Maximize Spanning Tree Stability with Upgrades
You are given an integer n, representing n nodes numbered from 0 to n - 1, and a list of edges, where edges[i] = [ui, vi, si, musti]:
uiandviindicate an undirected edge between nodesuiandvi.siis the strength of the edge.mustiis an integer (0or1). Ifmusti == 1, the edge must be included in the spanning tree. These edges cannot be upgraded.
You are also given an integer k, the maximum number of upgrades you can perform. Each upgrade doubles the strength of an edge, and each eligible edge with musti == 0 can be upgraded at most once.
The stability of a spanning tree is defined as the minimum strength score among all edges included in it.
Return the maximum possible stability of any valid spanning tree. If it is impossible to connect all nodes, return -1.
Note: A spanning tree of a graph with n nodes is a subset of the edges that connects all nodes together, meaning the graph is connected, without forming any cycles, and uses exactly n - 1 edges.
Example 1
Input
n = 3, edges = [[0,1,2,1],[1,2,3,0]], k = 1Output
2Edge [0,1] with strength 2 must be included, edge [1,2] can be upgraded from 3 to 6, and the minimum strength in the resulting spanning tree is 2.
Example 2
Input
n = 3, edges = [[0,1,4,0],[1,2,3,0],[0,2,1,0]], k = 2Output
6Upgrade edges [0,1] from 4 to 8 and [1,2] from 3 to 6, so the minimum strength in the resulting spanning tree is 6.
Constraints
- 2 <= n <= 10^5
- 1 <= edges.length <= 10^5
- edges[i] = [ui, vi, si, musti]
- 0 <= ui, vi < n
- ui != vi
- 1 <= si <= 10^5
- musti is either 0 or 1.
- 0 <= k <= n
- There are no duplicate edges.