Questions tagged [graphs]
Questions about graphs, discrete structures of nodes which are connected by edges, including trees and graphs with weighted edges.
4,838
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number of edges that appeared at least in one shortest path
In a simple weighted graph, with n vertices and m edges , for each pair of vertices we want to find the number of edges that appeared at least in one of the shortest paths between these two vertices.
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Assumption on SuccessiveShortestPaths
I read that one assumption for Successive Shortest Paths algorithm for computing the minimum cost flow problem is that every cost is non-negative.
I also read that this assumption can be removed with ...
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Placement of Tasks from Dataflow Graph on a Physical Graph
I have a dataflow graph where a set of different types of tasks are placed in its nodes. For example, I have tasks of types A, B, and C. A-type tasks are placed in all the leaf nodes of the dataflow ...
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Non-dominated maximal paths (DAG)
Let $D(V, A)$ be a DAG. We call a dominated path in $D$ a path $P$ such that
$\quad\quad P$ is maximal and $\exists_{P^{'} \in D}(P^{'} \text{ is maximal } \wedge V(P) \subset V(P^{'}))$
that is, $P$ ...
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Amortized analysis on skew heap arbitrary deletion
A practice problem in my textbook asks to proof the amortized complexity for a sequence of insert, delete min, and decrease-key operations on an initially empty skew heap. Insert and delete min both ...
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Identifying peaks at every cluster
I have a data set showing different clusters of data points and was attempting to find a way to get the peak of at every cluster. Comparing the moving average between a set of points does not work as ...
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Mapping relationtional data to hierarchical data (JSON, XML, etc.)
I'm working on a project that involves mapping relational data to a tree hierarchy. The hierarchical data is preferred to be JSON or XML. Is there an existing general algorithm for this?
Current ...
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Dynamic Program to find well formed set in a rooted tree
You are given a rooted tree $T=(V,E)$ with $n$ nodes and the root $r$. Each node $u\in V$ has an integer label $l(u)$. Suppose $S⊆V$ then $S$ is well-formed if for every $u,v\in S$ if $u$ is an ...
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The second shortest path on a directed graph
The question asks to write an algorithm using Dijkstra's algorithm with time complexity of
$\Theta(|E| \log |V|)$ that find the second shortest path between s∈V and t∈V.
The farthest I managed to get ...
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Borůvka's step in linear time
I am trying to understand this Expected linear time MST algorithm, and I have a problem in the implementation of the Borůvka's step.
My problem is with the removal of duplicate edges between merged ...
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Complexity of topological sorting with a special restriction
Let $G = (V, E)$ be a connected DAG (representing a circuit) with every vertex may be one of the following three types:
Input variable, with in-degree $0$ and out-degree $\geqslant 1$.
A gate, with ...
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Could this novel algorithm be qualified to be published in Nature or Science
I recently designed an algorithm for single-source shortest paths in graph structures, which can limit the number of edges as Bellman-Ford while approaching the performance of SPFA. Of course, it also ...
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O(n^3) Algorithm for Maximum Weighted Cycle Cover for Undirected Complete Graph with Triangle Inequality
I have been trying to implement an approximation algorithm for the max traveling salesman problem with triangle inequality, and each paper I've read references the step of finding the max cycle cover ...
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shortest path question
suppose directed graph $G = (V,E)$
Let $C$ be a cycle with weight 0. The weight of the minimum cycle in the graph is also 0. Prove that for every pair of nodes $u,v$ $(u \neq v)$ in the cycle $C$:
$\...
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NP-hardness of modified distance-colouring of graphs
Given a graph $G =(V,E)$, a set of colors $\mathcal{C}=\{0,1,2,3,...,c-1\}$, and an integer $r$, I want to know if I can find a coloring procedure that can assign a color to each nodes (all nodes must ...
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Fully Connected Graph to Lattice
I am looking for algorithms (or at least something similar to the problem definition):
Given a fully-connected weighted graph $G$ with $n$ nodes, find a subset $S$ of edges that form a square lattice ...
