# Evaluate the complexity of this graph coloring algorithm

• BRN
In summary, the conversation discusses the evaluation of the complexity of a graph coloring algorithm. The algorithm uses the adjacency matrix to determine the nodes to color and follows a set of steps to color them. The equation of recurrence for the worst case is determined to be $$T(n)=(1+\sqrt n)n^3+\sqrt n n^2 + n$$ and it can be concluded that the algorithm has a complexity of ##O(n^3)##.
BRN
Thread moved from the technical forums to the schoolwork forums
Hello everybody,
I should evaluate the complexity of this graph coloring algorithm.
To do this, I use the adjacency matrix in which the graph nodes are the elements on the diagonal, while the elements outside the diagonal indicate if a node is adjacent to another ##(A_{i,j} = 1)## or not Adjacent ##(A_{i,j} = 0)##.
The algorithm fulfill these steps:
1. - The number of coloring nodes is counted (NoDesNotCol function);
2. - For each not colored node, calculating the degree of saturation and note the row index of the node with maximum saturation degree (satDeg function);
3. - In case the degree of saturation is equal to that of another node, then consider the maximum node degree (nodeDeg function);
4. - The row index found is passed as a argument to the function that colores the node (setColor function);
5. - All this is repeated until all the nodes are colored.
Code:
int nColoredNodes = 0.;
while( nColoredNodes < nodesToColor)
{
int max = -1, index = -1;
for (int i = 0; i < adjMat.size(); ++i)
{
{
int sDegree = satDeg(i, adjMat, matOrder);
if (sDegree > max) { max = sDegree; index = i ; }
else if (sDegree == max)
{
}
}
}
nColoredNodes++;
}

Calculating the equation of recurrence considering that:
• - noesnotcol is ##O(n)##;
• - satDeg is ##O(n^2)##;
• - nodeDeg is ##O(n)##;
• - setColor is ##O(n^2)##;
in case ##n=1##:
the cost is 1 for NodesNotCol, 1 for Satdeg, 1 for Nodedeg, 1 for SetColor. Then $$T(1) = 1 + 1 * (1 * (1 + 1) +1) = 4$$

In case ##n>1##:
$$T(n) = n + n (n (n + n^2) + n^2) = n^4 + 2n^3 + n$$

So this algorithm has complexity ##O(n^4)##? Is right what I did?

OK, some correction:
setDeg is ##O(\sqrt n n)##
setColor is ##O(\sqrt n n)##

The equation of recurrence should be

##
T(n)=\left\{\begin{matrix}
4 & n=1\\
n+n(n(n+\sqrt n n)+\sqrt n n)& n > 1
\end{matrix}\right.
##

On this I'm pretty sure. Now the question is:
With an equation of recurrence for the worst case

##T(n)=(1+\sqrt n)n^3+\sqrt n n^2 + n##

Is it possible to say that asymptotically is ##O(n^3)##?

## 1. What is a graph coloring algorithm?

A graph coloring algorithm is a method used to assign colors to the vertices of a graph such that no adjacent vertices have the same color.

## 2. How is the complexity of a graph coloring algorithm evaluated?

The complexity of a graph coloring algorithm is evaluated by analyzing the time and space requirements for the algorithm to run, as well as the number of colors needed to properly color the graph.

## 3. What factors contribute to the complexity of a graph coloring algorithm?

The complexity of a graph coloring algorithm is influenced by the size and structure of the graph, the chosen coloring strategy, and the efficiency of the algorithm implementation.

## 4. How does the complexity of a graph coloring algorithm impact its performance?

The complexity of a graph coloring algorithm can greatly affect its performance, with higher complexity leading to longer run times and potentially requiring more resources to complete.

## 5. Are there any known efficient graph coloring algorithms?

Yes, there are several efficient graph coloring algorithms such as the Greedy Coloring Algorithm, the Welsh-Powell Algorithm, and the Recursive Largest First Algorithm. However, the most efficient algorithm may vary depending on the specific graph being colored.

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