#it allows for the most efficient sorting algorithms at O(0)

[mystery-incorporated-whore]

BRB gonna invent hypercube C

It's a time-agnostic programming language

You can run code last week

[welderofwords]

(In which there are several really stupid sorting alogirthms, handwavings of correctness proofs of same, and musings on general ways to construct algorithms of various "intelligence")

sorting algos

Let's talk about sorting algorithms.

A "sorting algorithm" is supposed to solve the following task: you have an array of n strictly ordered objects; you can perform constant-time access, comparison, swaps; the goal is to have the array sorted. Equivalently: a given array has n! possible permutations; in a single move we can compare any two elements, or apply any single swap-permutation, or stop; our goal is to reach the one permutation in which the array is sorted and stop there.

brutally simple

A particularly simple (and stupid) solution to the task is Bogosort: force the array into every single permutation in order, stop if it is sorted. That's O(n) to reorder the array into any given permutation1, O(n) to check if it's ordered this time, O(n!) permutations to go through, so O(n×n!) altogether. On the other hand, it's really easy to see that it succeeds: it can check directly whether the current point in the solution-space is the solution, it checks so for every single point, and so of course it will eventually stumble upon the solution itself and report it. (In general, brute force algorithms never rely on any ~internal structure~ of the problem, always obviously "work", but aren't any good at searching through huge solution spaces.)

do no harm

A slightly less stupid algorithm could at least "try to move in the right direction". For example: choose a random pair of elements, swap them if they're in the wrong order, repeat until the array is sorted. This is not quite but almost Spin-the-Bottle sort, which works in O(n² log n); much better.

How do we prove that this works?

Well, we can define some2 measure of "sortedness" of the array, demonstrate that every such swap decreases it, demonstrate that it is 0 (or otherwise has a minimal possible value) iff the array is sorted, and that'd be enough. That is: if every swap decreases the sortedness, if there are no infinitely decreasing sequences of sortednesses, and if 0-sortedness is the victory condition, then we can always keep doing swaps, and will eventually arrive at a sorted array.

A suitable measure of sortedness is the count of array inversions; the count of pairs (a_i, a_j) that are misordered: a_i > a_j but i<j.

count of inversions is a nonnegative integer; as long as it decreases, it will eventually hit 0

an inversion-free array is indeed sorted; every pair of elements in it is in located in the correct order

suppose we try to swap elements a and b, such that a > b and there are some elements between them, [..., a, x_i, ..., x_j, b, ...]. Let's count all changes in the inversion count.

if some x_k < b < a, then (x_k, a) was misordered and now isn't, and (x_k, b) wasn't misordered and now is; inversion count is unchanged.

if some x_k > a > b, then it's the other way around.

if some b < x_k < a, then both (x_k, a) and (x_k, b) were misordered, and now aren't; -2 inversions.

finally, (a, b) ceased to be misordered; -1 inversion.

so, swapping a and b definitely decreased the inversion count. QED.

there's a way

If we have an idea of a "right direction" to move in, we can try to be cleverer about it. Say, we could start from the beginning, search for an adjacent misordered pair [.., a, b, ..] and swap it into [.., b, a, ..]. Then, why not check for obvious improvements: maybe [.., c, b, ..] is now misordered. And so on, floating b as far left as obviously necessary, then continuing the search for misordered adjacents from a onwards. Oh look it's Gnome sort. (We're down to O(n²) sorts!) The proof of correctness is significantly easier3 here:

by construction, there are no misordered adjacent pairs to the left of the "current position"; every time a swap might introduce one, we check up on it and correct that

every step to the right requires only so many "propagating" steps back to the left

after we've stepped all the way to the right, there are no misordered pairs at all, and we're done.

step by step

There's another good principle we could use: pick a small subproblem, solve it entirely, never touch it again, repeat until done. A convenient small subproblem for sorting is "what element would be first in the sorted array?"; repeatedly doing that is Selection sort, also of O(n²). Correctness of that one is comparatively easy: again, we define some measure of "complexity" of the rest of the problem, we demonstrate that we can always pick a subproblem that leaves the whole problem less complicated unless we're done, we repeatedly pick a subproblem, we're done. In particular, a complexity measure for selection sort is "how many rightmost elements of the array are not yet guaranteed to be sorted?"; it decreases by 1 with every selection, until we're at 0, a sorted array.

the holy grail

Finally, we could split the problem into several strictly smaller instances of the same problem, and somehow combine them into a solution to the original. For sorting this means quicksort or mergesort; in general those are divide and conquer algorithms. Formal correctness proofs of these are somewhat finicky, relying on:

careful definition of "strictly smaller"

ability to combine smaller solutions into a larger solution

correctness in a number of "trivial" cases: at some point we need to stop recursing and solve the problem directly

what if we tried more power?

