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Problem solution · Python

Minimum Runes to Add to Cast Spell

Minimum Runes to Add to Cast Spell: a Python solution using depth-first search. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Depth-first search
Source
walkccc LeetCode Solutions
Length
76 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Depth-first search

For Minimum Runes to Add to Cast Spell, the implementation follows one branch at a time, making it suitable for components, trees, backtracking, or dependency exploration.

  1. Define the state carried into one recursive or stack frame.
  2. Mark or choose the current state before exploring children.
  3. Combine child results or undo the choice when the branch finishes.

Code notes

  • 76 lines of Python from the credited upstream file 3383.py.
  • The implementation visibly relies on sequence storage, ordered lookup.
  • No explicit loop blocks detected, together with recursive traversal.

Complexity

Count unique states for graph traversal; for backtracking, count the branching factor and maximum depth.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMinimum Runes to Add to Cast Spell · PythonPython
Use this to learn the idea, then write your own version.
class Solution:  def minRunesToAdd(      self,      n: int,      crystals: list[int],      flowFrom: list[int],      flowTo: list[int]  ) -> int:    graph = [[] for _ in range(n)]    reversedGraph = [[] for _ in range(n)]     for u, v in zip(flowFrom, flowTo):      graph[u].append(v)      reversedGraph[v].append(u)     # Identify Strongly Connected Components (SCC) using Kosaraju's Algorithm.    seen = set()    orderStack = []    componentIds = [-1] * n    componentCount = 0     for i in range(n):      if i not in seen:        self._kosaraju(graph, i, seen, orderStack)     while orderStack:      u = orderStack.pop()      if componentIds[u] == -1:        self._identifySCC(reversedGraph, u, componentIds, componentCount)        componentCount += 1     # Track crystal-containing components and inter-component edges.    hasCrystal = [False] * componentCount    hasInterComponentEdge = [False] * componentCount     for u in crystals:      hasCrystal[componentIds[u]] = True     for u, v in zip(flowFrom, flowTo):      id1 = componentIds[u]      id2 = componentIds[v]      if id1 != id2:  # Edge is inter-component.        hasInterComponentEdge[id2] = True     return sum(not hasCrystal[i] and not hasInterComponentEdge[i]               for i in range(componentCount))   def _kosaraju(      self,      graph: list[list[int]],      u: int,      seen: set[int],      orderStack: list  ) -> None:    """Creates a topological order stack using Kosaraju's Algorithm."""    seen.add(u)    for v in graph[u]:      if v not in seen:        self._kosaraju(graph, v, seen, orderStack)    orderStack.append(u)   def _identifySCC(      self,      reversedGraph: list[list[int]],      u: int,      componentIds: list[int],      componentId: int  ) -> None:    """Assigns component IDs during SCC identification in the second DFS."""    if componentIds[u] != -1:      return    componentIds[u] = componentId    for v in reversedGraph[u]:      if componentIds[v] == -1:        self._identifySCC(reversedGraph, v, componentIds, componentId)

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