Alternative proof of the fact that heapify can be linear-time

As an exercise, I’m trying to prove by myself that constructing a binary heap from an array in-place can be $ O(N)$ . I’ve come up with an idea, but I’m not sure about its correctness.

Firstly, I define the following procedure (pseudocode):

process_node():     if the node has children:         select the child node with the maximum value         if the selected node's value is greater than the value of current node:             swap current node with the selected node 

Then I run this procedure on all non-leaf nodes going down level by level in the binary tree view of the array. So the root is processed first, then comes the first level etc. After doing this, a half of leaf nodes become smaller than all of their ancestors. Another run of the procedure on the same nodes settles the other half of the leaves. Thus, they have been put in place in the heap and can be removed from the processing.

The number of leaves is halved, and so is the number of non-leaf nodes. The procedure is run again on new non-leaves, and the process is repeated until all the elements of the array are processed.

The number of nodes on which the procedure is run starts with $ N/2$ and later is halved. We get a decreasing geometric series with the sum being $ O(N)$ .

Are there any mistakes in my reasoning?

Meterpreter reverse shell alternative

What can you do if you are behind NAT but can’t do port forwarding and don’t want to pay for a virtual server with internet-facing IP? I read the answer to this question which suggests some tunneling software (e.g. ionide, pwnat) but those have to be installed on the target machine too and you can’t use them with Meterpreter. That question is four years old so are there better alternatives now?

Subsites are redirecting to localhost after alternative access mapping in sharepoint production Environment

Subsites are redirecting to localhost after alternative access mapping in sharepoint production Environment.

    Because of this the public url is redirecting to the default port url.      Please help any one regarding this issue. 



Alternative “File Open” (not stack “DocumentsUi”) for WhatsApp?

DocumentsUI, the AOSP standard “file manager” is quite buggy here (I use Lineage 15.1), especially in the context of multi-user and work profile (meaning it does not show files for other users then the main user and is hence not usable).

For “simple” file browsing, I therefore substituted DocumentsUI app ( with which works much better.

My issue and the core of the question is, that the glorious “facebook” WhatsApp Manager when attempting to attach files, resorts to opening the intent specifically for and fails even to open even in the case that it is the only filemanager still installed. Hence I cannot attach files in other users in Whatsapp, due to the buggy DocumentsUI and WhatsApp not being willing/capable/well-enough-designed to make a general “FileOpen” intent, instead of referring directly to

I understand that WhatsApp is particularly made very little IT-able persons, who do not care about privacy very much, therefore it is clear that the WhatApp is not geared to allow/accomodate for educated and more able users which might desire to use not the standard

An answer to this question would be to provide an insight which would be able to integrate well with WhatsApp Messenger

Alternative Lie bracket on $\chi^{\infty}(M)$ where $M$ is a Riemannian manifold with a symplectic structure

Let $ (M,\omega, g)$ be a Riemannian symplectic manifold. The poisson bracket of two functions $ f,g$ is denoted by $ \{f,g\}$ . The Hamiltonian vector field associated to a function $ f$ is denoted by $ H_f$ .

We define a Lie algebra structure on $ \chi^{\infty}(M)$ as follows:

$ $ [X,Y]=H_{\{div(X),div(Y)\} }$ $

Question 1: Is the Lie algebra $ \chi^{\infty}(M)/I$ a simple Lie algebra where $ I$ is the ideal $ $ I=\{X\in \chi^{\infty}(M)\mid div(X)=0\}$ $

As the second step for this question we would like to change the definition of the lie algebra operation by replacing the Hamiltonian vector field with gradient vector field. But we are not sure that it satisfies the Jackobi identity. So we ask the following question:

Question 2: Under which compatibilty condition between $ \omega$ and $ g$ , the following operation is a Lie algebra bracket on $ \chi^{\infty}(M)$ :

$ $ [X,Y]=\nabla_{\{div(X),div(Y)\}}$ $ is there an example of this situation with satisfication of the Jackobi identity?