Question

1. (25) [Single-source shortest-path: algorithm tracing] Show the tracing of Dijkstra's shortest path search algorithm on the weighted directed graph shown below. Do the tracing it twice, first starting with the nodes and, second, starting with the node z. For each tracing, each time the shortest path to a new node is determined, show the graph with the shortest path to the node clearly marked and show inside the node the shortest distance to the node. 6 3 2. 18 6 Z 2. (25) [Spanning tree: proof of properties] Consider an undirected connected graph G = (V, E) and a connected subgraph G = (V, E) of G where E' is a subset of E. Prove that G' is a spanning tree of G if and only if G is acyclic. TAs: Make sure there are both If part and Only-if part of the proof. The proof can be built on the properties that a tree has one fewer edges than the vertices and that a tree has no cycle in it.

please answer one of the two

Answer 1

Algorithm Step 1: Take distance Vector Carray) and Eritilize it to Enfinity previous vector and Critialize it to t Take Take an another Vector called visited and initiale it to Zoo Step 2 : Update the distance of source Zero with minimum distance which Step 3 : Find the verten not intested Step 4: Make visited of the verten with minimum dritance as Step 5:- if distance [i]> distance [u] + weight (le, i) then distance [1] = distance [u] + weight (4,7) previous [1] - u stop 6 :- Repeat the Step 3 to 5 times

Single-source shortest path: algorithm traving. b 6 3 H 8 2 5 3. 6 е с c & starting with a Distance Veetoh to Centralizing Dimance Vector to infinity (0) 00 00 Distance b c d е e to Initial previous vector previous a d Initialize visined vector to (zuo) visited. Lo o o e b d Source distance 000 8 000 Initialize Distance of souru to o Distance [a] = a со b cd e

find the veater with distance minimum = (verter with minimum distance) Condition to update dretance. zi if dist Li] > dist [m] + voright (1) weight (u, i) then dist [?] = dict [u] + weight (u) check for all the vertius. prev[i] = u if dutence [6] > diet [a] + weilight (a, b) <> 0 + 3 distance o 3 co b с c a d е distance [b]= 3 forevious [6] - a previous a b с d e distancu [c] > distance [a] + weight (a, c) os > Ot 5 distance 3 55 So dittance [] - 5 previous [c]=a 6 с c e prevlous b c d e $ distance [d] > die tana [a] + Weight (and)" 000+ if distance [e] > distanu [a] + weight (a,e) o > 0 + 80 vighted 1 O DO b c d е e а a

Next minimum ☺ 3 lub % distance [c] > distance (6] + weight (b, c) 5 > 3+1 update drerance [c] distance [c] o distance [6] + weight (bc) distance [c dish 3 H brev [c] ab prev 0 es a b d e b - a b c d e if distance [d] > distance [b] + weight (6, d) 00> 3+ 6 7 distance 3. 4 9 distance [d] =9 Brev [d] = b 0 Q 6 с c d e prev a b ь b-1 c d e L y deelance [e] > destance [b] + weight Cb, e) > 3+ s x visited Tol O a b c d e Find minimum min-4 if destane [d] > distance [c] + weight (c,d) 9 4 + 4 distance [d] =8 [update blev [d] 2c

of distance [e] > distance [e] + aright (c, e) 4t 6 distance 03 4 8 10 destance [e] to e a 5 с c d prev [e] ec prev -1 а a b с с c a b c d e visited 1 1 0 d e Minimum 8 u=d of distance [e] > distance [d] it weight (d, e) 10 > 8+ 2 distance 3 4 2 TO b ь е e d е e previous о a b с c C a 5 d visited + 1 1 1 0 b c d e a Shortest path distance od Oo 2 ab 3 a b c a b c d 8 6 o e abcte 10

2 Starting See the figure 1 for reference distance Os Od a b с c d e prev -1-1 b c d e 6. visited o o 0 O b c d e fource u = e distance 0 00 00 00 distance [source] =0 a b d e Minimum distance to Uue % distance [a] > distance [e] + weight (e,a) as >ot 3 distance [a] previous [a] = e prev = 3 der 3 O [update distance 5 e d e e - + b d e f distance [6]> dieteenu [e]+ weight (eb) >0+ X it distance [c] > destanu [e] tweght (es) >O+ if desitance [d] > distance [d] + weight (ed) a> 0+8

destance 3 O distance [d] =8 prev [d]=e a b c с d e previous е e 1 е e d au b с e visited ololo oli a b с d e min = 3 u= a if distance [b] > distance [a] + weight (a, b) > 3 + 3 distance [6]=6 frer [6] za il distance [c] > distance [a] + weight (a, c) oo > 3 + 5 distance [c]=8 pares [c] = a of distance [a] > distance (a]t weight (as d) 8 > 3+ Box distance ش) 6. 8 8 . a 5 d e previous е ala a b c e d e visited t a b c d e

Minimum u= b distance [d] > dutancu (6] + weight (6, d) 8 > 6+ 6 y distance [c] > distance [ 6] + + auright (b, c) 8 > 6+1 distance [c frer [c]=b distance 316 H80 b c d e previous е. a b ь --- е e d a b с е visited 1 b е e d е e Minimum it. lloc distance [d] & distance (c]t weight (ed) 8 > I+2 destance 3 6 7 O prev е e b е e visited 11 lo b Ь d e

b Shortest path distanu 3 e a 5 e → ab eta be ed 8 Z