Sorting algorithms can't really do better than O(n log n). An argument goes as follows: at the start, you have absolutely no idea which one of n! orders is the right one. As you start comparing elements, every comparison allows you to ignore at most half of conceivable orders as incorrect; and to get n! down to 1 by repeated divisions, you need at least O(log(n!)) = O(n log n) comparisons, and that's that.

One big assumption the argument relies on is that all possible orders are a) possible, b) equiprobable. This might or might not hold in real life, and if you know something about the distribution of possible orders, you can exploit that to get better results than O(n log n)4.

Another one is that all constraints are as described: all comparisons are strict, there's no other way to extract ordering information than comparisons, there's no other way to reorder than swaps, etc.

you can be more efficient by choosing close-by permutations, but really, if you have intelligence to spare to optimize the order of permutations in bogosort, you can just as well do not-bogosort. ↩︎

well-founded ↩︎

To be fair, this is also a proof by finitely decreasing measure; it's just that the relevant measure is slightly awkward. ↩︎

at least in expectation. ↩︎

[perfectwayfx]

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#button-48566 .elButtonSubtle:hover background-color: #005707 !important;

#button-48566 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-48566 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-48566 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-48566 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-48227 .elButtonFlat:hover background-color: #005707 !important;

#button-48227 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-48227 .elButtonSubtle:hover background-color: #005707 !important;

#button-48227 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-48227 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-48227 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-48227 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-21337 .elButtonFlat:hover background-color: #005707 !important;

#button-21337 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-21337 .elButtonSubtle:hover background-color: #005707 !important;

#button-21337 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-21337 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-21337 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-21337 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-83014-117 .elButtonFlat:hover background-color: #005707 !important;

#button-83014-117 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-83014-117 .elButtonSubtle:hover background-color: #005707 !important;

#button-83014-117 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

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#button-83014-117 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-83014-117 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-60538 .elButtonFlat:hover background-color: #005707 !important;

#button-60538 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-60538 .elButtonSubtle:hover background-color: #005707 !important;

#button-60538 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-60538 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-60538 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-60538 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#headline-84158-189 .elHeadline b color: rgb(255, 255, 255);

#headline-63060-117 .elHeadline b color: rgb(255, 255, 255);

#headline-92681 .elHeadline b color: rgb(255, 255, 255);

#headline-67482 .elHeadline b color: rgb(255, 255, 255);

#button-19604 .elButtonFlat:hover background-color: #005707 !important;

#button-19604 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-19604 .elButtonSubtle:hover background-color: #005707 !important;

#button-19604 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-19604 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-19604 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-19604 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-93947 .elButtonFlat:hover background-color: #005707 !important;

#button-93947 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-93947 .elButtonSubtle:hover background-color: #005707 !important;

#button-93947 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-93947 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-93947 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-93947 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-18315 .elButtonFlat:hover background-color: #005707 !important;

#button-18315 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-18315 .elButtonSubtle:hover background-color: #005707 !important;

#button-18315 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-18315 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-18315 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-18315 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-21023 .elButtonFlat:hover background-color: #005707 !important;

#button-21023 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-21023 .elButtonSubtle:hover background-color: #005707 !important;

#button-21023 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-21023 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-21023 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-21023 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-43497 .elButtonFlat:hover background-color: #005707 !important;

#button-43497 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-43497 .elButtonSubtle:hover background-color: #005707 !important;

#button-43497 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-43497 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-43497 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-43497 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#tmp_button-39971 .elButtonFlat:hover background-color: #ccb600 !important;

#tmp_button-39971 .elButtonBottomBorder:hover background-color: #ccb600 !important;

#tmp_button-39971 .elButtonSubtle:hover background-color: #ccb600 !important;

