You are an interior decorator, confronted with a dark living room. To lighten the room up, you have n candles and want to build k trees of candles. First, the candles {1, . . . , k} are already put on the floor, to serve as roots of the k trees. You would like to put the remaining n − k candles {k + 1, . . . , n} on top of them to form k trees {T1, · · · , Tk} of candles, where each Ti is rooted at candle i. You can put an arbitrary number of candles on top of the same candle. The cost to putting candle i on top of candle j is C(i, j). We assume that C(i, j) ≥ 0 and C(i, j) = C(j, i)) for i 6= j. No candle can be reused or put on top of itself. You cannot use the candles {1, . . . , k} anymore, as they have already been fixed to the floor.(a) Describe an algorithm to find a solution of minimum cost. (b) Prove the correctness of your algorithm. (c) Give a runtime analysis of your algorithm in the previous part.

a)

Algorithm to find a solution of min. cost

Function:cost(Graph G,Graph G1);

GC --> empty graph

for i in edges E:

if E(i,j) in G:

c(i,j)=c(i,j)

else if E(i,j) in G1:

c(i,j)=-c(i,j)

Function:Mincost(Graph G):

GC=Cost(G,G1)

while(negativecycle(GC)):

Update residal graph(G1)

GC=Cost(G,G1)

mincost=sum of Cij*F(i,j)

return mincost;

Explanation:

a)

1) Start the program

2) Read the no. of edges and vertices and also read the cost of the two nodes.

3) Find the min cost by travelling to the destination i.e.. finding all possible min. cost values.

4) Compare the all possible min.cost values

5) And display the least min. cost

6) Stop the program

b)

Correctness of algorithm

1)Here in these algorithm we are calculating all the possible cases.

2)These algorithm also supports the negative cost.

3)These algorithm occupies more space.

4)Takes less time

5)so,these algorithm provides the cost efficient solution very effectively.

c)

Run Time Analysis

1) While reading the values during the run time the program execution will stop until user provides the values.

2) Based on the User input Output vary.

3) Time consumption and space consumption is depends on the no. of inputs the user is given.