25
votes
Accepted
Why can't we find shortest paths with negative weights by just adding a constant so that all weights are positive?
Adding a weight to every edge adds more weight to long paths than short paths. (Long in the sense of having many edges.)
For example, suppose the lowest-cost edge has weight $-2$ and there are ...
- 81.1k
23
votes
What is the meaning of 'breadth' in breadth first search?
Consider the data structure used to represent the search. In a BFS, you use a queue. If you come across an unseen node, you add it to the queue.
The “frontier” is the set of all nodes in the search ...
- 967
17
votes
Minimum spanning tree vs Shortest path
Though Minimum Spanning Tree and Shortest Path algorithms computation looks similar they focus on 2 different requirements.
In MST, requirement is to reach each vertex once (create graph tree) and ...
- 281
16
votes
Accepted
A* graph search time-complexity
These are basically two different perspectives or two different ways of viewing the running time. Both are valid (neither is incorrect), but $O(b^d)$ is arguably more useful in the settings that ...
D.W.♦
- 152k
13
votes
Why does Dijkstra's algorithm fail on a negative weighted graphs?
Adding a constant amount to each edge length can change the shortest path for the simple reason that it increases the length of a path with many edges by more than it increases the length of a path ...
- 81.1k
12
votes
How does consistency imply that a heuristic is also admissible?
To proof the statement in your question, let us proof that consistency implies admissibility whereas the opposite is not necessarily true. This would make consistency a stronger condition than the ...
- 3,433
11
votes
Accepted
Shortest path between two points with n hops
If vertices can be visited more than once, then yes: you can create $n+1$
copies of the graph, with each vertex $v$ in the original graph becoming the $n+1$ vertices $v_1, \dots, v_{n+1}$ and each ...
- 5,354
11
votes
Accepted
Find shortest path between two vertices that uses at most one negative edge
You can use Dijkstra twice to find in your $G'$ the cost for each vertex $v \in V$, the cost of the optimal $s$-$v$-path and the cost of the optimal $v$-$t$-path. Store this in a table creatively ...
- 14.5k
11
votes
Find shortest path between two vertices that uses at most one negative edge
Another approach is to create a single graph $H$ as follows:
each vertex in $G$ has two counterparts in $H$: vertex $s$ becomes $s_A$ and $s_B$, vertex $t$ becomes $t_A$ and $t_B$, and so on.
each ...
- 611
10
votes
Minimum spanning tree vs Shortest path
I think an example will make it clearer..
The spanning tree looks like below. This is because if we add up the edges in this configuration, we get the least total cost possible: 2+5+14+4=25.
...
- 363
10
votes
Accepted
Finding all edges on any shortest path between two nodes
Off the top of my head, you could do this. (Let's say you want to find all edges on a shortest path from $s$ to $t$.)
Run an All-Points Shortest Path (APSP) algorithm to store the shortest-path ...
- 7,018
9
votes
Accepted
What role is the set, S playing in Dijkstra's algorithm given in the book CLRS?
No, you are not missing anything if you remove $S$ completely. You could implement and run Dijkstra's algorithm correctly still.
Set $S$ is used later in the book to help explain the algorithm and ...
- 38.2k
8
votes
Minimum spanning tree vs Shortest path
The difference lies in what is the ultimate goal of this algorithms-
Dijkstra's - Here the goal is to reach from start to end. You are concerned about only this 2 points, and optimize your path ...
8
votes
Why can't DFS be used to find shortest paths in unweighted graphs?
You can!!!
Mark the nodes as visited while you are going depth and unmark while you return, while returning as you find another branch(es) repeat same.
Save cost/path for all possible search where ...
- 181
8
votes
Accepted
Why do we need to run the bellman-ford algorithm for n-1 times?
Answering your first question is simple. Try out a few graphs and see for yourself why it doesn't work. As to your second question, $n-1$ is the maximal length of a shortest path in the graph. After $...
- 274k
7
votes
Accepted
How to find the shortest path from some vertex in set $S$ to set $S'$
If all edge lengths are non-negative, then this can be solved in $O(|E| \lg |V|)$ time using a single invocation of Dijkstra's algorithm.