#tmp_button-39971 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(255, 227, 0)), color-stop(1, #ccb600)); background-image: -o-linear-gradient(bottom, rgb(255, 227, 0) 0%, #ccb600 100%); background-image: -moz-linear-gradient(bottom, rgb(255, 227, 0) 0%, #ccb600 100%); background-image: -webkit-linear-gradient(bottom, rgb(255, 227, 0) 0%, #ccb600 100%); background-image: -ms-linear-gradient(bottom, rgb(255, 227, 0) 0%, #ccb600 100%); background-image: linear-gradient(to bottom, rgb(255, 227, 0) 0%, #ccb600 100%);

#tmp_button-39971 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(255, 227, 0)), color-stop(0, #ccb600)); background-image: -o-linear-gradient(bottom, rgb(255, 227, 0) 100%, #ccb600 0%); background-image: -moz-linear-gradient(bottom, rgb(255, 227, 0) 100%, #ccb600 0%); background-image: -webkit-linear-gradient(bottom, rgb(255, 227, 0) 100%, #ccb600 0%); background-image: -ms-linear-gradient(bottom, rgb(255, 227, 0) 100%, #ccb600 0%); background-image: linear-gradient(to bottom, rgb(255, 227, 0) 100%, #ccb600 0%);

#tmp_button-39971 .elButtonBorder border: 3px solid rgb(255, 227, 0) !important; color: rgb(255, 227, 0) !important;

#tmp_button-39971 .elButtonBorder:hover background-color:rgb(255, 227, 0) !important; color: #FFF !important;

#button-17705 .elButtonFlat:hover background-color: #005707 !important;

#button-17705 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-17705 .elButtonSubtle:hover background-color: #005707 !important;

#button-17705 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-17705 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-17705 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-17705 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-11496 .elButtonFlat:hover background-color: #005707 !important;

#button-11496 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-11496 .elButtonSubtle:hover background-color: #005707 !important;

#button-11496 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-11496 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-11496 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-11496 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#tmp_headline1-79944 .elHeadline b color: rgb(255, 255, 255);

#tmp_headline1-79944-131 .elHeadline b color: rgb(255, 255, 255);

#headline-61797 .elHeadline b color: rgb(255, 255, 255);

#headline-25082 .elHeadline b color: rgb(255, 255, 255);

#headline-72600 .elHeadline b color: rgb(255, 255, 255);

#headline-90115 .elHeadline b color: rgb(255, 227, 0);

#button-38290 .elButtonFlat:hover background-color: #005707 !important;

#button-38290 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-38290 .elButtonSubtle:hover background-color: #005707 !important;

#button-38290 .elButtonGradient background-image: -webkit-gradient( linear, left top, left bottom, color-stop(0, rgb(0, 138, 11)), color-stop(1, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 0%, #005707 100%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 0%, #005707 100%);

#button-38290 .elButtonGradient:hover background-image: -webkit-gradient( linear, left top, left bottom, color-stop(1, rgb(0, 138, 11)), color-stop(0, #005707)); background-image: -o-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -moz-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -webkit-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: -ms-linear-gradient(bottom, rgb(0, 138, 11) 100%, #005707 0%); background-image: linear-gradient(to bottom, rgb(0, 138, 11) 100%, #005707 0%);

#button-38290 .elButtonBorder border: 3px solid rgb(0, 138, 11) !important; color: rgb(0, 138, 11) !important;

#button-38290 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

#button-72749 .elButtonFlat:hover background-color: #005707 !important;

#button-72749 .elButtonBottomBorder:hover background-color: #005707 !important;

#button-72749 .elButtonSubtle:hover background-color: #005707 !important;

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#button-72749 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

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#button-18315-165 .elButtonBorder:hover background-color:rgb(0, 138, 11) !important; color: #FFF !important;

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[sirlennon]

