scholarly journals Unique Sequences Containing No $k$-Term Arithmetic Progressions

10.37236/3007 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tanbir Ahmed ◽  
Janusz Dybizbánski ◽  
Hunter Snevily
Keyword(s):  
Greedy Algorithm ◽  
Arithmetic Progressions ◽  
Famous Conjecture ◽  
Basic Inequality

In this paper, we are concerned with calculating $r(k, n)$, the length of the longest $k$-AP free subsequences in $1, 2, \ldots , n$. We prove the basic inequality $r(k, n) \le n − \lfloor m/2\rfloor$, where $n = m(k − 1) + r$ and $r < k − 1$. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdős and Turán) and describe a simple greedy algorithm that appears to give an optimal $k$-AP free sequence infinitely often. We provide many exact values of $r(k, n)$ in the Appendix.

scholarly journals The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

2008 ◽  
Vol 51 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Ernie Croot
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Famous Conjecture ◽  
Positive Integers

AbstractHow few three-term arithmetic progressions can a subset S ⊆ ℤN := ℤ/Nℤ have if |S| ≥ υN (that is, S has density at least υ)? Varnavides showed that this number of arithmetic progressions is at least c(υ)N2 for sufficiently large integers N. It is well known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös's famous conjecture about whether a subset T of the naturals where Σn∈T 1/n diverges, has a k-term arithmetic progression for k = 3 (that is, a three-term arithmetic progression).We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density υ as N runs through the primes, and as N runs through the odd positive integers.

scholarly journals Greedy algorithm, arithmetic progressions, subset sums and divisibility

1999 ◽  
Vol 200 (1-3) ◽  
pp. 119-135 ◽  
Author(s):  
P. Erdős ◽  
V. Lev ◽  
G. Rauzy ◽  
C. Sándor ◽  
A. Sárközy
Keyword(s):  
Greedy Algorithm ◽  
Arithmetic Progressions ◽  
Subset Sums

Implementasi Algoritma Dijkstra Pada Game Pacman

CCIT Journal ◽  
2019 ◽  
Vol 12 (2) ◽  
pp. 170-176
Author(s):  
Anggit Dwi Hartanto ◽  
Aji Surya Mandala ◽  
Dimas Rio P.L. ◽  
Sidiq Aminudin ◽  
Andika Yudirianto
Keyword(s):  
Artificial Intelligence ◽  
Greedy Algorithm ◽  
Dijkstra Algorithm ◽  
Testing Phase ◽  
A Value ◽  
Route Problem ◽  
Starting Point ◽  
The Greedy Algorithm

Pacman is one of the labyrinth-shaped games where this game has used artificial intelligence, artificial intelligence is composed of several algorithms that are inserted in the program and Implementation of the dijkstra algorithm as a method of solving problems that is a minimum route problem on ghost pacman, where ghost plays a role chase player. The dijkstra algorithm uses a principle similar to the greedy algorithm where it starts from the first point and the next point is connected to get to the destination, how to compare numbers starting from the starting point and then see the next node if connected then matches one path with the path). From the results of the testing phase, it was found that the dijkstra algorithm is quite good at solving the minimum route solution to pursue the player, namely by getting a value of 13 according to manual calculations

Parallelism of adaptive Hungary greedy algorithm for biomolecular networks alignment

2013 ◽  
Vol 33 (12) ◽  
pp. 3321-3325
Author(s):  
Jin MA ◽  
Jiang XIE ◽  
Dongbo DAI ◽  
Jun TAN ◽  
Wu ZHANG
Keyword(s):  
Greedy Algorithm ◽  
Biomolecular Networks
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