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Yudong Jin
104 lines
4.2 KiB
Markdown
104 lines
4.2 KiB
Markdown
# Exercises
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## Concept Review
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### Represent the Same Graph in Two Ways
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An undirected graph has four vertices, `A, B, C, D`, and the edges
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`A-B, A-C, B-C, C-D`.
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<!-- numbered-subquestions -->
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1. Write its adjacency list.
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2. Fill in its adjacency matrix using only 0s and 1s.
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3. To determine whether `A` and `D` are directly connected, which graph representation requires checking only one stored entry?
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4. If a graph has many vertices but few edges, which representation usually uses less space?
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??? success "Answer"
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1. The adjacency list is:
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```text
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A: B, C
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B: A, C
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C: A, B, D
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D: C
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```
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2. The adjacency matrix is:
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| | A | B | C | D |
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| --- | --- | --- | --- | --- |
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| A | 0 | 1 | 1 | 0 |
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| B | 1 | 0 | 1 | 0 |
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| C | 1 | 1 | 0 | 1 |
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| D | 0 | 0 | 1 | 0 |
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3. In an adjacency matrix, you can directly check row `A`, column `D`, making it well suited to determining whether any two vertices are directly connected.
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4. When a graph has many vertices but few edges, an adjacency list records only the edges that actually exist. It usually uses less space than an adjacency matrix, which reserves a position for every pair of vertices.
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### Breadth-First and Depth-First Traversal Orders
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An undirected graph has vertices `A, B, C, D, E` and edges
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`A-B, A-C, B-D, C-D, D-E`.
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Start at A. Whenever there are several unvisited adjacent vertices, choose them in alphabetical order:
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<!-- numbered-subquestions -->
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1. Write the visit order of breadth-first traversal (BFS).
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2. Write the visit order of recursive depth-first traversal (DFS).
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3. Why must both traversals record which vertices have already been visited?
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??? success "Answer"
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1. The BFS visit order is `A, B, C, D, E`. It first visits B and C, which are one edge away from A,
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and then visits the more distant D and E.
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2. The DFS visit order is `A, B, D, C, E`. It repeatedly enters an unvisited adjacent vertex,
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first following `A → B → D → C`. When C has no new adjacent vertex, it returns to D and then visits E.
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3. The graph contains a cycle, such as `A-B-D-C-A`. Without recording visited vertices,
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a traversal could repeatedly visit the same vertices around the cycle and fail to terminate normally.
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### Can One BFS Visit the Entire Graph?
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An undirected graph has vertices `A, B, C, D, E, F` and only the edges
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`A-B, B-C, D-E`.
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<!-- numbered-subquestions -->
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1. Which vertices can one BFS starting at A visit?
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2. Based on Question 1, has this BFS visited every vertex in the graph? Why or why not?
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3. Suppose you scan all vertices in alphabetical order and start a new BFS whenever you reach an unvisited vertex.
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What is the starting vertex of each BFS? Into how many mutually disconnected parts (connected components) is the graph divided?
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??? success "Answer"
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1. Starting from A, the traversal can visit only `A, B, C`.
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2. It has not visited every vertex. `D, E` form another connected part, while F is an isolated vertex.
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None of them has a path to A, so they cannot be reached from A.
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3. The three BFS traversals start at `A, D, F`, and visit
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`{A, B, C}`, `{D, E}`, and `{F}`, respectively. Therefore, the graph has 3 connected components.
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## Programming Exercises
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### Determine Whether a Path Exists in an Undirected Graph
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You are given an undirected graph with $n$ vertices numbered from $0$ to $n-1$. Each entry `[u, v]` in the array `edges` represents an undirected edge between vertices `u` and `v`.
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You are also given a starting vertex `source` and a destination vertex `destination`. First build an adjacency list from `edges`, then use BFS or DFS
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to determine whether a path exists from `source` to `destination`. Return `true` if one exists and `false` otherwise.
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The graph may contain cycles and may be disconnected.
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??? tip "Hints"
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1. Add every undirected edge in both directions
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2. The graph may contain cycles, so you must record which vertices have already been visited
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3. Starting from source, return true if you encounter destination; if the traversal ends without reaching it, return false
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[LeetCode](https://leetcode.com/problems/find-if-path-exists-in-graph/){ .rounded-button .exercise-button target="_blank" rel="noopener noreferrer" }
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