The Three Most Important SEO Success Factors

There should not end up being any haste in selecting the SEO service providing company. Keyword research is still the critically-important element of SEO. Finally SEARCH ENGINE OPTIMIZATION Smart links allows you in order to set nofollow attribute and open up links in new window. Page titles and descriptions have an effect on what individuals see in lookup results, so it's necessary in order to check these out in any kind of SEO audit. Mobile user experience may enjoy a key role in the particular way sites are ranked, therefore not having a mobile-friendly edition could greatly hinder your SEARCH ENGINE OPTIMIZATION efforts. SEO is actually a cross-functional marketing activity that will gives visibility to content within the organic search channel. 15 Upon Page SEO Techniques To Position High In Search engine outcomes. There are several technical modifications that web developers and SEARCH ENGINE OPTIMIZATION companies can do to repair this, so do not overlook it. If your competitors have a fast mobile-friendly site plus you never, you will shed traffic and business to all of them. Improve the customer experience by offering the thorough assistance and provide your SEO ranking a increase. SEO, today, means what often has: creating a website that will can be properly understood simply by a search engine. Many growing SEO trends could infuse large difference to future SEO technique of businesses. Plus that's why search engine optimization (SEO) - the earning associated with traffic through search engines -- has been this kind associated with important topic for site proprietors for more than 20 many years. One particular of the biggest mistakes manufacturers make when it comes in order to their SEO is focusing upon metrics that misrepresent the whole success, such as keyword rating and keyword search volume. Some SEO software equipment give you the option in order to spy on your competition, in order to see where they are obtaining their links from, what key phrases they are competing for, plus so much more. I'm curious : sometimes with your chapter-based manuals, you might have them most on a single page such as this one, but sometimes they may separate pages by having an overview web page as a sort of introduction + table of contents, like as the Keyword Research regarding SEO Definitive Guide. Plus 2018, making your website cellular friendly is the biggest pattern in SEO. When we talk about SEARCH ENGINE OPTIMIZATION or Local SEO, most companies limit it to Google. A successful SEO technique in 2018 will gravitate in the direction of mobile-friendly websites, content material relevance, brand and relationship advancement. Based upon the previous point, SEOs may think about various landing webpages with proper keywords, meta labels and the overall website construction from the SEO viewpoint. RankBrain can force a much closer examination of user experience, and SEOs will need to monitor jump rate and average time upon page much more closely compared to they have previously. We all concentrated within the keyword-based aspect associated with all the SEO tools that will included the capabilities, because which where most business users may primarily focus. Hence, it's essential to keep track of every Google algorithm change to enhance rankings, organic website traffic plus finally boost SEARCH ENGINE OPTIMIZATION. Filter the keywords within line with the search quantity and try to see when you can find any content ideas through it. Let's imagine - you select a keyword Local SEO Trends”. It can be Local SEARCH ENGINE OPTIMIZATION Checklist, Local SEO tips plus tricks, Local SEO trends with regard to 2018 and so forth. A person can then either use this particular article as being a guest post or even publish it on your very own blog. "Search engine friendly" websites are web pages that have apparently had SEARCH ENGINE OPTIMIZATION techniques applied to them. Since Google emerged out with its Pigeon” upgrade, local SEO has gained substantial prominence forcing businesses and online marketers to pay attention to the local tangent. Lastly, a typical issue such as penalization simply by Google for spam links or even content duplication is a permanently issue for SEOs to prevent cautiously. The takeaway here is that will if you have LOTS associated with location pages serving ONE ONE business in a single area, then those are very possibly classed as some sort associated with doorway pages, and probably old-school SEO tips for these type associated with pages will see them categorised as lower-quality - or maybe : spammy pages. Subscribe to the Individual Grain blog now for your most recent content on SEO, PPC, compensated social, and the future associated with online marketing. 2018 (I believe) will be a large amount of catch up on current SEARCH ENGINE OPTIMIZATION themes, with the biggest trends” occurring around voice queries in order to search results. Off-page SEO really efficiently in promoting your company where social media, bookmarking websites, forums, blog directory, Q&A, content articles, videos, image and infographic posting, and document sharing play nicely. Apart from the particular ethical questions raised over littering the web with rehashed plus useless content, this black head wear SEO tactic makes it challenging for search engines to physique out which page is initial and relevant to the provided search query. Visual search is modifying the world of search plus a good SEO strategy to think about this 2018 is to really think about the way individuals consume visual content. Tracking your website and keyword rankings is a part of your SEO process that allows you to make content decisions to get the most away from SEO. I believe that will 2018 is going to be the particular year where voice search changes how users search and SEOs need to optimize. The purpose of off-page SEO would certainly be to get links in order to your information. There are usually many tools on the internet to assist with basic key word research (including the Google Key word Planner tool and there are even more useful third party SEO tools in order to help you do this). Another important point to add up to the particular checklist for local SEO within 2018 is UX. The consumer experience for Local SEO will certainly become even more important within 2018. In the older times of SEO, Google could not understand modifiers, meaning if We put in more keywords within like best” or new, ” I could outrank someone that failed to use those key phrases in the meta title. While the real difference, for the snippets, is not the method the information continues to end up being fetched by Google (as this particular works much like the great old SEO) but how the particular result is presented towards the particular user: these snippets answer the lot of questions, triggered simply by our voice.

