Question

2699. Modify Graph Edge Weights

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Description

You are given an undirected weighted connected graph containing n nodes labeled from 0 to n - 1, and an integer array edges where edges[i] = [ai, bi, wi] indicates that there is an edge between nodes ai and bi with weight wi.

Some edges have a weight of -1 (wi = -1), while others have a positive weight (wi > 0).

Your task is to modify all edges with a weight of -1 by assigning them positive integer values in the range [1, 2 * 109] so that the shortest distance between the nodes source and destination becomes equal to an integer target. If there are multiple modifications that make the shortest distance between source and destination equal to target, any of them will be considered correct.

Return _an array containing all edges (even unmodified ones) in any order if it is possible to make the shortest distance from _source_ to _destination_ equal to _target_, or an empty array if it's impossible._

Note: You are not allowed to modify the weights of edges with initial positive weights.

 

Example 1:

Input: n = 5, edges = [[4,1,-1],[2,0,-1],[0,3,-1],[4,3,-1]], source = 0, destination = 1, target = 5
Output: [[4,1,1],[2,0,1],[0,3,3],[4,3,1]]
Explanation: The graph above shows a possible modification to the edges, making the distance from 0 to 1 equal to 5.

Example 2:

Input: n = 3, edges = [[0,1,-1],[0,2,5]], source = 0, destination = 2, target = 6
Output: []
Explanation: The graph above contains the initial edges. It is not possible to make the distance from 0 to 2 equal to 6 by modifying the edge with weight -1. So, an empty array is returned.

Example 3:

Input: n = 4, edges = [[1,0,4],[1,2,3],[2,3,5],[0,3,-1]], source = 0, destination = 2, target = 6
Output: [[1,0,4],[1,2,3],[2,3,5],[0,3,1]]
Explanation: The graph above shows a modified graph having the shortest distance from 0 to 2 as 6.

 

Constraints:

• 1 <= n <= 100
• 1 <= edges.length <= n * (n - 1) / 2
• edges[i].length == 3
• 0 <= ai, bi < n
• wi = -1 or 1 <= wi <= 107
• ai != bi
• 0 <= source, destination < n
• source != destination
• 1 <= target <= 109
• The graph is connected, and there are no self-loops or repeated edges

Solutions

Solution 1: Shortest Path (Dijkstra's Algorithm)

First, we ignore the edges with a weight of $-1$ and use Dijkstra's algorithm to find the shortest distance $d$ from $source$ to $destination$.

• If $d < target$, it means there is a shortest path composed entirely of positive weight edges. No matter how we modify the edges with a weight of $-1$, we cannot make the shortest distance from $source$ to $destination$ equal to $target$. Therefore, there is no modification scheme that satisfies the problem, and we can return an empty array.

• If $d = target$, it means there is a shortest path composed entirely of positive weight edges, and its length is $target$. In this case, we can modify all edges with a weight of $-1$ to the maximum value $2 \times 10^9$.

• If $d > target$, we can try to add an edge with a weight of $-1$ to the graph, set the weight of the edge to $1$, and then use Dijkstra's algorithm again to find the shortest distance $d$ from $source$ to $destination$.

  • If the shortest distance $d \leq target$, it means that after adding this edge, the shortest path can be shortened, and the shortest path must pass through this edge. Then we only need to change the weight of this edge to $target-d+1$ to make the shortest path equal to $target$. Then we can modify the remaining edges with a weight of $-1$ to the maximum value $2 \times 10^9$.
  • If the shortest distance $d > target$, it means that after adding this edge, the shortest path will not be shortened. Then we greedily keep the weight of this edge as $-1$ and continue to try to add the remaining edges with a weight of $-1$.

The time complexity is $O(n^3)$, and the space complexity is $O(n^2)$, where $n$ is the number of points in the graph.

class Solution:
  def modifiedGraphEdges(
    self, n: int, edges: List[List[int]], source: int, destination: int, target: int
  ) -> List[List[int]]:
    def dijkstra(edges: List[List[int]]) -> int:
      g = [[inf] * n for _ in range(n)]
      for a, b, w in edges:
        if w == -1:
          continue
        g[a][b] = g[b][a] = w
      dist = [inf] * n
      dist[source] = 0
      vis = [False] * n
      for _ in range(n):
        k = -1
        for j in range(n):
          if not vis[j] and (k == -1 or dist[k] > dist[j]):
            k = j
        vis[k] = True
        for j in range(n):
          dist[j] = min(dist[j], dist[k] + g[k][j])
      return dist[destination]

    inf = 2 * 10**9
    d = dijkstra(edges)
    if d < target:
      return []
    ok = d == target
    for e in edges:
      if e[2] > 0:
        continue
      if ok:
        e[2] = inf
        continue
      e[2] = 1
      d = dijkstra(edges)
      if d <= target:
        ok = True
        e[2] += target - d
    return edges if ok else []

class Solution {
  private final int inf = 2000000000;

  public int[][] modifiedGraphEdges(
    int n, int[][] edges, int source, int destination, int target) {
    long d = dijkstra(edges, n, source, destination);
    if (d < target) {
      return new int[0][];
    }
    boolean ok = d == target;
    for (var e : edges) {
      if (e[2] > 0) {
        continue;
      }
      if (ok) {
        e[2] = inf;
        continue;
      }
      e[2] = 1;
      d = dijkstra(edges, n, source, destination);
      if (d <= target) {
        ok = true;
        e[2] += target - d;
      }
    }
    return ok ? edges : new int[0][];
  }

