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Assume that I have a set of coplanar points $P = \{p_1, p_2, ... , p_n\}$ The equation of the plane is unknown. $\forall p_i,p_j \in P$, pairwise euclidian distance $d(p_ip_j)$ is known. And I have …

Given a weighted, undirected graph $G = (V,E)$, how can I compute the average weight of edges? It seems an easy problem (divide the total weight to the number of edges!) but I couldn't manage to find …

In a sensor network graph $G = (V,E)$ $V = \{1,2,...,n\}$ is the set of sensors and the edge $(i,j)$ denotes that sensor $i$ and sensor $j$ are inside each other's sensing range. The weight of that ed …

This is the question I asked four months ago and took very satisfactory answers. However, I tackle a new problem now. Here, I summarize the original problem: We have points in 3D space. We do not …

Suppose that we have a number $x \in \mathbb{Z}^+$. I am seeking an algorithm to find three numbers $a, b, c \in \mathbb{Z}^+$ such that $a \times b \times c = x$ and $a + b + c$ is minimum. Is this …

Suppose that we have $n$ points in 3D. I want to find a plane $ax + by + cz + d$ such that sum of all the orthogonal distances to the plane is minimum. I read this article. However, I need an algori …

Suppose that there are a set of $n$ points $P = \{(x_1,y_1), \dots, (x_n,y_n)\}$ in 2D. Given two coordinates $(a,b)$ and a number $r \in \mathbb{R}$, is there an algorithm with $O(|Q| + \log n)$ run …

Almost all sorting algorithms can be implemented both ways. Regarding to your question, I think what you ask is which sorting algorithms run in $O(n \log n)$ in the worst case? … As can be seen here, there are several sorting algorithms that work in $O(n \log n)$ in the worst case. …

Suppose that $G = (V, E)$ is an undirected graph, and $G$ is promised not to contain $C_{k-1}$. Goal is to find any $C_k$ in $G$ using an efficient algorithm. Efficient, in this scope, meaning that f …

Given a connected graph $G = (V,E)$, assume that there are partitions $\{p_0, p_1, ..., p_k\}$. Denote the partition set of a vertex $v \in V$ as $p(v)$. The neighborhood of a vertex $v$ is denoted as …

Consider an undirected weighted graph $G = (V,E)$, where $V \subset \mathbb{R}^3$ so the points are 3D, and the weight of an edge equals the (Euclidean) distance between its endpoints. Note that we'r …

Given a graph $G = (V,E)$, assume that we have two disjoint vertex sets $N = \{n_1, n_2 ...\} \subset V$ and $P = \{p_1, p_2, ...\} \subset V$ such that $N \bigcup P \neq V$. I want to find if there …

Assume that we have 10 points. If all those points are on the same plane, they all are coplanar. But some of them might be at a different place. That disrupts the structure of the plane if we were to …

Consider four points $i,j,k,l$ and their pairwise Euclidiean distances $d(ij)$ $d(ik)$ $d(il)$ $d(jk)$ $d(jl)$ $d(kl)$ Say that, we know the coordinates of the points $j$, $k$ and $l$. However, we …

Suppose that there are $k$ sets $S_1, S_2, S_3, \dots, S_k$. The numbers $N = \{1, 2, \dots,n\}$ are distributed into these sets equally. Say that we partition $N$ into $m$ sets $P_1, P_2, \dots, P_ …