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Independent sets generation in a graph
Is there an algorithm that, given an undirected graph and one independent set IS1, finds an other independent set IS2 by adding and deleting vertices from the first IS1?
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Algorithm for constructing a numbering reflecting the order of activities
I'm following a book about graphs and they introduce a concept called 'activity network'. In an activity network, each vertex represents an activity in a project (like building a house for example) ...
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Algorithm question - check if there exists a path that touches A nodes exactly once and can revisit all other nodes
I am having trouble with a problem where I am given an adjacency list and a list of the nodes that must be visited exactly once to connect two nodes. What is the most efficient way of doing this? This ...
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Graph Algorithms (SSSP vs MST)
I am currently facing a question "Can a graph with a unique MST product a different spanning graph using Dijkstra vs using Prim's algorithm?" The answer is false and I am struggling to ...
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Can Dijkstra's algorithm be used this way?
Let us say that I wanted to solve a Hamiltonian path problem by treating it as a Hamiltonian cycle(on a weighted graph). I use a TSP solver, and implement a dummy node of edge weight zero, whose ...
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epsilon-optimality in cycle-cancelling for min cost flow
I'm learning about the (min-mean) cycle-cancelling alg for min-cost flow in Ahuja, Magnanti, and Orlan's Network Flows book (Chapters 9 and 10). When talking about the alg, they prove this fact ...
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Prove vertex limit for cluster editing kernel
A graph G is a cluster graph if every connected component of G is a clique. In the Cluster Editing problem, we are given as input a graph G and an integer k, and the objective is to check whether one ...
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How to modify Dijkstra's algorithm to model the path of an electric car?
I know that Dijkstra's algorithm is used to find the shortest path between nodes in a weighted graph. And I know that this can be used to model road networks.
Somebody online asked (but nobody ...
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Why not n^2 comparisons in the Alien Dictionary problem on leetcode?
Here's the problem statement (as given on GeeksForGeeks website):
Given a sorted dictionary of an alien language having N words and k starting alphabets of standard dictionary, find the order of ...
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A $O(|E||V|)$ algorithm to determine if a graph is singly connected?
In exercise 22.3-13 of CLRS (Intro to Algorithms 3rd edition), the authors provide the following problem:
A directed graph $G = (V, E)$ is singly connected if the existence of a path from $u$ to $v$ ...
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Edge connectivity using flow network
Find an algorithm for edge connectivity in undirected graph using flow networks. Try to use $O(m)$ edges.
So basically the flow network should be used as a "helper function" and the graph ...
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Optimizing Delivery Routes in a Graph-Based Network to Minimize Maximum Delivery Time
In a graph with N nodes, where each node represents a house and is labeled from 0 to N-1, an ...
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Expected maximum matching size in a random bipartite graphs
What is the expected maximum matching size of a bipartite graph $(A\cup B, V)$ where $\lvert A\rvert = n$ and $\lvert B\rvert = n$ and the probability of a edge existing between $A$ and $B$ is a fixed ...
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Dinitz’ algorithm in simple unit-capacity networks
I am studying for an algorithm design course, and can't understand this demonstration about how Dinitz’ algorithm computes a maximum flow in $O(m \sqrt{n})$ time.
This is what is written on the slides ...
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Can a trie or DAWG loop?
I am looking at DAWGs, which are compressed tries, like this:
It is an acyclic graph though, and I'm wondering if you are allowed to create loops or cycles in such a data structure.
For example, I am ...
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In Johnson's algorithm for enumerating elementary circuits in a directed graph, why do B-lists have a different structure than the adjacency list?
I have a question on the implementation of the Johnson's algorithm in C that is actually at the interface of theory and practice.
Link to Johnson's paper
I already implemented this algorithm in R ...
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Increasing families of expander graphs
I would like to know if there is any research dealing with the problem of constructing an increasing family of expander graphs.
The goal is to find a family of expander graphs $(G_i)_{i \in \mathbb{N}}...
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Lookup Using Path Matrix in Floyd Warshall Algorithm
How is the path matrix created by the Floyd Warshall algorithm used for path lookup?
