I've been reading about a randomized BST called Zip Tree.
Please see a link here to the article.
The following lemma is proven in the article:
without telling us what k is. I thought at first that k refers to x.rank but that doesn't make sense.
Can anyone clarify that?
Related
I am currently reading about an algorithm from Tarjan on how to get the Least Common Ancestor of two nodes in a Binary Tree.
I have read the pseudo code from Wikipedia, but I'm not getting the gist of it. I mean I am not able to apply the algorithm on any given Binary Tree. I also tried to find some explanation of each steps on Google but i did not get anything worth. So, if anybody can help me in understanding how this algorithm works on a Binary Tree, it would be really appreciable.
Besides the given binary tree , you should implement another data structure named disjoint set to apply this algorithm . There are three main methods along with this data structure , MAKE-SET、UNION、FIND-SET . I strongly recommend you to read Chapter 21 "Data Structures for Disjoint Sets" of 《Introduction to algorithms》 for a better understanding.
I am currently following an algorithms class and we have to solve a sudoku.
I have already a working solution with naive backtracking but I'm much more interest in solving this puzzle problem with a tree data structure.
My problem is that I don't quite understand how it works. Is anyone can explain to me the basic of puzzle solving with tree?
I don't seek optimization. I looking for explanation on algorithms like the Genetic algorithm or something similar. My purpose only to learn at this point. I have hard time to take what I read in scientific articles and translate it on real implementation.
I Hope, I've made my question more clear.
Thank you very much!
EDIT: I edit the post to be more precise.
I don't know how to solve Sudoku "with a tree", but you're trying to mark as many cells as possible before trying to guess and using backtrace. Therefore check out this question.
Like in most search problems, also in Sudoku you may associate a tree with the problem. In this case nodes are partial assignments to the variables and branches correspond to extensions of the assignment in the parent node. However it is not practical (or even feasible) to store these trees in memory, you can think of them as a conceptual tool. Backtracking actually navigates such a tree using variable ordering to avoid exploring hopeless branches. A better (faster) solution can be obtained by constraint propagation techniques since you know that all rows, columns, and 3x3 cells must contain different values. A simple algorithm for that is AC3 and it is covered in most introductory AI books including Russell & Norvig 2010.
I've often been slightly stumped by recursive algorithms that seem to require magical leaps (with a large dose of shrunken notation born of an ink shortage) of logic.
I realize that the alternative is to simply memorize the Big O notation for all the common algorithms but at a certain point, that approach fails. For example, I am happy to disclose the performance for bubble sort, insertion sort, binary tree insertion/removal, mergesort, and quicksort but don't ask me to come up with the performance of AVL trees or Djikstra's shortest path algorithm off the top of my head.
Where can I go to get:
A discussion of recursive algorithm analysis that uses words instead of a profusion of symbols
Practice problems to confirm that my newly-obtained understanding is actually correct
Example:
Bad:
Sigma v e T (1+cv)
Possible 'good' equivalent:
The amount of work required for 1 node in the tree (which is 1+the # of children of a node), which is then executed once for every element in the tree where the original node is the root.
Side commentary:
I could simply watch a video for every single algorithm because there's no way to make one's voice turn into a subscript (or any of the other contortions) but I suspect that would take an inordinate amount of time compared to reading textual descriptions.
Update:
Here's 1 source of solved problems: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/ (this tackles #2 above)
TopCoders has a great source of tutorials and thorough explanations. Have you tried them out?
http://www.topcoder.com/tc?d1=tutorials&d2=alg_index&module=Static
To 'officially' answer this question...
Source of sample problems: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/
An explanation of analysis for CS: http://en.wikipedia.org/wiki/Concrete_Mathematics.
Sometimes I come across interview questions like this: "Find the common parent of any 2 nodes in a tree". I noticed that they ask LCA questions also at Google, Amazon, etc.
As wikipedia says LCA can be found by intersection of the paths from the given nodes to the root and it takes O(H), where H is the height of the tree. Besides, there are more advanced algorithms that process trees in O(N) and answer LCA queries in O(1).
I wonder what exactly interviewers want to learn about candidates asking this LCA question. The first algorithm of paths intersection seems trivial. Do they expect the candidates to remember pre-processing algorithms ? Do they expect the candidates to invent these algorithms and data structures on the fly ?
They want to see what you're made of. They want to see how you think, how you tackle on problem, and handle stress from deadline. I suppose that if you solve the problem because you already know the solution then they will just pose you another problem. It's not the solution they want, it's your brainz.
With many algorithms questions in an interview, I don't care (or want) you to remember the answer. I want you to be able to derive the answer from the main principles you know.
Realistically: I could give two cares less if you already know the answer to "how to do X", if you are able to quickly construct an answer on the fly. Ultimately: in the real world, I can't necessarily assume you have experience tackling problems of domain X, but if you run across a problem in said domain, I will certainly hope you'll have the analytical ability to figure out a reasonable answer, based on the general knowledge you hold.
A tree is a common data structure it's safe to assume most will know - if you know the answer offhand, you should be able to explain it. If you don't know the answer, but understand the data structure, you should be able to fairy easily derive the answer.
I'm trying to improve my current algorithm for the 8 Queens problem, and this is the first time I'm really dealing with algorithm design/algorithms. I want to implement a depth-first search combined with a permutation of the different Y values described here:
http://en.wikipedia.org/wiki/Eight_queens_puzzle#The_eight_queens_puzzle_as_an_exercise_in_algorithm_design
I've implemented the permutation part to solve the problem, but I'm having a little trouble wrapping my mind around the depth-first search. It is described as a way of traversing a tree/graph, but does it generate the tree graph? It seems the only way that this method would be more efficient only if the depth-first search generates the tree structure to be traversed, by implementing some logic to only generate certain parts of the tree.
So essentially, I would have to create an algorithm that generated a pruned tree of lexigraphic permutations. I know how to implement the pruning logic, but I'm just not sure how to tie it in with the permutation generator since I've been using next_permutation.
Is there any resources that could help me with the basics of depth first searches or creating lexigraphic permutations in tree form?
In general, yes, the idea of the depth-first search is that you won't have to generate (or "visit" or "expand") every node.
In the case of the Eight Queens problem, if you place a queen such that it can attack another queen, you can abort that branch; it cannot lead to a solution.
If you were solving a variant of Eight Queens such that your goal was to find one solution, not all 92, then you could quit as soon as you found one.
More generally, if you were solving a less discrete problem, like finding the "best" arrangement of queens according to some measure, then you could abort a branch as soon as you knew it could not lead to a final state better than a final state you'd already found on another branch. This is related to the A* search algorithm.
Even more generally, if you are attacking a really big problem (like chess), you may be satisfied with a solution that is not exact, so you can abort a branch that probably can't lead to a solution you've already found.
The DFS algorithm itself does not generate the tree/graph. If you want to build the tree and graph, it's as simple building it as you perform the search. If you only want to find one soution, a flat LIFO data structure like a linked list will suffice for this: when you visit a new node, append it to the list. When you leave a node to backtrack in the search, pop the node off.
A book called "Introduction to algorithms" by anany levitan has a proper explanation for your understanding. He also provided the solution to 8 queens problem just the way you desctribed it. It will helpyou for sure.
As my understanding, for finding one solution you dont need any permutation all you need is dfs.That will lonely suffice in finding solution