• 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 $$$O(N^2)$$$), Goal is to perform the Dijkstra algorithm on this dense graph and find out the shortest path cost 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. Edges are uni-directional.

Example:

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

Constraints:

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

Do spend some time thinking about this problem, before proceeding to the solution.

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 give TLE. Also, it is not possible to store all edges in the adjacency list, giving MLE.

• So we need to seek out some ways to reduce the number of edges so that time complexity can be improved.

• Our solution proceeds like this, first we try to convert this dense graph to a sparse graph with less number of edges and then perform the Dijkstra Algorithm to find our answer.
• Since the input of edges is also given in compressed format, intuitively we can think that 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.
• In our solution, We only use the structure and ideas of segment tree (complete binary tree). We dont calculate or store anything in the segment tree nodes.
• 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 $$$G$$$ and add edges from parent to its left and right children with 0 cost.

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

• Since from any intermediate node, we can visit the leaf nodes in its subtree with 0 cost, this graph $$$G'$$$ preserves the same information of graph $$$G$$$.
• Total number of edges is $$$2N$$$ (intial graph) + $$$M*logN$$$ (for each edge-group), So total edges are $$$E = O(Mlog(N))$$$.
• Performing dijkstra in this graph $$$G'$$$ would give time complexity $$$O(Mlog^2(N))$$$ which would pass under the given constraints.

Problems using this idea:

• Problem 1

• Problem 2