ITEC 2620A
Introduction to Data Structures
Instructor: Prof. Z. Yang
Course Website:
http://people.math.yorku.ca/~zyang/itec
2620a.htm
Office: Tel 3049
Stacks and Queues
Stack-based Recursion
• Computers use stacks to manage function
calls/function returns.
– When a function is called, data is pushed onto the
stack
– When a function finishes, data is popped off the
stack
• Rather than using the computer’s stack, we
can use our own to implement recursion!
• Binary tree traversal
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Stacks and Recursion
• Any recursive algorithm can be
implemented non-recursively.
– Do you have to use a function that calls
itself?
•No. You can use a stack to implement recursion
– Do you have to use a recursive algorithm?
•Yes. How do you traverse a binary tree without a
recursive algorithm?
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Queues
• A queue is a “First-In, First-Out” = “FIFO” buffer.
– e.g. line-ups
• people enter from the back of the line
• people are served (exit) from the front of the line
• When elements are added, they are enqueued to the
back.
• When elements are removed, they are dequeued from
the front.
• enqueue() and dequeue() are the two defining
functions of a queue.
• Example
• Link-based Queue
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Priority Queues
• Previous queues were based on entry order
(e.g. LIFO, FIFO).
• Priority queues are based on item value.
• Stacks and queues aren’t designed for
searches.
• BST could work, but there is more overhead
than we need.
– don’t need completely sorted queue – only need
first element
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Heaps and Heapsort
Key Points
• Heaps
• Make a heap
• Heapsort
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Heaps
• Complete binary tree
• Partially ordered
– max heap – the value of every node is
greater than its children
– min heap – the value of every node is
smaller than its children
• No relationship between siblings, only
ancestors (up-down)
• BST has left-right relationships.
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Removing the Minimum Value
• Minimum value is at root, so we should
remove this element.
• How do we maintain a complete binary
tree?
– Put last value at the root and sift down
– Switch the value with the smallest of its
children – continue
• Heap property is maintained.
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Inserting a new value
• How do we maintain a complete binary
tree?
– Put value at the bottom and sift up
• Heap property is maintained.
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Complexity Analysis
• Remove minimum
– Best  value stays at depth 1  O(1)
– Worst  value goes back to bottom  O(logn)
– Average  value goes half way O(logn)
• Insert
– Best  value stays at bottom  O(1)
– Worst  value goes to top  O(logn)
– Average  value goes half way O(logn)
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Making a Heap
• Array representation
• Make a heap
• Analysis of building a heap
• Heapsort
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