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# Questions tagged [graphs]

Questions about graphs, discrete structures of nodes which are connected by edges, including trees and graphs with weighted edges.

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34 views

### Understanding line in Ramsey's Theorem proof

The following is from lecture notes for Concepts in Mathematics from CMU (William Gunther): I am confused about the bolded sentence in the last line quoted here: Theorem 4 (Ramsey's Theorem). For any ...
36 views

### Understanding the last algebra step of this proof

Could someone help me understand why in the below proof, if $k > log_{2}n$, then $P(N_i=1)< \frac{1}{n}$?
1 vote
92 views

### Running time of modified BFS algorithm to find shortest path in weighted DAG

While the shortest path can be calculated with $O(V+E)$ time over a weighted directed acyclic graph using topological sort, I wonder about the running time of the following BFS type algorithm I ...
• 113
1 vote
25 views

### CHK graph dominance algorithm proof

The algorithm developed by Keith D. Cooper, Timothy J. Harvey, and Ken Kennedy is widely used in industry for building dominator tree, but I cannot follow the proof in the “Engineering the data ...
1 vote
31 views

### Understanding Arc Consistency 4 in the context of Wave Function Collapse

Following this blog post about the Arc Consistency algorithms. Namely, the explanation on AC4. Consider the following example: If I understand AC problems correctly. Then: We have two known ...
• 111
28 views

### Distance to specific node incremental addition

Let us say I have an empty graph G and a list of nodes N to add to the graph one-by-one. Let us say that I will have a node <...
• 101
20 views

### flow network, class and classroom matching

Problem: given a set of classes and classrooms, then given a set M of pairs (a,b), which means it is valid assignment from class a to classroom b(ex:(c,2), (c,3), (d,2), means class c can be assigned ...
1 vote
24 views

### Mapping a grammar to a graph - Relation of Terminal Symbol and Node Coverage in Software Testing

I am currently trying map a grammar to a graph in the context of syntax- and graph-based testing. However, until now I haven't had an enlightenment how would I map non-terminals and terminals to nodes ...
• 111
25 views

### Name of graph family defined by modular sum

In the context of finite, simple, undirected graphs, associate with each node $v\in V$ an integer $n(v)$ (you can limit this to positive integers without loss of generality). Create the set of edges ...
• 849
166 views

### Can we solve $\mathrm{MFVS} \leq 1$ in linear (or subquadratic) time?

$\mathrm{MFVS} \leq 1$ is a concise way of writing the following decision problem: Let $G = (V, E)$ be a directed graph. Is there a $v \in V$ such that every cycle in $G$ passes through $v$? (More ...
88 views

### Optimal graph data structure for set of points that allows dynamic updates

We aim to optimize the execution of a specific task. Consider a set, P, containing N 2D points. A new query point, p1, is introduced, and the objective is to identify the nearest point in P to p1. If ...
1 vote
57 views

### Choose edges in fully connected weighted graph that minimize sum of weights

I was looking at the Advent of Code 2023 day 11 problem and misread it to think that the question ended up asking something akin to what's below (after transforming the positions of the "galaxies&...
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1 vote
115 views

### 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. ...
1 vote
60 views

### 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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22 views

### 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 corresponding types of nodes. Say the task types are called A, B, and C. A-type tasks are placed in all the leaf nodes of ...
• 1
61 views

### Non-dominated maximal paths in a DAG

Let $D(V, A)$ be a DAG. We call a dominated path in $D$ a path $P$ such that $P$ is maximal and $\exists P^{'} \in D . (P^{'} \text{ is maximal } \wedge V(P) \subset V(P^{'}))$ that is, $P$ is a ...
38 views

### 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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13 views

### 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 ...
• 101
34 views

### 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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1 vote
244 views

### The second shortest path on a directed graph [closed]

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 ...
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63 views

### 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 ...
• 15.7k
1 vote
51 views

### 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 ...
209 views

### 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 ...
1 vote
21 views

### 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 ...
1 vote
115 views

### 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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34 views

### 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 ...
• 113
1 vote
31 views

### 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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1 vote
32 views

### 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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35 views

### 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 ...
1 vote
29 views

### 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 ...
1 vote
84 views

### 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 ...
1 vote
19 views

### 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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64 views

### 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 ...
134 views

### 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 ...
161 views

### 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 $v$ ...
23 views

### 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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65 views

### 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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1 vote
105 views

### 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 ...
80 views

### 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 ...
39 views

### 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 ...
• 2,243
70 views

### 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 ...
37 views

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1 vote