Of course. This is the perfect way to understand UCS. The handwritten notes on the image actually show the algorithm in action, which is fantastic. Let's break it down, step by step, and see exactly how those tables were filled out.
The Goal: Find the cheapest path from the Initial State: Sibiu to the goal state, Bucharest.
The Tool: Uniform Cost Search (UCS). Its rule is simple: always expand the node in the Frontier that has the lowest total path cost from the start.
Step 0: Initialization
• Action: We start at Sibiu. The cost to get here is 0.
• Frontier: We add our starting point. The Frontier is a priority queue, sorted by cost. It now contains: { (0, Sibiu) }
• Explored: This list is empty. {}
Step 1: Expand Sibiu (S)
• Action: Pop the cheapest node from the Frontier. That's (0, Sibiu).
• Frontier: {} (It's empty for a moment).
• Explored: We've now finished exploring Sibiu, so we add it to the Explored list. { S }
• Discovery: Now, find Sibiu's neighbors: Rimnicu Vilcea (RV) and Fagaras (F).
○ Cost to RV = (Cost to Sibiu) + (Edge S->RV) = 0 + 80 = 80.
○ Cost to Fagaras = (Cost to Sibiu) + (Edge S->F) = 0 + 99 = 99.
• Action: Add these new nodes to the Frontier.
• Frontier: { (80, RV), (99, F) }. It's sorted by cost, so RV is first.
This matches the first two entries in your handwritten Frontier table!
Step 2: Expand Rimnicu Vilcea (RV)
• Action: Pop the cheapest node from the Frontier. That's (80, RV). (This is why RV/80 is crossed out in your notes).
• Frontier: { (99, F) }
• Explored: { S, RV }
• Discovery: Find RV's neighbors: Sibiu (already explored, ignore) and Pitesti (P).
○ Cost to Pitesti = (Cost to RV) + (Edge RV->P) = 80 + 97 = 177.
• Action: Add (177, Pitesti) to the Frontier.
• Frontier: { (99, F), (177, P) }. Fagaras is still cheaper than Pitesti.
This matches your notes perfectly. You crossed out RV and added P with a cost of 177.
Step 3: Expand Fagaras (F)
• Action: Pop the cheapest node from the Frontier. That's (99, F). (This is why F/99 is crossed out).
• Frontier: { (177, P) }
• Explored: { S, RV, F }
• Discovery: Find F's neighbors: Sibiu (explored, ignore) and Bucharest (B).
○ Cost to Bucharest = (Cost to F) + (Edge F->B) = 99 + 211 = 310.
• Action: Add (310, Bucharest) to the Frontier.
• Frontier: { (177, P), (310, B) }. Pitesti is still the cheapest path overall.
The algorithm now knows a path to Bucharest, but it's very expensive (310). It won't explore it yet because there's a more promising, cheaper path to explore first (the one through Pitesti).
Step 4: Expand Pitesti (P) - The "Aha!" Moment
• Action: Pop the cheapest node from the Frontier. That's (177, P). (This is why P/177 is crossed out).
• Frontier: { (310, B) }
• Explored: { S, RV, F, P }
• Discovery: Find P's neighbors: Rimnicu Vilcea (explored, ignore) and Bucharest (B).
○ Cost to Bucharest = (Cost to P) + (Edge P->B) = 177 + 101 = 278.
• Action: We found a path to Bucharest. We check if it's already in the Frontier. Yes, it is, with a cost of 310.
• UPDATE: Since our new path (cost 278) is cheaper than the old one (cost 310), we update the Frontier. We replace (310, B) with our better path.
• Frontier: { (278, B) }.
This is the key step shown in your notes! The old, expensive path to Bucharest is replaced by this new, much better one. That's why the number 278 is written in and circled.
Step 5: Goal Reached
• Action: Pop the cheapest node from the Frontier. It's (278, Bucharest).
• Check: Is this the goal? YES!
• Result: The algorithm terminates and returns the path that resulted in this entry.
The final, cheapest path is: Sibiu -> Rimnicu Vilcea -> Pitesti -> Bucharest, with a total cost of 278.
---Here is the complete and accurate extraction of all content from the image in a structured plain text format:
---
**Section 1: Graph and Initial State**
**Title:** Graph
**Chart Type:** Network diagram (or graph) depicting cities as nodes and roads/paths between them as edges, with associated costs/distances.
