When we look at different graph algorithms, it's important to know how much time and space they need. This helps us understand how well these algorithms work, especially when we have a lot of data. Let’s break it down into simpler parts.

1. Depth-First Search (DFS)

• Time Complexity: $O(V + E)$
  Here, $V$ is the number of points (vertices) and $E$ is the number of connections (edges). This means we check every point and connection one time.

• Space Complexity: $O(V)$
  This is about how much space we need to remember our steps, either through a stack in the computer's memory or a list of points we visited.

2. Breadth-First Search (BFS)

• Time Complexity: $O(V + E)$
  Just like DFS, BFS also looks at every point and connection once.

• Space Complexity: $O(V)$
  This is because we use a queue to remember which points we need to look at next.

3. Dijkstra’s Algorithm

• Time Complexity:
  • Using a simple list: $O(V^2)$
  • Using a priority list (heap): $O((V + E) \log V)$
• Space Complexity: $O(V)$
  This is for storing the distances and the previous points we used to get to those distances.

4. Kruskal’s Algorithm

• Time Complexity:
  • $O(E \log E)$ or $O(E \log V)$ based on how we sort the connections.
• Space Complexity: $O(E)$
  This means we need space to keep track of the connections and other tools we use to keep things organized.

5. Prim’s Algorithm

• Time Complexity:
  • Using a simple list: $O(V^2)$
  • Using a priority list: $O((V + E) \log V)$
• Space Complexity: $O(V)$
  This is similar to Dijkstra's, where we remember distances and which points came before the ones we checked.

Knowing these time and space needs can help you pick the best algorithm for your project. You can balance how quickly it runs with how much memory it uses!