Segment tree for graph problems

Правка en5, от ASHWANTH_K, 2024-11-01 08:31:48
  • Recently I came across a new idea to solve graph-related problems using segment trees, I would like to discuss the same with a problem.

Problem:

Given a graph $$$G$$$ with $$$N$$$ nodes and many edges (around $$$O(N^2)$$$), the goal is to perform Dijkstra algorithm on this dense graph and find out the shortest path from node 1 to all other nodes.

The edges are given in a compressed format. The input follows $$$M$$$ lines. Each of the M lines consists of 4 integers U Lx Rx C meaning there are edges from node U to all other nodes in range [Lx , Rx] with cost C.

Example:

For N = 6 , the edge-group U = 1 , Lx = 3 , Rx = 5 , C = 10lLooks like:

Constraints:

  • $$$1 \le N \le 10^5$$$
  • $$$1 \le M \le 10^5$$$
  • $$$1 \le U \le N$$$
  • $$$1 \le Lx \le Rx \le N$$$
  • $$$1 \le C \le 10^9$$$

Initial Thought Process:

  • The main problem faced here is that in the worst case, we have plenty of edges $$$O(N^2)$$$, performing Dijkstra in this dense graph would lead to time complexity $$$O(N^2 log(N))$$$ which would lead to TLE. Also, it is not possible to store all edges in adjacency list, which would lead to MLE.
  • So we have to seek out some ways to reduce the number of edges so that time complexity can be improved.
  • Our solution proceeds like this, we first try to convert this dense graph to a sparse graph with less number of edges and then perform Dijkstra Algorithm to find our answer.
  • Since the input of edges is given in compressed format, there might be some ways to represent a group of edges as a single edge so that we can reduce the number of edges and solve our problem.

Solution: Segment Tree as a structure:

  • One main observation in this problem is that we add edges from node U to a range of nodes [Lx, Rx]. This gives an intuition that we deal with ranges and we can use segment trees to solve our problem.
  • First, we build a segment tree $$$G'$$$ with $$$N$$$ leaf nodes, ($$$2N$$$ nodes in total). These N leaf nodes represent the N nodes of the original graph $$$G$$$.
    Example N = 8
  • It is a well known fact that any range [Lx , Rx] can be covered by $$$O(log(N))$$$ nodes in a segment tree. So for every edge-group U Lx Rx C, we add edges from node U to these $$$O(log(N))$$$ nodes with cost C.

Example: N = 8, U = 1, [Lx,Rx] = [3,8], C = 10

История

 
 
 
 
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en10 Английский ASHWANTH_K 2024-11-01 12:05:17 0 (published)
en9 Английский ASHWANTH_K 2024-11-01 12:04:45 805 Tiny change: 'ch of the M lines con' -> 'ch of the $M$ lines con'
en8 Английский ASHWANTH_K 2024-11-01 09:00:59 51 Tiny change: 'ke: \n[Imgur](https://' -> 'ke: \n[ ](https://'
en7 Английский ASHWANTH_K 2024-11-01 08:44:57 96 Tiny change: ' \n ![ ](https://' -> ' \n ![](https://'
en6 Английский ASHWANTH_K 2024-11-01 08:37:06 454
en5 Английский ASHWANTH_K 2024-11-01 08:31:48 486
en4 Английский ASHWANTH_K 2024-11-01 08:26:10 444
en3 Английский ASHWANTH_K 2024-11-01 08:16:41 967 Tiny change: ' \n \nExample: \n \n' -> ' \n \n**Example:** \n \n'
en2 Английский ASHWANTH_K 2024-11-01 08:04:24 508
en1 Английский ASHWANTH_K 2024-11-01 07:58:37 187 Initial revision (saved to drafts)