  private long dijkstra(int[][] edges, int n, int src, int dest) {
    int[][] g = new int[n][n];
    long[] dist = new long[n];
    Arrays.fill(dist, inf);
    dist[src] = 0;
    for (var f : g) {
      Arrays.fill(f, inf);
    }
    for (var e : edges) {
      int a = e[0], b = e[1], w = e[2];
      if (w == -1) {
        continue;
      }
      g[a][b] = w;
      g[b][a] = w;
    }
    boolean[] vis = new boolean[n];
    for (int i = 0; i < n; ++i) {
      int k = -1;
      for (int j = 0; j < n; ++j) {
        if (!vis[j] && (k == -1 || dist[k] > dist[j])) {
          k = j;
        }
      }
      vis[k] = true;
      for (int j = 0; j < n; ++j) {
        dist[j] = Math.min(dist[j], dist[k] + g[k][j]);
      }
    }
    return dist[dest];
  }
}

using ll = long long;
const int inf = 2e9;

class Solution {
public:
  vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
    ll d = dijkstra(edges, n, source, destination);
    if (d < target) {
      return {};
    }
    bool ok = d == target;
    for (auto& e : edges) {
      if (e[2] > 0) {
        continue;
      }
      if (ok) {
        e[2] = inf;
        continue;
      }
      e[2] = 1;
      d = dijkstra(edges, n, source, destination);
      if (d <= target) {
        ok = true;
        e[2] += target - d;
      }
    }
    return ok ? edges : vector<vector<int>>{};
  }

  ll dijkstra(vector<vector<int>>& edges, int n, int src, int dest) {
    ll g[n][n];
    ll dist[n];
    bool vis[n];
    for (int i = 0; i < n; ++i) {
      fill(g[i], g[i] + n, inf);
      dist[i] = inf;
      vis[i] = false;
    }
    dist[src] = 0;
    for (auto& e : edges) {
      int a = e[0], b = e[1], w = e[2];
      if (w == -1) {
        continue;
      }
      g[a][b] = w;
      g[b][a] = w;
    }
    for (int i = 0; i < n; ++i) {
      int k = -1;
      for (int j = 0; j < n; ++j) {
        if (!vis[j] && (k == -1 || dist[j] < dist[k])) {
          k = j;
        }
      }
      vis[k] = true;
      for (int j = 0; j < n; ++j) {
        dist[j] = min(dist[j], dist[k] + g[k][j]);
      }
    }
    return dist[dest];
  }
};

func modifiedGraphEdges(n int, edges [][]int, source int, destination int, target int) [][]int {
  const inf int = 2e9
  dijkstra := func(edges [][]int) int {
    g := make([][]int, n)
    dist := make([]int, n)
    vis := make([]bool, n)
    for i := range g {
      g[i] = make([]int, n)
      for j := range g[i] {
        g[i][j] = inf
      }
      dist[i] = inf
    }
    dist[source] = 0
    for _, e := range edges {
      a, b, w := e[0], e[1], e[2]
      if w == -1 {
        continue
      }
      g[a][b], g[b][a] = w, w
    }
    for i := 0; i < n; i++ {
      k := -1
      for j := 0; j < n; j++ {
        if !vis[j] && (k == -1 || dist[j] < dist[k]) {
          k = j
        }
      }
      vis[k] = true
      for j := 0; j < n; j++ {
        dist[j] = min(dist[j], dist[k]+g[k][j])
      }
    }
    return dist[destination]
  }
  d := dijkstra(edges)
  if d < target {
    return [][]int{}
  }
  ok := d == target
  for _, e := range edges {
    if e[2] > 0 {
      continue
    }
    if ok {
      e[2] = inf
      continue
    }
    e[2] = 1
    d := dijkstra(edges)
    if d <= target {
      ok = true
      e[2] += target - d
    }
  }
  if ok {
    return edges
  }
  return [][]int{}
}

function modifiedGraphEdges(
  n: number,
  edges: number[][],
  source: number,
  destination: number,
  target: number,
): number[][] {
  const inf = 2e9;
  const dijkstra = (edges: number[][]): number => {
    const g: number[][] = Array(n)
      .fill(0)
      .map(() => Array(n).fill(inf));
    const dist: number[] = Array(n).fill(inf);
    const vis: boolean[] = Array(n).fill(false);
    for (const [a, b, w] of edges) {
      if (w === -1) {
        continue;
      }
      g[a][b] = w;
      g[b][a] = w;
    }
    dist[source] = 0;
    for (let i = 0; i < n; ++i) {
      let k = -1;
      for (let j = 0; j < n; ++j) {
        if (!vis[j] && (k === -1 || dist[j] < dist[k])) {
          k = j;
        }
      }
      vis[k] = true;
      for (let j = 0; j < n; ++j) {
        dist[j] = Math.min(dist[j], dist[k] + g[k][j]);
      }
    }
    return dist[destination];
  };
  let d = dijkstra(edges);
  if (d < target) {
    return [];
  }
  let ok = d === target;
  for (const e of edges) {
    if (e[2] > 0) {
      continue;
    }
    if (ok) {
      e[2] = inf;
      continue;
    }
    e[2] = 1;
    d = dijkstra(edges);
    if (d <= target) {
      ok = true;
      e[2] += target - d;
    }
  }
  return ok ? edges : [];
}