Off-Page SEO refers in order to the techniques that are utilized to improve the ranking associated with website or blog. In this situation, in, the article ranks soon after the official Google guidelines (and it seems sensible that Search engines should be number 1 upon their own branded query) yet Smashing magazine is shown since a position 0” snippet associated with text within the query Google place up guidelines” in. Search Motor Land, a high-quality SEO blog page that is a pillar associated with the community is ranking right after Smashing (which happens to end up being more of a design blog page than an SEO one). SEARCH ENGINE OPTIMIZATION is the discipline that enables people and websites to gain access to the potential of the internet. Search motor optimization (SEO) is the procedure by which a website will be optimized for making sure this has the highest position position around the search engines results web page. Create new and fascinating blogs, have them optimized regarding SEO, link to other related pages, and share them upon social media marketing. Today, some website themes furthermore come with a built-in SEARCH ENGINE OPTIMIZATION tool kit which allows a person to definitely monitor the key phrases you use. SEO will be conducted around the understanding that websites rank because of how related a webpage is to analysis online query and how numerous links point out that web page. You might be which owns a growing and flourishing business, the webmaster of the dozen sites, the SEO expert in a Web agency or even a DIY SEO ninja fascinated in the mechanics of Lookup: information is meant for a person. In truth, this SEO method is regarded deceptive since it tricks lookup engines in order to obtain the desired ranks for focus on keywords. Black Head wear SEO's tend to look regarding loopholes and for you to game” the particular search engines in a method that causes massive results, plus fast results. In order in order to maintain the right track, all of us have come up with the particular new SEO Trends that will certainly be necessary to overcome your own competition in the year 2018. Subscribe in order to the Single Grain blog right now for the latest content upon SEO, PPC, paid social, plus the future of internet advertising. Traditional SEO techniques are usually still effective, but taking straight into account these trends could considerably increase your relevance on the particular search landscape. As digital assistants exceed smart phone devices, there is a excellent opportunity both for SEO plus content, taking advantage of the growing market that connects the particular brand with a user within a unique but still relevant plus useful way. SEO plus SEM both heavily rely upon keywords to operate a vehicle traffic to company websites and webpages. The first issue raised is that, when people expect SEO to become a magic potion for immediate on line success and profit, ” they have a tendency to be sorely disappointed and frustrated. ” Other issues discussed are the time it takes to focus on SEO, the technical aspects that can seem intimidating, the possibility of receiving a penalty for doing something inappropriate and more. Google Lookup Console is, slowly but certainly, getting better Google's getting the much more aggressive about producing rank tracking more difficult (some rank tracking folks I'm helpful with told me that Q4 2017 was particularly gut-punching), plus the SEO software field is usually way, way more densely loaded with competitors than ever just before. In this post, we all would be talking about On-Page SEO and some of the very effective ways of boosting your ratings on Search Engines. Some Webmasters will pay lots of money to a so-called professional SEARCH ENGINE OPTIMIZATION expert to have their Sites on top within the ratings. Meta keywords would end up being the least priority in standing factors in SEO. The SEO Trends associated with 2018 are turning to be able to impact approach on various other procedures related to Digital Marketing. These types of advanced SEO techniques might not really be the easiest to apply - they actually require even more effort than basic keyword analysis and link building - yet they are incredibly effective. Whenever it comes to WordPress SEARCH ENGINE OPTIMIZATION, it is important to keep in mind that Google is still the pc algorithm, and that is definitely why keywords still matter within 2018. ย You definitely should end up being writing for your visitors plus customers as Google recommends, yet write smart. In the event that you want to excel in SEO in 2018, you ought to start thinking of long-form content. Search Engine Marketing (SEO) is one of the particular hardest parts of running the website. The history of SEO times back to the 90s whenever the search engines like search engines emerged for the first period. One particular reason is because of device learning, however take is that will the major search engines utilizes machine learning to help all of them identify great content, identify articles that users like the many, identify one of the most authoritative content plus better understand user intent plus queries. ” All of that will keeps SEOs in business, quite much as usual. (Note that will this is figure was used at the time of composing - in case you click on on the link you'll probably find a different number, because data from this source is definitely collected in real time. ) This is a lot associated with activity for any given company to compete with and is definitely why SEO is such a good important (but tricky) section of electronic marketing to know. This will also help you spy on the competition's content marketing and advertising efforts, determine what queries your own website ranks on in numerous countries and what your best SEO pages are. Let's imagine a search motor optimization company wants to position for the term "Best SEARCH ENGINE OPTIMIZATION Company", the title of the particular webpage or post that is definitely being optimized should start with those keywords. White Hat (and successful) SEO is not about ‘gaming' Google, or other search motors for that matter. Mobile SEO is mobile research engine optimization or optimizing articles for a better search motor ranking. So how should businesses make use of SEO as a marketing technique? SEO efforts may involve the site's coding, presentation, and construction, as well as difficulties that will could prevent search engine indexing programs from fully spidering the site. I am a good expert SEO and in this particular article I will highlight several important SEO tips that you need to stick to in order to secure great search engine ranking positions. Mobile users are increasing therefore it you must aware with the mobile SEO tactics and also updated that Google making. 1- I am aware that do backlinks with robots or programs, that not great for SEO and Big Sibling Google do not love this i mention to Fiverr Gig”. Knowledge associated with SEO techniques can assist a company to promote its online existence effectively. Therefore, as a good online marketer, you should have got a clear regarding the current SEO trends. Along with SEO becoming more and even more important, search engines had in order to develop a way to utilize this with greater accuracy. Given the rankings plus search volume, SEO can generate considerable traffic and leads intended for Grainger. Google's Featured Thoughts and quick answers can become another important SEO trend that will you should watch in 2018. Many business people find examining up on the "moving target" of SEO distracts them through daily priorities more than they will ever imagined, therefore it is definitely good to look closely in why is sense for every business. The next SEO Techniques in 2018 is to adhere to Voice Search. I believed I'd offer some advice in order to stay ahead of SEO developments that are bound to create a big impact in 2018. Understanding consumer behavior - and how in order to create a great website knowledge for those users - can solve most, if not almost all, of your SEO problems shifting forward. While the lookup engines try and deliver the particular right answer to their consumer, it really is our work as SEOs to comprehend exactly what that user is looking regarding and deliver that answer intended for them with the search motor. Another reason to enhance your images for SEO: this gives search engines like search engines another clue of what your own page is about…which can assist it position in organic search. SEO is looking futuristic along with popular search engines like Search engines, Bing and Yahoo getting better, answering more queries related in order to page results and increasing the particular trend the of voice lookup.

