You are hired by a game design company and one of their most popular games is The Journey. The game has a ton of quests, and for a player to win, the player must finish all the quests. There are a total of N quests in the game. Here is how the game works: the player can arbitrarily pick one of the N quests to start from. Once the player completes a quest, they unlock some other quests. The player can then choose one of the unlocked quests and complete it, and so on. For instance, let's say that this game had only 4 quests: A, B, C, and D. Let's say that after you complete
• quest A, you unlock quests [B, D). .
• quest B, you unlock quests [C, D).
• quest C, you unlock nothing [ ].
• quest D, you unlock quest [C].
Is this game winnable?
Yes, because of the following scenario:
The player picks quest A to start with. At the end of the quest A, the unlocked list contains [B, D). Say that player chooses to do quest B, then the unlocked list will contain [C, D). Say that player chooses to complete quest C, then the unlocked list will contain quest [D]. Finally, player finishes quest D.
Note that if the player had started with quest C instead of quest A, they would have lost this game, because they wouldn't have unlocked any quest and would be stuck. But the game is still winnable, because there is at least one starting quest which makes the player win. The Journey has N quests, enumerated as {nj, N2, . ng}. We construct a directed graph G for the game, where each node in G represents a quest. If completing quest n; will unlock quest n; then there is a directed edge from n; to n; in the graph. Suppose the total number of edges in the graph is M. (1) We call a quest as a lucky quest if starting from this quest will allow the player to win the game. In the example above, quest A is a lucky quest. Show that (a) All lucky quests are in the same strongly connected component of G, and that (b) Every quest in that component is a lucky quest. [We are expecting a short but rigorous proof for both claims] (2) Suppose there is at least one lucky quest. Give an algorithm that runs in time O(N+M) and finds a lucky quest. You may use any algorithm we have seen in class as subroutine. [We are expecting pseudocode or a description of your algorithm, and a short justification of the runtime]