31
votes
Accepted
What is dynamic programming about?
Dynamic programming gives you a way to think about algorithm design. This is often very helpful.
Memoization and bottom-up methods give you a rule/method for turning recurrence relations into code. ...
D.W.♦
- 143k
12
votes
What is dynamic programming about?
Your understanding of dynamic programming is correct (afaik), and your question is justified.
I think the additional design space we get from the kind of recurrences we call "dynamic programming" can ...
- 71.1k
11
votes
Accepted
Deformation of algorithms
There is no general way to do this. The "space of algorithms" is not a nice one, with a natural metric or other nice properties, unlike e.g. the real numbers. Note that even in the case of trying to ...
- 13.2k
8
votes
Accepted
Kernels in parameterized complexity
Intuitively, a kernelization algorithm is an algorithm which in polynomial time preprocesses a given instance and outputs an instance whose size is bounded in the parameter. The goal of kernelization ...
- 13.3k
8
votes
Accepted
Finding an element in a sorted array with at most three queries to larger elements
If you have only one life only safe way is to check every element starting from minimal. It's $O(n)$
If you have two lives and limited with $k + 1$ comparisons the minimal element of array you can ...
- 251
7
votes
Accepted
Hash-Table in Practice
SHA1 or SHA256, whichever you use, is for any practical purpose a random function. What you are observing is that random allocation is not as good as deterministic allocation. If you knew all the ...
- 270k
6
votes
What is dynamic programming about?
Dynamic Programming allows you to trade memory for computation time. Consider the classic example, Fibonacci.
Fibonacci is defined by the recurrence $Fib(n)=Fib(n-1)+Fib(n-2)$. If you solve using ...
- 608
6
votes
What is dynamic programming about?
Here is another slightly different way of phrasing what dynamic programming gives you. Dynamic programming collapses an exponential number of candidate solutions into a polynomial number of ...
- 577
6
votes
Accepted
How to give an approximation algorithm for this unusual bin packing problem?
Your problem is known as Multi-Capacity Bin Packing. One of the foundational papers in the area is by Leinberger, Karypis and Kumar, who state a result of Garey, Graham, Johnson and Yao that in the ...
- 270k
6
votes
A general algorithm for greedy algorithms
There is no such thing as the correct generalization of the greedy selection technique, because it's an informal technique. That said, there has been some effort at modeling the greedy heuristic, with ...
- 270k
6
votes
Accepted
Implementation of QuickSort to handle duplicates
The simple implementation idea is to separate the values into three groups: values less than the pivot, values equal to the pivot, and values greater than the pivot.
In pseudocode, the algorithm ...
- 34.8k
5
votes
Accepted
Data structure for efficient searching, when insertions and removals are only one-sided
Store the elements as a sequence, sorted by increasing timestamp. Use binary search to find the location where $\tilde{t}$ would occur if it were in the array; then you can easily find the two ...
D.W.♦
- 143k
5
votes
Any algorithm better that O(N*logN) for a problem of finding student with largest average score in a list of N scores of the form StudentID, score
You are on a good path, but seem to be confused for no reason. After step 2 you have created an array with average scores. This is a worst case O(N) size array. What is the complexity of finding the ...
- 506
4
votes
Accepted
"Searching and sorting" algorithm to find the natural logarithim of a number?
Suppose you have an array $A = [e^0, e^1, e^2, \dots]$. You do a search in this array, and try to find the biggest value in the array that's smaller than or equal to $x$.
You find this value at ...
- 12.4k
3
votes
Beginner help with Arrays & Complexity
Let's try to solve a simpler problem first:
Given an array that only has 1's and 0's in it, the method turns every number into the amount of 'steps' it is away from the closest 0 to the left.
Now ...
- 86
3
votes
Why is reduction mostly associated with proving hardness?
Actually, people do commonly use reductions for both purposes: both for proving lower bounds, and for designing algorithms to handle a certain problem.
For instance, it's very common to reduce a ...
D.W.♦
- 143k
3
votes
Finding one of 2/3 of all array elements in constant expected time
One approach would be to randomly pick a large constant $k$ indices and test them. The exact probability of at least one of them being $A[i] = X$ would be:
$$\begin{align}
P(\text{at least }1\; X) &...
- 4,371
2
votes
Do compression tables exist and where are they used?
The big advantage of Lempel-Ziv-style compression is that
it creates its own dictionary on the fly and
tailors it to the data. Therefore, they are
general-purpose compression algorithms which
can be ...
- 71.1k
2
votes
Accepted
How to synchronize a 2d cellular automaton in $\Theta(\sqrt{n} \log n)$ steps
A naive solution could be that each general first launch a binary counter in its row and counts the current time until $k\log(k)$ steps and then run the FSSP. To do this you just need to detect the ...
- 156
2
votes
Accepted
Why isnt node checked for nil value in start when transplanting binary tree
If $v$ is NIL then transplanting $v$ just transplants an empty tree. This is fine. It is handled in the exact same way as transplanting an actual tree, the only difference being that we don't need to ...
- 270k
2
votes
How to improve the binomial algorithm?
The answer for your question How to improve with an implementation that would be based on saving computed values in an array? is to use memoization (sometimes known as dynamic programming), which ...
- 270k
2
votes
Could my algorithm be considered Divide-and-Conquer?
TL;DR
Divide and concur doesn't have any rigorous definition. So any argument you make for or against will be just an emotional argument. The situation is made worse by the fact that the task ...
- 1,358
2
votes
Algorithm and proof for combining a list of node connections into groups of nodes efficiently
You should look into DFS and BFS, which can solve this problem in $O(n+m)$ time where $n$ is the number of nodes and $m$ is the number of edges.
The performance of the algorithm you described is much ...
2
votes
Why is reduction mostly associated with proving hardness?
The need to define reduction formally arises only when defining classes of hard problems. Consider polynomial time reducibility as an example. The same notion is used for both "positive" and "negative"...
- 270k
2
votes
Why is reduction mostly associated with proving hardness?
I've seen reduction used implicitly. A good example is the problem of finding a maximum spanning tree. In that case, you reduce the problem to the minimum spanning tree problem by multiplying all edge ...
- 61
2
votes
Find all rational roots of a polynomial equation
If you only want to find all rational roots, you can simply use the rational root theorem. This theorem states that, given a polynomial $a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1x+a_0$, for any rational ...
- 6,998
2
votes
Find all rational roots of a polynomial equation
The paper Computing Real Roots of Real Polynomials by Sagraloff and Mehlhorn from 2015 provides an almost optimal algorithm and references for simpler algorithms that might be used in practice. The ...
- 5,910
2
votes
Beginner help with Arrays & Complexity
Just some hints that will tell you how to solve it efficiently:
If you know where the first zero is, then you know what numbers to write down at all positions before and up to that zero. For example,...
- 25.5k
2
votes
Is there a formalization of a general version of the sweep line algorithm? If not is it easy to derive?
Perhaps this is too broad a question for a coherent answer, so let me first try to say something about line segment intersection, at least. The main reason the sweep-line algorithm is efficient is ...
- 6,998
2
votes
improve the running time
You have a sorted array of n elements.
If the first element is > k then no elements have absolute value ≤ k. (Why ?)
If the last element is < -k then no elements have absolute value ≤ k. (Why ?)
...
- 25.5k
Only top scored, non community-wiki answers of a minimum length are eligible