Prior to you implement SEO to your own website, you must know exactly what key words or keywords your focus on audience is definitely to lookup for the product or assistance that you simply offer. Reverse image research is an effective off-page SEO practice that helps link building professionals to find out blogs exactly where the image has been inserted. Keywords are important for local SEO too. While there are therefore many opportunities in the entire world of SEO in 2018, content material and links are just mainly because important as ever. To obtain into the minds from the particular SEO specialists, read the comprehensive research below to get serious insights concerning the top priorities, issues, practices and their impacts upon latest SEO trends. In 2018, the particular key to SEO is in order to provide content that is fascinating and directly relevant for your own community to become visible over and above Google's influence. As cellular usage is around the increase -- 58% of online searches are usually carried out that way, instead than the 40% on the particular desktop - it seems practical that SEO will continue carrying out work in that direction. For that reason, I would say that 2018 is a challenge for Search engines, as much as it may be for SEOs. This also ties straight into another hot SEO trend : voice search. If you click upon the link for his or her post, The reason why SEO Is Not Going Away…But Will Change - Here's Exactly why #155, ” you can select to pay attention to the particular site's 155th digital marketing video clip or read the transcripts. If you don't take treatment of SEO, you can obtain lost among the trillions associated with links out there. Search engine methods are closer than ever in order to mimicking a human understanding associated with query intent based on synthetic intelligence; therefore the focus associated with all facets of SEO want to be biased toward customers with an emphasis on relevance, quality, and engagement. If you started 2018 along with a strong SEO campaign, yet aren't too sure if this is worth your time plus effort, it's time to contact Knoxweb. For hyperlink building in 2016 and further than, the ability to understand these types of external website concepts and furthermore to use effective off-page SEARCH ENGINE OPTIMIZATION techniques can help you along with in influencing a website's standing power.

