Maximal Network Rank — Python

# Time:  O(m + n + k^2), k is the number of values greater or equal to top2
# Space: O(m + n)

# optimized from Solution2 with counting sort
class Solution(object):
    def maximalNetworkRank(self, n, roads):
        """
        :type n: int
        :type roads: List[List[int]]
        :rtype: int
        """
        MAX_N = 100
        MAX_NUM = MAX_N-1
        def counting_sort(arr, key=lambda x:x, reverse=False):  # Time: O(n), Space: O(n)
            count = [0]*(MAX_NUM+1)
            for x in arr:
                count[key(x)] += 1
            for i in xrange(1, len(count)):
                count[i] += count[i-1]
            result = [0]*len(arr)
            if not reverse:
                for x in reversed(arr):  # stable sort
                    count[key(x)] -= 1
                    result[count[key(x)]] = x
            else:
                for x in arr:  # stable sort
                    count[key(x)] -= 1
                    result[count[key(x)]] = x
                result.reverse()
            return result

        degree = [0]*n
        adj = collections.defaultdict(set)
        for a, b in roads:
            degree[a] += 1
            degree[b] += 1
            adj[a].add(b)
            adj[b].add(a)
        sorted_idx = counting_sort(xrange(n), key=lambda x:degree[x], reverse=True)
        m = 2
        while m < n:
            if degree[sorted_idx[m]] != degree[sorted_idx[1]]:
                break
            m += 1
        result = degree[sorted_idx[0]] + degree[sorted_idx[1]] - 1  # at least sum(top2 values) - 1
        for i in xrange(m-1):  # only need to check pairs of top2 values
            for j in xrange(i+1, m):
                if degree[sorted_idx[i]]+degree[sorted_idx[j]]-int(sorted_idx[i] in adj and sorted_idx[j] in adj[sorted_idx[i]]) > result:  # if equal to ideal sum of top2 values, break
                    return degree[sorted_idx[i]]+degree[sorted_idx[j]]-int(sorted_idx[i] in adj and sorted_idx[j] in adj[sorted_idx[i]])                                                 
        return result


# Time:  O(m + nlogn + k^2), k is the number of values greater or equal to top2
# Space: O(m + n)
import collections


# optimized from Solution3
class Solution2(object):
    def maximalNetworkRank(self, n, roads):
        """
        :type n: int
        :type roads: List[List[int]]
        :rtype: int
        """
        degree = [0]*n
        adj = collections.defaultdict(set)
        for a, b in roads:
            degree[a] += 1
            degree[b] += 1
            adj[a].add(b)
            adj[b].add(a)
        sorted_idx = range(n)
        sorted_idx.sort(key=lambda x:-degree[x])
        m = 2
        while m < n:
            if degree[sorted_idx[m]] != degree[sorted_idx[1]]:
                break
            m += 1
        result = degree[sorted_idx[0]] + degree[sorted_idx[1]] - 1  # at least sum(top2 values) - 1
        for i in xrange(m-1):  # only need to check pairs of top2 values
            for j in xrange(i+1, m):
                if degree[sorted_idx[i]]+degree[sorted_idx[j]]-int(sorted_idx[i] in adj and sorted_idx[j] in adj[sorted_idx[i]]) > result:  # if equal to ideal sum of top2 values, break
                    return degree[sorted_idx[i]]+degree[sorted_idx[j]]-int(sorted_idx[i] in adj and sorted_idx[j] in adj[sorted_idx[i]])                                                 
        return result


# Time:  O(m + n^2)
# Space: O(m + n)
import collections


class Solution3(object):
    def maximalNetworkRank(self, n, roads):
        """
        :type n: int
        :type roads: List[List[int]]
        :rtype: int
        """
        degree = [0]*n
        adj = collections.defaultdict(set)
        for a, b in roads:
            degree[a] += 1
            degree[b] += 1
            adj[a].add(b)
            adj[b].add(a)
        result = 0
        for i in xrange(n-1):
            for j in xrange(i+1, n):
                result = max(result, degree[i]+degree[j]-int(i in adj and j in adj[i]))
        return result