Update README.md

README.md

@@ -18,7 +18,7 @@ This project discusses the main differences between the different types of AI Se
 - Data Structure: Stack <br>
 - Completeness: Not complete <br>
 - Optimality: Not optimal <br>
-- Time: O(b ^ m) <br>
+- Time: O(b<sup>m</sup>) <br>
 - Space: O(b * m) <br>
 - b is the branching factor <br>
 - m is the maximum possible search tree depth <br>
@@ -29,8 +29,8 @@ This project discusses the main differences between the different types of AI Se
 - Data Structure: Queue <br>
 - Completeness: Complete <br>
 - Optimality: Optimal <br>
-- Time: O(b ^ s) <br>
-- Space: O(b ^ s) <br>
+- Time: O(b<sup>s</sup>) <br>
+- Space: O(b<sup>s</sup>) <br>
 - b is the branching factor <br>
 - s is the level where the goal state is present <br>
 - BFS is not very well dealing with space <br>
@@ -43,7 +43,7 @@ This project discusses the main differences between the different types of AI Se
 - Optimality: Optimal <br>
 - A* Search Always finds the optimal solution <br>
 - A* Search considers the past cost and the upcoming heuristic cost <br>
-- Manhattan distance = |x1 - x2| + |y1 - y2|
+- Manhattan distance = |x<sub>1</sub> - x<sub>2</sub>| + |y<sub>1</sub> - y<sub>2</sub>|
 
 # A* Search with Euclidean heuristic
 
@@ -52,12 +52,59 @@ This project discusses the main differences between the different types of AI Se
 - Optimality: Optimal <br>
 - A* Search Always finds the optimal solution <br>
 - A* Search considers the past cost and the upcoming heuristic cost <br>
-- Euclidean distance = sqrt((x1 - x2) ^ 2 + (y1 - y2) ^ 2)
+- Euclidean distance = &radic;((x<sub>1</sub> - x<sub>2</sub>)<sup>2</sup> + (y<sub>1</sub> - y<sub>2</sub>)<sup>2</sup>)
 
 # Test Cases & Comparisons
-<img src="images/t1.png" alt="">
-<img src="images/t2.png" alt="">
-<img src="images/t3.png" alt="">
-<img src="images/t4.png" alt="">
-<img src="images/t5.png" alt="">
-<img src="images/o.png" alt="">
+<h3>Test #1 (Input: 1 2 3 4 5 6 8 7 0) (Unsolvable test case)</h3>
+
+|          | DFS     | BFS     | A* M    | A* E    |
+| -------- | ------- | ------- | ------- | ------- |
+| Cost     |         |         |         |         |
+| Depth    | 66906   | 31      | 34      | 32      |
+| Ex nodes | 181440  | 181440  | 181440  | 181440  |
+| Time     | 0.38774 | 0.67555 | 1.06707 | 1.32079 |
+
+<h3>Test #2 (Input: 1 2 5 3 4 0 6 7 8)</h3>
+
+|          | DFS     | BFS     | A* M    | A* E    |
+| -------- | ------- | ------- | ------- | ------- |
+| Cost     | 157     | 3       | 3       | 3       |
+| Depth    | 157     | 3       | 3       | 3       |
+| Ex nodes | 158     | 10      | 4       | 4       |
+| Time     | 0.0     | 0.0     | 0.0     | 0.0     |
+
+<h3>Test #3 (Input: 1 0 2 7 5 4 8 6 3)</h3>
+
+|          | DFS     | BFS     | A* M    | A* E    |
+| -------- | ------- | ------- | ------- | ------- |
+| Cost     | 17881   | 23      | 23      | 23      |
+| Depth    | 17881   | 24      | 23      | 23      |
+| Ex nodes | 18601   | 122117  | 2714    | 4306    |
+| Time     | 0.03459 | 0.42513 | 0.01753 | 0.03551 |
+
+<h3>Test #4 (Input: 8 0 6 5 4 7 2 3 1)</h3>
+
+|          | DFS     | BFS     | A* M    | A* E    |
+| -------- | ------- | ------- | ------- | ------- |
+| Cost     | 57329   | 31      | 31      | 31      |
+| Depth    | 57329   | 31      | 31      | 31      |
+| Ex nodes | 67802   | 181439  | 16582   | 35007   |
+| Time     | 0.13396 | 0.77618 | 0.10551 | 0.28074 |
+
+<h3>Test #5 (Input: 1 4 2 6 5 8 7 3 0)</h3>
+
+|          | DFS     | BFS     | A* M    | A* E    |
+| -------- | ------- | ------- | ------- | ------- |
+| Cost     | 51618   | 8       | 8       | 8       |
+| Depth    | 51618   | 9       | 8       | 8       |
+| Ex nodes | 59137   | 203     | 13      | 13      |
+| Time     | 0.11829 | 0.001   | 0.0     | 0.0     |
+
+<h3>Observation</h3>
+
+- DFS has Longest paths and the least time in case of no solution
+
+- A* Manhattan time is less than time of A* Euclidean
+  - Reasoning: that Manhattan distance and Euclidean are both admissible but Manhattan distance is larger than Euclidean distance
+
+- A* Manhattan/Euclidean heuristics decreases the number of expanded nodes so much than in BFS.