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1 ..- Give an adjacency-li repreenaion for a complee binary ree on verice. Give an equivalen adjacency-marix repreenaion. Aume ha verice are numbered from o a in a binary heap. (Edge are direced from paren o child) / / / / / / / Adjacency Marix:..- he quare of a direced graph G = (V, E) i he graph G = (V, E ) uch ha (u,w) E if and only if for ome v V, boh (u,v) E and (v, w) E. ha i, G conain an edge beween u and w whenever G conain a pah wih exacly wo edge beween u and w. Decribe efficien algorihm for compuing G from G for boh he adjacency-li and adjacency-marix repreenaion of G. Analyze he running ime of your algorihm. G for an adjacency marix: - ompuing G may be done in V ime by marix muliplicaion: for i = o V for j = o V { G [i][j] = ; for k = o V if (g[i][k] == && g[k][j] == ) { G [i][j] == ; break; } }

2 G for an adjacency li: Procedure G-quare (V[G], E[G]) V[G ] ß V[G] for each u V[G] for each v Adj[u] for each w Adj[v] E[G ] ß {(u, w)} E[G ] Run ime = O(V )..- how he d and Π value ha reul from running breadh-fir earch on he direced graph of Figure.(a), uing verex a he ource. -/-/- // //- // // //..- Wha i he running ime of BF if i inpu graph i repreened by an adjacency marix and he algorihm i modified o handle hi form of inpu? Each verex can be explored once and i adjacen verice mu be deermined oo. hi ake Θ(V ) ime.. Do a DF on figure. (p.). laify each edge baed on he DF ree you deermine. / q / F B v w / / / / x z B B y / / r u / /. Find he rongly conneced componen in figure.. From., he fir DF give he li R U Q U W (revere order of urning black)

3 / q / B v w / / B / / x B y / / r u / z / rongly conneced componen are {, w, v}, {q, y, }, {x, z}, {r}, {u}..- how he ordering of verice produced by OPOLOGIAL-OR when i i run on he dag of Figure., under he aumpion of Exercie.-. / / / / m n o p / F q r / / u v w / / / / x y z / / / p n o m r y v x w z u q..- How can he number of rongly conneced componen of a graph change if a new edge i added? he number of rongly conneced componen can be reduced.

4 ..- Profeor Bacon claim ha he algorihm for rongly conneced componen can be implified by uing he original (inead of he ranpoe) graph in he econd deph-fir earch and canning he verice in order of increaing finihing ime. I he profeor correc? onider For hi algorihm, he fir DF will give a li for he econd DF. All verice will be incorrecly repored o be in he ame. For he LR algorihm, he fir DF will give a li for he econd DF. Afer revering edge, he correc {, } and {} will be repored...- Prove ha for any direced graph G, we have ((G ) ) = G. ha i, he ranpoe of he componen graph of G i he ame a he componen graph of G. ince he rongly conneced relaionhip i an equivalence relaion, G and G will alway have he ame rongly conneced componen. If wo verice and are no rongly conneced, hen here i a unique pah from o in G iff here i a unique pah from o in G. A imilar propery will alo hold for he componen graph and ranpoe of he componen graph, i.e. If wo verice and are no rongly conneced in G, hen here i a unique pah from (G,) o (G,) in (G) iff here i a unique pah from (G,) o (G,) in (G ) )...- Le (u, v) be a minimum-weigh edge in a graph G. how ha (u, v) belong o ome minimum panning ree of G. Le A be a ube of ome M uch ha (u, v) A. o chooe an edge o be added o A, all he edge on he cu are conidered and an edge wih lowe weigh i eleced. ince (u, v) i he minimum weigh edge in he graph G, i ge eleced on ome cu.

5 ..- Run Dijkra algorihm on he direced graph of Figure., fir uing verex a he ource and hen uing verex z a he ource. In he yle of Figure., how he d and Π value and he verice in e afer each ieraion of he while loop. Black verice are in he e. he black, hick arrow are he value of Π and he value of d are included are lied inide each node. A. B.. D. E. F.

6 Uing Verex a he ource. A. B.. D.

7 E. F. G.

8 . a. Deermine he raniive cloure of he following Boolean marix by uing Warhall algorihm. () = ()= ()= ()= ()= b. onver he marix o indicae ucceor and ue he verion of Warhall algorihm ha allow pah racing

9 ..- In Figure.(b), which i he flow acro he cu ({, v, v }, {v, v, })? Wha i he capaciy of hi cu? / v / v / / / / / v v / / Ne flow acro he cu i f(, v) + f(v,v) + f(v,v) + f(v,v) + f(v,) = = and i capaciy i = =.- v v / v / v / v v v v v v / / v v / v v / v / v / v v v v / / / v v / v v / / /

10 v v v v v v v v.- Augmening pah??? Augmening pah??? Flow N/W: / / / / / / Reidual N/W: Augmening pah??? Augmening pah???

11 Augmening pah??? Augmening pah??? Final Reidual graph:

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