# Posts published in “Graph”

There are n cities numbered from 0 to n-1. Given the array edges where edges[i] = [fromi, toi, weighti] represents a bidirectional and weighted edge between cities fromi and toi, and given the integer distanceThreshold.

Return the city with the smallest numberof cities that are reachable through some path and whose distance is at most distanceThreshold, If there are multiple such cities, return the city with the greatest number.

Notice that the distance of a path connecting cities i and j is equal to the sum of the edges’ weights along that path.

Example 1:

Input: n = 4, edges = [[0,1,3],[1,2,1],[1,3,4],[2,3,1]], distanceThreshold = 4
Output: 3
Explanation: The figure above describes the graph.
The neighboring cities at a distanceThreshold = 4 for each city are:
City 0 -> [City 1, City 2]
City 1 -> [City 0, City 2, City 3]
City 2 -> [City 0, City 1, City 3]
City 3 -> [City 1, City 2]
Cities 0 and 3 have 2 neighboring cities at a distanceThreshold = 4, but we have to return city 3 since it has the greatest number.


Example 2:

Input: n = 5, edges = [[0,1,2],[0,4,8],[1,2,3],[1,4,2],[2,3,1],[3,4,1]], distanceThreshold = 2
Output: 0
Explanation: The figure above describes the graph.
The neighboring cities at a distanceThreshold = 2 for each city are:
City 0 -> [City 1]
City 1 -> [City 0, City 4]
City 2 -> [City 3, City 4]
City 3 -> [City 2, City 4]
City 4 -> [City 1, City 2, City 3]
The city 0 has 1 neighboring city at a distanceThreshold = 2.


Constraints:

• 2 <= n <= 100
• 1 <= edges.length <= n * (n - 1) / 2
• edges[i].length == 3
• 0 <= fromi < toi < n
• 1 <= weighti, distanceThreshold <= 10^4
• All pairs (fromi, toi) are distinct.

## Solution: Floyd-Warshall

All pair shortest path

Time complexity: O(n^3)
Space complexity: O(n^2)

## C++

There are n computers numbered from 0 to n-1 connected by ethernet cables connections forming a network where connections[i] = [a, b] represents a connection between computers a and b. Any computer can reach any other computer directly or indirectly through the network.

Given an initial computer network connections. You can extract certain cables between two directly connected computers, and place them between any pair of disconnected computers to make them directly connected. Return the minimum number of times you need to do this in order to make all the computers connected. If it’s not possible, return -1.

Example 1:

Input: n = 4, connections = [[0,1],[0,2],[1,2]]
Output: 1
Explanation: Remove cable between computer 1 and 2 and place between computers 1 and 3.


Example 2:

Input: n = 6, connections = [[0,1],[0,2],[0,3],[1,2],[1,3]]
Output: 2


Example 3:

Input: n = 6, connections = [[0,1],[0,2],[0,3],[1,2]]
Output: -1
Explanation: There are not enough cables.


Example 4:

Input: n = 5, connections = [[0,1],[0,2],[3,4],[2,3]]
Output: 0


Constraints:

• 1 <= n <= 10^5
• 1 <= connections.length <= min(n*(n-1)/2, 10^5)
• connections[i].length == 2
• 0 <= connections[i], connections[i] < n
• connections[i] != connections[i]
• There are no repeated connections.
• No two computers are connected by more than one cable.

## Solution 1: Union-Find

Time complexity: O(V+E)
Space complexity: O(V)

## Solution 2: DFS

Time complexity: O(V+E)
Space complexity: O(V+E)

## C++

You are given a string s, and an array of pairs of indices in the string pairs where pairs[i] = [a, b] indicates 2 indices(0-indexed) of the string.

You can swap the characters at any pair of indices in the given pairs any number of times.

Return the lexicographically smallest string that s can be changed to after using the swaps.

Example 1:

Input: s = "dcab", pairs = [[0,3],[1,2]]
Output: "bacd"
Explaination:
Swap s and s, s = "bcad"
Swap s and s, s = "bacd"


Example 2:

Input: s = "dcab", pairs = [[0,3],[1,2],[0,2]]
Output: "abcd"
Explaination:
Swap s and s, s = "bcad"
Swap s and s, s = "acbd"
Swap s and s, s = "abcd"

Example 3:

Input: s = "cba", pairs = [[0,1],[1,2]]
Output: "abc"
Explaination:
Swap s and s, s = "bca"
Swap s and s, s = "bac"
Swap s and s, s = "abc"



Constraints:

• 1 <= s.length <= 10^5
• 0 <= pairs.length <= 10^5
• 0 <= pairs[i], pairs[i] < s.length
• s only contains lower case English letters.

## Solution: Connected Components

Use DFS / Union-Find to find all the connected components of swapable indices. For each connected components (index group), extract the subsequence of corresponding chars as a string, sort it and put it back to the original string in the same location.

e.g. s = “dcab”, pairs = [[0,3],[1,2]]
There are two connected components: {0,3}, {1,2}
subsequences:
1. 0,3 “db”, sorted: “bd”
2. 1,2 “ca”, sorted: “ac”
0 => b
1 => a
2 => c
3 => d
final = “bacd”

Time complexity: DFS: O(nlogn + k*(V+E)), Union-Find: O(nlogn + V+E)
Space complexity: O(n)

## C++/Union-Find

There are n servers numbered from 0 to n-1 connected by undirected server-to-server connections forming a network where connections[i] = [a, b] represents a connection between servers a and b. Any server can reach any other server directly or indirectly through the network.

critical connection is a connection that, if removed, will make some server unable to reach some other server.

Return all critical connections in the network in any order.

Example 1:

Input: n = 4, connections = [[0,1],[1,2],[2,0],[1,3]]
Output: [[1,3]]
Explanation: [[3,1]] is also accepted.


Constraints:

• 1 <= n <= 10^5
• n-1 <= connections.length <= 10^5
• connections[i] != connections[i]
• There are no repeated connections.

## Solution: Tarjan

Time complexity: O(v+e)
Space complexity: O(v+e)

## C++

Given an N x N grid containing only values 0 and 1, where 0 represents water and 1 represents land, find a water cell such that its distance to the nearest land cell is maximized and return the distance.

The distance used in this problem is the Manhattan distance: the distance between two cells (x0, y0) and (x1, y1)is |x0 - x1| + |y0 - y1|.

If no land or water exists in the grid, return -1.

Example 1:

Input: [[1,0,1],[0,0,0],[1,0,1]]
Output: 2
Explanation:
The cell (1, 1) is as far as possible from all the land with distance 2.


Example 2:

Input: [[1,0,0],[0,0,0],[0,0,0]]
Output: 4
Explanation:
The cell (2, 2) is as far as possible from all the land with distance 4.


Note:

1. 1 <= grid.length == grid.length <= 100
2. grid[i][j] is 0 or 1

## Solution: BFS

Put all land cells into a queue as source nodes and BFS for water cells, the last expanded one will be the farthest.

Time compleixty: O(n^2)
Space complexity: O(n^2)

## C++

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