5 inteiros cada um representando as porções que os convidados de dona Chica vão consumir. O primeiro inteiro representa as porções do Chico, o segundo do Bento, o terceiro do Bernardo, o quarto da Marina e o quinto a da Iara.

Saída
A saída consiste de um único inteiro que representa quanta mandioca Marlene deve preparar em gramas.

let bento = 1500 * parseInt(gets());
let bernardo = 600 * parseInt(gets());
let marina = 1000 * parseInt(gets());
let iara = 150 * parseInt(gets());
let marlene = 225 * parseInt(gets());

// Digite aqui o calculo total
console.log(total);

np complete – Reduction from SUBSET-SUM to 0-1-INT-PROG

The 0-1-INT-PROG problem is given an integer $m times n$ matrix $A$ and an integer $m$-vector $b$, is there an integer $n$-vector $x$ with $A cdot x leq b$.

I am trying to prove that 0-1-INT-PROG is NP-Hard by reducing SUBSET-SUM to it.

My attempt:

Given an instance of SUBSET-SUM $<{a_1, a_2, cdots, a_n}, k>$ we construct an instance of 0-1-INT-PROG $<[a_1, a_2, cdots, a_n], [k]>$.

Claim: $<{a_1, a_2, cdots, a_n}, k> in $ SUBSET_SUM if and only if $<[a_1, a_2, cdots, a_n], [k]> in $ 0-1-INT-PROG.

proof of forward direction. When there exists a subset $S’ subseteq S$ whose sum is $k$, for each $a_i in S’$, let $x_i = 1$. If $x_i not in S’$, then $x_i = 0$. Then $x$ is an $n$-vector which achieves $[a_1, a_2, cdots, a_n] cdot x = k$. So $<[a_1, a_2, cdots, a_n], [k]> in $ 0-1-INT-PROG.

Question:

When I have tried to prove the other direction I run into a problem. Namely, just because there exists an $x$ vector with $[a_1, a_2, cdots, a_n] cdot x leq k$, does not imply that there exists a subset $S’ subseteq S$ whose sum is $k$, because of the less than or equals to.

So how can I approach the backwards direction?