The heap, deque, tree, and graph operations you actually write in an interview, with the time complexity that explains why.

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What's on the sheet

• Heap. The smallest (or largest) item, ready in O(1), whenever you do not need the rest sorted.
• Deque. A queue or a stack, whichever end you need O(1) at, without two different classes to remember.
• Tree. Parent and child links you walk top-down, one level or one branch at a time.
• Graph. Nodes and edges you look up by key, not by index, an adjacency list first.

Watch out

• Heap: Tuples compare left to right, so two equal priorities fall through to comparing the second value, and a TypeError follows if that value cannot be ordered, like a dict. Push a counter in the middle, (priority, count, item), so the tie never reaches it.
• Deque: A list only tracks one end efficiently, so popping from the front means sliding every remaining item over, an O(n) cost. A deque tracks both ends directly, so popleft() finishes in O(1) no matter how long the queue is.
• Tree: BFS holds a whole level in its queue at once, O(w) space for the widest level. DFS only holds the current branch, O(h) space for the height. A wide, shallow tree favors DFS; a deep, narrow one favors BFS.
• Graph: A tree only has one path in; a graph can loop back on itself. Skip the visited set and a cycle spins the traversal forever, so mark a node visited the moment you enqueue or recurse into it, not once you get around to processing it.

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