graph theory a branch of mathematics

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Manu · Accepted Answer

Step-by-Step Dijkstra's Algorithm Step 1: Initialize the table Start by setting the distance to the start vertex, J, as 0, and all others as infinity (∞). No vertex has been visited yet. Vertex Distance from J Previous Vertex PATH J 0 None J K ∞ - - L ∞ - - M ∞ - - N ∞ - - P ∞ - - Q ∞ - - R ∞ - - S ∞ - - Explanation: Initialization : Set the distance from J to itself as 0 and to all other nodes as infinity. Set J's path to itself. Step 2: Update from J From J, we can reach K with a distance of 4, and L with a distance of 8. Vertex Distance from J Previous Vertex PATH J 0 None J K 4 J J → K L 8 J J → L M ∞ - - N ∞ - - P ∞ - - Q ∞ - - R ∞ - - S ∞ - - Explanation: Update from J : Set the distance from J to K as 4 and to L as 8 based on direct connections, updating their paths accordingly. Step 3: Update from K and L Choosing the smallest, not yet permanent vertex, which is K (distance 4): From K to M is 8, updating M to 12 (J → K → M). Now update from L (distance 8): L to P is 1, updating P to 9 (J → L → P). L to N is 7, updating N to 15 (J → L → N). Vertex Distance from J Previous Vertex PATH J 0 None J K 4 J J → K L 8 J J → L M 12 K J → K → M N 15 L J → L → N P 9 L J → L → P Q ∞ - - R ∞ - - S ∞ - - Explanation: Choose K for updates : As K has the smallest distance among all unvisited nodes, update the distance from K to M to 12 (4 from J to K plus 8 from K to M), updating the path. Step 4: Update from P Now P (distance 9) has two updates: P to R is 2, updating R to 11 (J → L → P → R). P to N is 6, but existing distance to N is 15; hence, no update is made. Vertex Distance from J Previous Vertex PATH J 0 None J K 4 J J → K L 8 J J → L M 12 K J → K → M N 15 L J → L → N P 9 L J → L → P Q ∞ - - R 11 P J → L → P → R S ∞ - - Explanation: Choose L for updates : Next, consider L for its connections, updating the distance to P to 9 (8 from J to L plus 1 from L to P) and to N to 15 (8 from J to L plus 7 from L to N), updating their paths. Step 5: Update from R Now R (distance 11) has updates: R to M is 4, but existing distance to M is 12; hence, no update is made. R to Q is 14, updating Q to 25 (J → L → P → R → Q). R to S is 10, updating S to 21 (J → L → P → R → S). Final table after updating from R: Vertex Distance from J Previous Vertex PATH J 0 None J K 4 J J → K L 8 J J → L M 12 K J → K → M N 15 L J → L → N P 9 L J → L → P Q 25 R J → L → P → R → Q R 11 P J → L → P → R S 21 R J → L → P → R → S The shortest path from J to S is "J → L → P → R → S" with a distance of 21. Explanation: Choose P for updates : With P now the closest unvisited node, update the distance to R to 11 (9 from J to L to P plus 2 from P to R), updating the path. Choose R for final updates : Finally, from R, update the distances to Q to 25 (11 from J to L to P to R plus 14 from R to Q) and to S to 21 (11 from J to L to P to R plus 10 from R to S), completing the paths. Final Answer: Dijkstra's algorithm, we calculated the shortest path from vertex J to S in a step-by-step manner. The algorithm successfully found that the shortest route is "J → L → P → R → S" with a total distance of 21.

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