**Main Elements:**
* **Nodes (Cities):**
* Sibiu (Square node, top left)
* Rimnicu Vilcea (Square node, middle left, below Sibiu)
* Fagaras (Square node, top right)
* Pitesti (Square node, middle right, below Rimnicu Vilcea)
* Bucharest (Square node, bottom right)
* **Edges (Connections) and Weights (Costs/Distances):**
* Sibiu - Rimnicu Vilcea: 80
* Sibiu - Fagaras: 99
* Rimnicu Vilcea - Pitesti: 97
* Fagaras - Bucharest: 211
* Pitesti - Bucharest: 101
**Labels and Annotations:**
* The nodes are labeled with city names: "Sibiu", "Fagaras", "Rimnicu Vilcea", "Pitesti", "Bucharest".
* Edge weights are labeled numerically on each edge.
* A red circle is drawn around the '211' label on the Fagaras-Bucharest edge.
* Red checkmarks are present near the Sibiu and Bucharest nodes.
* Red wavy lines are drawn under the names Sibiu, Rimnicu Vilcea, Pitesti, and Bucharest.
* A red wavy line is drawn under the '101' label.
**Initial State Information:**
* **Text:** Initial state: Sibiu, path cost= 0.
---
**Section 2: Frontier Table**
**Title:** Frontier
**Table Content:**
This appears to be a two-row table, likely representing cities/states and their associated path costs in a search algorithm. Many elements are crossed out, indicating steps in a process.
| Column Header (Implied/Labelled) |
| :------------------------------- |
| S (crossed out) |
| RV (crossed out) |
| F (crossed out) |
| P (crossed out) |
| B (crossed out) |
| Row Content (Likely Path Costs) |
| :------------------------------ |
| 0 (crossed out) |
| 80 (crossed out) |
| 99 (crossed out) |
| 177 (crossed out) |
| 260 (crossed out) |
**Annotations within/around the table:**
* All values in the first row (S, RV, F, P, B) are crossed out with a red diagonal line.
* All values in the second row (0, 80, 99, 177, 260) are crossed out with a red diagonal line.
* A red horizontal line is drawn under '177'.
* The text "H P" is written in red above the 'P' and 'B' columns.
* The number "278" is written in red below the '260' (which is crossed out).
* A large "278" is written in red to the right of the table, with a red curved arrow pointing from it towards the area of the 'B' column in the table.
---
**Section 3: Explored Table**
**Title:** Explored
**Table Content:**
This table is partially visible and appears to be a single row (or the first row of a multi-row table).
| Column Header (Implied/Labelled) |
| :------------------------------- |
| S |
| RV |
| F |
| P |
---
视频信息
答案文本
视频字幕
Uniform Cost Search is a graph search algorithm that finds the optimal path by always expanding the node with the lowest total path cost. Our goal is to find the cheapest path from Sibiu to Bucharest in this Romanian cities graph. We start with Sibiu at cost zero, an empty frontier containing only our starting node, and an empty explored set.
Now we execute the first two expansion steps. Step 1: We expand Sibiu, which has the lowest cost of zero. We add its neighbors Rimnicu Vilcea at cost 80 and Fagaras at cost 99 to the Frontier, then move Sibiu to the Explored set. Step 2: We expand Rimnicu Vilcea since it has the lowest cost in the Frontier at 80. We add Pitesti at cost 177 to the Frontier and move Rimnicu Vilcea to Explored. The Frontier now contains Fagaras at 99 and Pitesti at 177.
Step 3: We expand Fagaras since it has the lowest cost of 99 in the Frontier. We discover the first path to our goal Bucharest with a total cost of 99 plus 211 equals 310. We add this path to the Frontier. However, UCS doesn't stop here because it's not greedy. Notice that Pitesti still has a lower cost of 177 compared to Bucharest at 310, so we must continue exploring. The algorithm systematically ensures we find the optimal path by always expanding the lowest cost node first.
Step 4 demonstrates the key optimization feature of UCS. We expand Pitesti with cost 177 and discover a better path to Bucharest. The new cost is 177 plus 101 equals 278, which is significantly better than our previous path cost of 310. This is where UCS shows its power: when we find a better path to a node already in the Frontier, we update it with the improved cost. The old expensive path through Fagaras is replaced by this new, cheaper path through Pitesti. Our Frontier now contains only the optimized entry with cost 278.
Step 5 completes our UCS algorithm. We expand Bucharest at cost 278 and reach our goal, so the algorithm terminates successfully. The optimal solution is the path Sibiu to Rimnicu Vilcea to Pitesti to Bucharest with a total cost of 278. Compare this to the alternative path through Fagaras which would cost 310, saving us 32 units. UCS guarantees this is the lowest-cost solution because it systematically explores nodes in order of their path cost, ensuring no cheaper path exists. This demonstrates the power of UCS in finding optimal solutions through methodical, cost-based exploration.