Minimum Runes to Add to Cast Spell

Minimum Runes to Add to Cast Spell

What is this problem about?

The Minimum Runes to Add to Cast Spell problem models a spell-casting scenario as a directed graph. You have spell components that require other components to be activated first. To cast the final spell, you may need to place additional "rune" activators at components with no prerequisites. The goal is to find the minimum number of runes to add so that all components of the spell chain can be activated. This Minimum Runes to Add to Cast Spell coding problem is fundamentally about counting unreachable source nodes in a directed graph.

Why is this asked in interviews?

DE Shaw asks this hard problem to test deep graph reasoning — specifically about strongly connected components, topological ordering, and source node detection. The array, union find, BFS/DFS, and topological sort interview pattern all play a role. Understanding which nodes can "self-activate" versus which need external runes requires structural insight into directed graphs.

Algorithmic pattern used

The approach uses Strongly Connected Components (SCC) via Kosaraju's or Tarjan's algorithm, followed by condensation into a DAG. In the condensed DAG, count the number of SCCs with in-degree zero — these are the components that need runes placed. Each zero-in-degree SCC that isn't reachable from another SCC requires at least one rune. The answer is the count of such source SCCs.

Example explanation

Components: A→B, B→C, D→A, E (standalone). SCCs: {A,B,C,D} (if D→A creates a cycle with A→B→C), {E}. Condensed DAG: node for {A,B,C,D}, node for {E}. Both have in-degree 0. Minimum runes = 2 (one for the {A,B,C,D} cluster, one for {E}).

Common mistakes candidates make

• Treating this as a simple DFS reachability problem without SCC condensation.
• Counting individual nodes with zero in-degree instead of SCCs.
• Incorrectly implementing Kosaraju's two-pass DFS.
• Forgetting to build the condensed DAG after finding SCCs.

Interview preparation tip

Hard graph problems often require a two-step approach: decompose the graph into simpler structures (SCCs → DAG), then apply a simpler algorithm on the result. Practice Kosaraju's and Tarjan's SCC algorithms until you can implement them reliably. Understanding graph condensation — the technique of collapsing SCCs into single super-nodes — is key for a whole class of hard graph interview questions involving reachability and dependency resolution.