CS%20280%20Data%20Structures

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CS 280Data Structures

• Professor John Peterson

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Best / Worst Case

• boolean m(int a)
• for (int i 0 i lt a.length i)
• for (int j i1 j lt a.length j)
• int s 0
• for (int k j1 k lt a.length k)
• s s ak
• if (ai aj s)
• return true
• return false
• Best case? Worst case?

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Best / Worst Case

• boolean sumIn(int s, int a)
• return (sumIn1(s,0,a))
• boolean sumIn1(int s, int which, int a)
• if (which gt a.length) return (s 0)
• if (sumIn1(s, which1, a)) return true
• if (sumIn1(s-awhich, which1, a) return
  true
• return false
• Best case? Worst case? General worst case data?

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Heapsort

• Definition a heap is a tree in which the heap
  condition holds at all points
• heap(n) an gt aleft(n)
• an gt aright(n)
• (this ignores the case where left / right are
  outside the heap we could add a condition to
  catch this)
• This is essentially an invariant something that
  we want to always be true.

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Arrays as a Heap

• Suppose we have an array
• 4, 2, 7, 1, 3, 6, 9, 0, 5
• Draw this as a tree.
• Is it a heap or not?
• 4
• 2 7
• 1 3 6 9
• 0 5

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More Heaps

• How about these? What do these look like in
  array notation?
• 6 4
• 6 1 3 4
• 6 3 0 1 2 4 2 4

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Creating a Heap

• Well start by turning a non-heap into a heap
  using exchanges.
• Well do this using recursion a method which
  calls itself.
• Lets start with a simple problem a headless
  heap one in which the heap condition holds
  everywhere except the root of the tree.

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heapify

• 4
• 6 3
• 2 5 1 0
• So what should we do with the 4?

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heapify

• 4
• 6 3
• 2 5 1 0
• So what should we do with the 4?
• Swap it with the larger of its two descendents.
• But that means that the subtree no longer
  satisfies the heap condition!

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heapify

• void heapify(int i, Sortable s, int n) // n is
  the limit of s
• int l left(i) // 2i1
• int r right(i) // 2i2
• if (right(i) lt n)
• if (s.gtr(l, i) s.gtr(r, i))
• if (s.gtr(l,r))
• s.swap(i,l)
• heapify(l)
• else
• s.swap(i,r)
• heapify(r)
• else if (left(i) lt n)
• if (s.gtr(i,l)) swap(i,l)