We're going to modify the graph slightly by adding a new ...
D.W.♦
- 152k
7
votes
Accepted
Linear time algorithm for finding $k$ shortest paths from $s$ to $t$
First of all, the answer that applies here was already given by Raphael in the comments to the question: "Given that we don't even know how to find one simple shortest path in linear time, I doubt it."...
- 3,433
7
votes
Accepted
Can we find k shortest paths between all pairs faster than solving the pairwise problem repeatedly?
First of all, a crucial difference in computing $k$-shortest paths is if the paths need to be simple or not. A path is called simple, if it does not contain nodes repeatedly. A path with a loop, for ...
- 86
7
votes
Can I run Dijkstra's algorithm using priority queue?
Yes, you can use priority queues to improve the complexity of the algorithm from $O(V^2)$ to $O(|E| + |V| \log|V|)$ where $E$ is the number of edges and $V$ is the number of nodes.
You ...
- 171
7
votes
Accepted
Shortest walk through a given subset of edges
This is NP-hard, so it's very unlikely that a polynomial-time algorithm exists.
Given any instance $G=(V, E)$ of Hamiltonian Path, create a new graph $G'=(V', E')$ in which every vertex $v \in V$ ...
- 5,354
7
votes
Accepted
Finding shortest paths in undirected graphs with possibly negative edge weights
I contacted one of the authors (Kevin Wayne; thanks) of the textbook "Algorithms, 4th Edition" and got a hint:
Try adding "t-joins" or "perfect matching" to your web searches.
Following this, I ...
- 9,431
7
votes
Accepted
Dijkstra with max instead of sum
Yes, it is true.
Let $w: E(G) \to \mathbb{R}$ be a weight function on the edges of $G$, $s \in V(G)$ be the start vertex. Let $p(v) = \min\{\max\{w(e_1), \ldots, w(e_k)\} \mid e_1, \ldots, e_k \text{...
- 231
7
votes
Accepted
Shortest paths between given red vertices and arbitrary blue vertices
I assume no negative weights in this solution (since you stated Dijkstra). My solution also uses Dijkstra. Let $G'$ be the graph resulting from $G$ by contracting all the blue vertices into one vertex ...
- 4,358
6
votes
Accepted
Finding all vertices on negative cycles
If you don't constrain yourself to simple cycles, you can actually use the Bellman-Ford algorithm to find all the relevant vertices.
Start by running a DFS on the graph to find its strongly connected ...
- 563
6
votes
Comparison between IDA* and Recursive best first search
Let me please start by succintly summarizing the behaviour of RBFS. For a thorough explanation of the algorithm refer to the original journal paper: Richard Korf. Linear-space best-first search. ...
- 3,433
6
votes
Accepted
Dijkstra's algorithm on huge graphs
There are libraries available to compute shortest paths on such graphs. How do they do this? More specifically, how do they load the required part of the graph to run Dijkstra's algorithm?
You can ...
- 176
6
votes
Dijkstra with bitwise OR of edge costs
I'm not sure if you can adapt Dijkstra specifically in any way to this problem, but there's a different efficient solution that's actually easier to come up with.
Because bitwise operators treat bits ...
- 416
6
votes
Accepted
Where does the heuristic come from in the A-star algorithm and how do we know it has the right properties?
(i) Where do the heuristic values for shortest distance come from? There is mention of "straight roads" but I don't understand this. I know "heuristic" is "informed guess" but why not values 1237, 978,...
- 81.1k
6
votes
Compute single-source shortest paths in O(n+m) time?
There are many ways to solve this problem in linear time. I'll write the one I found easiest to understand.
The main idea is to transform the weighted graph into an unweighted one. If the number of ...
- 474
6
votes
Accepted
How hard is finding the shortest path in a graph matching a given regular language?
This problem can be solved in polynomial time by a product construction. Construct the graph $G^\prime$ as follows:
The vertices of $G^\prime$ are $(V \times M) \cup \{\#\}$, i.e. all pairs of a ...
- 3,443
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