Color tree with minimum colors such that colors of edges incident to a vertex are different
Given a tree with N nodes. The task is to color the tree with the minimum number of colors(K) such that the colors of the edges incident to a vertex are different. Print K in first-line and then in next line print N – 1 space-separated integer represents the colors of the edges.
Examples:
Input: N = 3, edges[][] = {{0, 1}, {1, 2}}
0
/
1
/
2
Output:
2
1 2
0
/ (1)
1
/ (2)
2
Input: N = 8, edges[][] = {{0, 1}, {1, 2},
{1, 3}, {1, 4},
{3, 6}, {4, 5},
{5, 7}}
0
/
1
/ \ \
2 3 4
/ \
6 5
\
7
Output:
4
1 2 3 4 1 1 2
Approach: First, let’s think about the minimum number of colors K. For every vertex v, deg(v) ≤ K should meet (where deg(v) denotes the degree of vertex v). In fact, there exists a vertex with all K different colors. First, choose a vertex and let it be the root, thus T will be a rooted tree. Perform a breadth-first search from the root. For each vertex, determine the colors of edges between its children one by one. For each edge, use the color with the minimum index among those which are not used as colors of edges whose one of endpoints is the current vertex. Then each index of color does not exceed K.
Below is the implementation of the above approach:
CPP
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std; // Add an edge between the vertexes void add_edge(vector<vector<int> >& gr, int x, int y, vector<pair<int, int> >& edges) { gr[x].push_back(y); gr[y].push_back(x); edges.push_back({ x, y }); } // Function to color the tree with minimum // number of colors such that the colors of // the edges incident to a vertex are different int color_tree(int n, vector<vector<int> >& gr, vector<pair<int, int> >& edges) { // To store the minimum colors int K = 0; // To store color of the edges map<pair<int, int>, int> color; // Color of edge between its parent vector<int> cs(n, 0); // To check if the vertex is // visited or not vector<int> used(n, 0); queue<int> que; used[0] = 1; que.emplace(0); while (!que.empty()) { // Take first element of the queue int v = que.front(); que.pop(); // Take the possible value of K if (K < (int)gr[v].size()) K = gr[v].size(); // Current color int cur = 1; for (int u : gr[v]) { // If vertex is already visited if (used[u]) continue; // If the color is similar // to it's parent if (cur == cs[v]) cur++; // Assign the color cs[u] = color[make_pair(u, v)] = color[make_pair(v, u)] = cur++; // Mark it visited used[u] = 1; // Push into the queue que.emplace(u); } } // Print the minimum required colors cout << K << endl; // Print the edge colors for (auto p : edges) cout << color[p] << " "; } // Driver code int main() { int n = 8; vector<vector<int> > gr(n); vector<pair<int, int> > edges; // Add edges add_edge(gr, 0, 1, edges); add_edge(gr, 1, 2, edges); add_edge(gr, 1, 3, edges); add_edge(gr, 1, 4, edges); add_edge(gr, 3, 6, edges); add_edge(gr, 4, 5, edges); add_edge(gr, 5, 7, edges); // Function call color_tree(n, gr, edges); return 0; } |
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Python
# Python3 implementation of the approach from collections import deque as queue gr = [[] for i in range(100)] edges = [] # Add an edge between the vertexes def add_edge(x, y): gr[x].append(y) gr[y].append(x) edges.append([x, y]) # Function to color the tree with minimum # number of colors such that the colors of # the edges incident to a vertex are different def color_tree(n): # To store the minimum colors K = 0 # To store color of the edges color = dict() # Color of edge between its parent cs = [0] * (n) # To check if the vertex is # visited or not used = [0] * (n) que = queue() used[0] = 1 que.append(0) while (len(que) > 0): # Take first element of the queue v = que.popleft() # Take the possible value of K if (K < len(gr[v])): K = len(gr[v]) # Current color cur = 1 for u in gr[v]: # If vertex is already visited if (used[u]): continue # If the color is similar # to it's parent if (cur == cs[v]): cur += 1 # Assign the color cs[u] = cur color[(u, v)] = color[(v, u)] = cur cur += 1 # Mark it visited used[u] = 1 # Push into the queue que.append(u) # Print minimum required colors print(K) # Print edge colors for p in edges: i = (p[0], p[1]) print(color[i], end = " ") # Driver code n = 8 # Add edges add_edge(0, 1) add_edge(1, 2) add_edge(1, 3) add_edge(1, 4) add_edge(3, 6) add_edge(4, 5) add_edge(5, 7) # Function call color_tree(n) # This code is contributed by mohit kumar 29 |
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Output:
4 1 2 3 4 1 1 2
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Improved By : mohit kumar 29