To get exceptional SEO results, spend some period finding the best relevant content material. With the vast majority of search queries emerging through mobile, SEO professionals are focusing on it in relation to voice research to enhance their SEO. The SEO experts are usually depending more on videos in order to drive traffic. So, along with content material marketing, SEO now must be significantly aligned with your company's PAGE RANK efforts. These SEARCH ENGINE OPTIMIZATION crawler programs are similar in order to Google's own crawlers and will certainly give you an overview displaying how your page will execute SEO 2019 PDF in SEO rankings. In purchase to help you stay within front of the competition, beneath you will find the most recent SEO trends and ranking elements in 2018. The value associated with keyword studies still prolific within SEO, although we've been changing to a topic-based search globe for years now. SEO is definitely driven by Google Majorly and Other small search engines. What really matters within SEO in 2018 is precisely what you prioritise today therefore that in 3-6 months a person can see improvements within the particular quality of your organic visitors.

Ranking today provides more to do with significant and useful content than SEARCH ENGINE OPTIMIZATION voodoo link crazy linking methods and spending your time upon keyword density along with other technical products. SEO is making the most of findability, user experience, and wedding by making certain content plus messaging is aligned and arranged based on the language that will potential customers and visitors are usually most often using to explain the products, solutions, services or even information that a website is usually providing. With a couple of correct best WordPress SEO procedures you can really see the nice little boost both within SERPs and Google Picture Search. Essential whenever they will plan to expand their web business; they quickly need in order to take the help of SEARCH ENGINE OPTIMIZATION optimization services. First, understand that schema markup is one of the almost all effective, least used parts associated with SEO today Schema are essentially brief snippets of data that will can give extra information in order to search users and search motors like google.

You could expect a rise in video articles, page relevance, mobile SEO, plus site speed. Companies are already optimizing their particular SEO strategies to provide clean user experience to users. With 15 yrs of SEO experience and study under her belt, she understands what works and what in order to prevent and she shares huge useful and actionable information upon her blog. Today that you understand the significance of pleasing your users, allow me you Pillar #2 within this SEO strategy. Algorithmic chasers, technical SEOs, plus Google Doodle followers should develop their technical skills to pay out attention to emerging voice lookup technologies and AI applications. The particular SEO companies who know just how to handle their clients nicely do include these techniques associated with Content Marketing. Getting this right affects most you do in search, plus really helps with items that are usually becoming big in SEO at this time like: voice, answer boxes plus topic-based content (instead of the particular old-school keyword focused stuff).

[demorun]

Skiena Chapter 4 (sorry for the lack of images

_These are my notes for chapter 4 of Skiena’s Algorithm Design manual. They have images locally, but I’m not going to upload them here until i REALLY need something to do. If you have the book, they go in order and you can follow along._

## INTRO - Selection sort: repeatedly extracts the smallest remaining element from the unsorted part of the set. ```  SelectionSort(A)    for i = 1 to n do      Sort[i] = Find-Minimum from A      Delete-Minimum from A  Return(Sort) ``` - Easy. And has O(n^2) efficiency. So, not great. It performs n iterations with n/2 steps per iteration, and then goes again. - But we can speed it up to O(n log n) by replacing our generic data structure (the set A) with a better data structure, like a heap or a balanced binary tree. Each loop then takes O(log n), improving us to O(n log n).

