# Tag Info

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 ...

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 ...

### 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 ...
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### 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 ...

### 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 ...

### 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 ...
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### 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 ...
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### 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 ...

### 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 ...

### 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. ...
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### 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 ...
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### 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 ...

### 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 ...

### 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 ...
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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 ...
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### 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 ...

### 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. ...
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### 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 ...

### 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 ...
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### 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,...
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 ...