41
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How to prove greedy algorithm is correct
Ultimately, you'll need a mathematical proof of correctness. I'll get to some proof techniques for that below, but first, before diving into that, let me save you some time: before you look for a ...
D.W.♦
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14
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How to prove greedy algorithm is correct
I will use the following simple sorting algorithm as an example:
...
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9
votes
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Matrix Chain Multiplication Greedy Approach
You don't state why you think that your algorithm is correct. In fact, it is incorrect. Here is an example. Consider the problem of computing the product of matrices of dimensions $2\times 1$, $1\...
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8
votes
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Counterexample to this modified Dijkstra's
Your algorithm makes the wrong choice between the following two paths:
5 channels with a reliability of 50% (combined reliability 3.125%), weight $5 \cdot {1 \over 0.50} = 10$.
A single channel with ...
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7
votes
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Is there a polynomial time algorithm to determine whether an 'up down' language is 'emptible'?
Reduction from 3-SAT:
a variable in 3-SAT becomes a character in your problem and is paired with its negation. Each clause becomes a word.
e.g.
3 SAT: (a,b,-c) && (-b,c) =>
pairs: (a,-a), (...
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7
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Find the lexicographically smallest order of N numbers such that the total of N numbers <= Threshold value
You are on the right track.
It turns out the original question can be solved by a greedy algorithm. (A full blown solution by dynamic programming as I tried a while ago is both an overkill on coding ...
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6
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greedy algorithm for Maximum directed cut
The starting point is the trivial random algorithm that chooses $S$ completely at random. Each directed edge is cut with probability $1/4$ (why?), and so in expectation, this random algorithm gives a $...
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6
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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 ...
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5
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Greedy Algorithms for Non-monotone Submodular Maximization with Cardinality Constraints
Edit: See my answer to the same question on math stackexchange here. I've copy and pasted the answer again for ease of use.
There has been a recent line of work investigating algorithms for ...
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5
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set with maximum sum consisting of mutually co-prime numbers
Project Euler asks you to solve the problems yourself, without help. So dont read on if you want to submit a solution for Project Euler; that would be cheating.
Since the numbers are mutually co-...
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5
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Vertex cover algorithms for directed graphs?
Thanks for the edit! This isn't vertex cover; it's something different.
There are simple algorithms for this problem. Decompose the graph into a dag of strongly connected components. (The dag is ...
D.W.♦
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5
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Greedy algorithms: Minimum sum number pairing
Suppose that numbers are $x_1, \ldots x_{2n}$, and let us rename them as $a_1, \ldots a_n, b_1, \ldots, b_n$, where $a_i \geq b_j$ for any $i, j$, $a_1 \geq a_2 \geq \ldots \geq a_n$, and $b_1 \leq ...
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Prove that the greedy algorithm to remove k digits from a n-digit positive integer is optimal
The greedy algorithm is optimal.
The simple observation is that any optimal $k$ digits to remove must contain the rightmost digit in the initial non-decreasing digits of A, or one of its equivalents....
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5
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N numbers, N/2 pairs. Minimizing the maximum sum of a pairing. Proving greedy algorithm
Can you see why $\max((-2)+5, 3+4) \lt \max(-2+3, 4+5)$?
The reason is simple. Because on the right hand side, the maximum number 5 is not paired with the minimum number.
Let the numbers are $a_1\...
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5
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Dynamic programming: optimal order to answer questions to score the maximum expected marks
First, if any $p_i=0$, then immediately throw it away since you're guaranteed to lose. If any probability is $1$ then immediately ask it! (after all why risk not getting the reward when you're ...
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4
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Correctness of the greedy algorithm
The example provided by Valentin Lorentz can be slightly modified to break your solution for one order of traversal:
1 0 0 1
0 0 0 0
0 0 0 0
You can build a ...
- 416
4
votes
Coin Change Problem(Greedy Algorithm)
For the set of coins (2,3,11). $\frac{3}{2}<\frac{11}{3}$ so by your assumption we can be greedy here. Consider the value of 23. The greedy strategy would involve first taking 2 11 cent coins to ...
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Information about ε-greedy algorithms
ε-greedy is just a way to promote exploration in Reinforcement Learning. I would not classify SARSA or Q-Learning as ε-greedy algorithms.
The latter are very common reinforcement learning algorithms ...
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How to prove greedy algorithm is correct
Jeff Ericson in his "Algorithms" states three conditions:
Greedy choice: There is an optimal solution that includes the choice the algorithm makes.
Inductive structure: The smaller ...
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4
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Dynamic Programming vs Greedy - coin change problem
A dynamic approach would say that "x$ can be made out of change using, as the first coin, v1 or v2 or v3 ... or vn" and then build a table so that the second coin would be v1 or v2 or v3 ... or vn + ...
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Maximum cut using a 1/2 approximation greedy algorithm
Let us say that an edge $(v,w)$ belongs to $v$ if when $v$ is processed, $w \in A \cup B$, and that $(v_1,v_2)$ belongs to $v_2$. Denote by $N_v$ the number of edges belonging to $v$, and by $C_v$ the ...
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4
votes
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Correctness proof for greedy algorithm based on ratio
The strategy to prove your ratio greedy algorithm is what I called "unimprovable solution by exchange of elements".
Instead of proving that an algorithm produces the optimal solution, this strategy ...
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4
votes
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Merging balls interview problem
Keep all the values $\frac{D_i - D_{i+1}}{V_i - V_{i+1}}$ in a min-heap. At each step, remove the minimum value, say $T = \frac{D_i - D_{i+1}}{V_i - V_{i+1}}$. We would first like to update all ...
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4
votes
Accepted
Gas Station problem : Fixed path variation
The greedy strategy works in this case (fill up as much as you need to reach the next cheaper city, or fill up to K if no cheaper city is reachable with a full tank).
The proof is pretty much the ...
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Why this greedy algorithm fails in rod cutting problem?
Because a local maximum is not always a global maximum.
The length 3 rod is the most price-for-length effective single rod piece. But the optimization goal isn't to find the best price-to-length ...
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4
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Greedy heuristic for buying fewest fridges of set temperature for products that can be kept in some temp. ranges?
The formalized problem here is selecting the minimum number of points such that for each temperature interval $\ell_i\in L$ we have at least one point that covers it. Let each interval $\ell_i$ be ...
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Difficulty in understanding the proof of the lemma : "Matroids exhibit the optimal-substructure property"
The adjective "maximum-weight" should not appear in item (1) in that proof of the lemma. This is a minor bug of that famous book.
To be fully clear, item (1) should have been the following.
...
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Why this problem has such a simple solution? How would you avoid looking for more complex solutions first?
In this case: It is obvious that you start with some task immediately, start a second task as soon as the first task is finished, then start a third task immediately after this etc.
It is obviously ...
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4
votes
Accepted
Hamiltonian path greedy and anti-greedy algorithms
No.
In fact, we can prove the following stronger proposition.
Claim. Given a complete weighted undirected graph, a run of the greedy algorithm on it and a run of the anti-greedy algorithm on it, the $...
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