The 2 images show the graph (b) and the path matrix (c). Both are taken from the book: Foundations of Algorithms by ...
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Number of decycling sets in a 3-regular planar graph
Defintitions:
A graph is said to be r-regular if every of its vertex has a degree $r$. A graph is planar if it can be drawn in a plane without any of its edges intersecting each other except at their ...
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Reductions to perfect matching
Can we reduce any well-known problems to deciding whether a (possibly non-bipartite) graph $G$ has a perfect matching? I'm particularly interested in finding a reduction from deciding whether a ...
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Algorithmic Complexity of Recognizing Claw-Free Graphs
Let $H=\left(V_H, E_H\right)$ and $G=(V, E)$ be graphs. A subgraph isomorphism from $H$ to $G$ is a function $f: V_H \rightarrow V$ such that if $(u, v) \in E_H$, then $(f(u), f(v)) \in E$. $f$ is an ...
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Linked Lists, Ordered Pairs?
I would like to model linked lists using set theory similar to that in Scheme and LISP.
There is a set theoretic definition of the ordered pair:
$p = \{\{a, 1\}, \{b, 2\}\}$
My question is how does ...
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How to avoid global delaunay check in conforming triangulation?
I implemented a conforming (i.e. it creates Steiner points using Ruppert's algorithm) delaunay triangulator, which is working, but there is one step I am doing that I straight up don't understand and ...
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Fast way to know if deleting an edge will disconnect a fully connected undirected graph
Given a fully connected undirected graph, is there a quick way for me to know if the graph will remain fully connected if I delete a given edge from the graph?
Quick = O(lg V) or something like that. ...
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GNI public coin interactive proof: why randomize y?
I've read this scribe that provides a public coin interactive proof for graph non-isomorphism.
In the proof, the verifier samples both a pairwise-independent hash function and a target $y$. Then it ...
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How to actually implement ruppert's algorithm?
I have been scouting the internet for resource son how to properly implement Ruppert's algorithm and what I ahve found is always lacking in details. The best resources I have so far are these 2:
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An efficient way to find a pair of unrelated edges
I'm writing a program which uses an undirected graph to represent certain social connections, and I'm trying to check whether or not it's contains a specific induced subgraph.
Given a dense an ...
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Why is there no self-stabilizing determinsitic algorithm for vertex coloring in general graphs?
This question considers the design of a deterministic self-stabilizing algorithm for vertex coloring in uniform anonymous networks. Uniform anonymous networks do not have distinguished nodes and all ...
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Time complexity of using BFS to find shortest path within k stops
I'm referencing this Leetcode question: https://leetcode.com/problems/cheapest-flights-within-k-stops/solution/, which asks you to find the length of the shortest path from source to dest using less ...
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Probabilistic Pathfinding
Here an interesting graph problem I've recently saw:
After a heist in New York City, a group must reach Miami within a set timeframe to catch an escape boat. Their vehicle's GPS shows U.S. routes with ...
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Constructing a sparse subgraph of original weighted undirected graph with approximation algorithm
In Kruskal's algorithm, we sort edges from least weight to greatest weight and add the edge if and only if both endpoints are in different connected components in current iteration.
Now the question ...
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Is there a fully dynamic algorithm to find a minimum arborescence (like Chu–Liu/Edmonds' algorithm)?
Given a weighted directed graph with nonnegative edge weights and a vertex $r$ designated as root, at each step I will do one of the following:
Add a new edge to the graph.
Remove an edge from the ...
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A* (A-star) search algorithm including closest distance from a node to an obstacle in heuristic and step cost
I want to include the distance of a node to the closest obstacle in the cost function, so that the path length is not only minimal, but also not near obstacles.
We know that:
Dijkstra's algorithm uses ...
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Efficient intersection of multiple paths in a tree
Consider a graph tree $T$, where we are given $k > 1$ unique pairs of nodes $\{u_1,v_1\}\dots \{u_k,v_k\}$. Let $P_{i}$ denote the unique path on $T$ between $u_i$ and $v_i$. Then, my problem is ...
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