## HEAP SORT

- Heaps: a data structure that efficiently supports the priority queue operations insert and extract-min. They maintain a partial order on the set of elements which is weaker than the sorted order (so it's efficient to maintain because you don't need to keep it intact) but stronger than a random order (so you can find the minimum without sorting for forever). - a heap-labeled tree: a binary tree such that the key labeling of each node dominates the key labeling of each of its children.  - min-heap: a node dominates its children by containing a smaller key than they do  - max-heap: a node dominates its children by being bigger.  - a heap lets us represent binary trees without storing pointers or references anywhere. if you store the data as an array of keys, we can use their position to implicitly satisfy the role of the pointers.  - root goes to arr[0] and its left and right children go in arr[1] and arr[2] (the book use arrays starting at 1 but fuck that)  - p. 110, we have a heap-labeled tree of important years from American history and a corresponding implicit heap representation, which looks like this:  - Note how it's arranged. If it's on the same level in the tree, it's next to its neighbor with the same parent dominating it in the array. - this makes it easy to figure out the position of the children and parents of random key k.  - for the key at position k, left child is at 2k and right child is 2k + 1, and parent is at floor(k/2).    - why? as we add elements to the heap, they're inserted in order. mathematically, each new level (i.e. the depth of the parent-child relationship) doubles the number of elements we're looking at (it's a tree) and so we can always find them. this lets us dodge the need for pointers. - example they're using for a priority_queue struct (in C):  - it's got a queue, and it knows that n is the number of elements inside it.   - heapsort needs to pack things in tightly to avoid leaving holes in the array, because we need the full binary tree to make this work correclty. if number of nodes < 2^h, where h is the tree's height, we'd lose our ability to traverse the heap without pointers because we wouldn't have a fully packed tree. therefore, only the last level's allowed to be incomplete; everything else must exist in a dominance relationship with another node in the set. - we can and need to represent an n-key tree using exactly n elements of the array. without that, we might need an array of size 2^n, because we'd be leaving levels unfilled.  - but that doesn't make sense. what does "unfilled" mean here?    - it means that somewhere in the tree, some node ends early, or doesn't have as many children, or whatever.    - and in that situation, if we went through and added every child recursively, we would add a bunch of empty spaces inside it. - caveats for this: we can't store any binary tree using an implicit representation. we can't store any kind of tree topology without leaving giant holes. we can't move subtrees around without explicitly moving each element in the subtree. we can't just change a pointer to move things. it works fine for heaps, but it's not as great as some other approaches. - okay, so WHY can't we efficiently search for a particular key in the heap?  - we need to just keep traversing the whole thing until we find it, right? we can't look ahead or behind. linear search.

### OKAY HOW DO YOU BUILD IT:

- you insert each new element n into the array at pos k, and then check it against the parent at k/2. if the parent is unhappy, you swap them so that the parent is properly dominated, and check again. - example:      - pq_insert  :      - receive a queue and an item x      - make sure we haven't gone over the memory size allotted.      - up the number of elements n in the array by 1      - push in x at position k (which now == n because we upped the array size, which you need to do in c)      - start checking it against its parent at n/2. it's a recursive function, which will ultimately return here    - bubble up:      - you've got a queue and an integer p (which is k, position of the last element in the set).      - if the parent function returns -1, you're done. no more parents.      - if the dominance relation is off, swap the parent at k/2 and the element at k and test again. this time, pass it the position of the parent.      - otherwise return nothing. it's done and all set.

### EXTRACTING THE MINIMUM/MAXIMUM, AKA HEAPSORT

- so these are the last two priority queue operations, identifying and deleting the dominant element. - if we delete that top element, we need to maintain the order of the tree. we use a bubble_down function to traverse/keep it organized:   - extract min removes an element from the array, moves the last element in the array of size n to the front, decrements n by 1, and bubbles the element now at 1 down until it finds its correct level of dominance. eventually, it returns the minimum element.   - bubble\_down finds the child of an element and sets its min\_index to the current value. it checks if the element at either c (the child's position) or c + 1 is "lighter" than the element at min\_index, in which case min\_index = c + i (where i is 0 or 1), and we've located a heavier element.  - if we find it, swap the element in queue q at index p with the child at min\_index, then check again at that min\_index to make sure the tree stays identical.  - that's heapifying and finding the minimum in an array - so what's heapsort? repeatedly extracting the minimum from an unordered priority_queueq and storing it in an array s as we efficiently locate the minimum. as it goes, we reset minimum (or maximum, but this example uses minimum) over and over again (for this purpose) and return it, bubbling those values to the top of s as the minimum gets located according to where it sits in the dominance hierarchy  

- ok! features  - runs in O(n log n) time, which is the best we can expect for any sort.  - it's in-place and doesn't use extra memory over the array containing the elements to sort.

### BUILDING IT FASTER Say we stuff everything in (first n elements) into the array. The shape will work, but dominance will suck. How do you fix it?

- "Consider the array in reverse order...it represents a leaf of the tree and dominates its nonexistent children. The same is true for the last n/2 positions of the array, they're all leaves. If we keep walking backwards through the array, we'll find an internal node with children. It might not dominate, but its children are all well-formed heaps."  - so then we can bubble_sort! stuff it all in, start at the end, and walk back.    - remember: bubble_down is recursive. swaps it with the parent and reiterates.  - if we do this, we get O(n log n). Same as above.  - but it's an upper bound, because n/2 nodes in a binary tree with n nodes are leaves.  - there are at most [n/2^h+1] nodes of height h, so the cost of building a heap is:      - series converges to linear. so we can do the construction in constant time, which prob. doesn't matter