Documentation

Std.Data.PairingHeap

inductive Std.PairingHeapImp.Heap (α : Type u) :

A Heap is the nodes of the pairing heap. Each node have two pointers: child going to the first child of this node, and sibling goes to the next sibling of this tree. So it actually encodes a forest where each node has children node.child, node.child.sibling, node.child.sibling.sibling, etc.

Each edge in this forest denotes a le a b relation that has been checked, so the root is smaller than everything else under it.

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    instance Std.PairingHeapImp.instReprHeap :
    {α : Type u_1} → [inst : Repr α] → Repr (Std.PairingHeapImp.Heap α)
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    • Std.PairingHeapImp.instReprHeap = { reprPrec := Std.PairingHeapImp.reprHeap✝ }

    O(n). The number of elements in the heap.

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      A node containing a single element a.

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        O(1). Is the heap empty?

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          @[specialize #[]]

          O(1). Merge two heaps. Ignore siblings.

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            @[specialize #[]]

            Auxiliary for Heap.deleteMin: merge the forest in pairs.

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              O(1). Get the smallest element in the heap, including the passed in value a.

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                O(1). Get the smallest element in the heap, if it has an element.

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                  Amortized O(log n). Find and remove the the minimum element from the heap.

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                    Amortized O(log n). Get the tail of the pairing heap after removing the minimum element.

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                      Amortized O(log n). Remove the minimum element of the heap.

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                        A predicate says there is no more than one tree.

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                          def Std.PairingHeapImp.Heap.foldM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (le : ααBool) (s : Std.PairingHeapImp.Heap α) (init : β) (f : βαm β) :
                          m β

                          O(n log n). Monadic fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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                            @[inline]
                            def Std.PairingHeapImp.Heap.fold {α : Type u_1} {β : Type u_2} (le : ααBool) (s : Std.PairingHeapImp.Heap α) (init : β) (f : βαβ) :
                            β

                            O(n log n). Fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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                              def Std.PairingHeapImp.Heap.toArray {α : Type u_1} (le : ααBool) (s : Std.PairingHeapImp.Heap α) :

                              O(n log n). Convert the heap to an array in increasing order.

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                                def Std.PairingHeapImp.Heap.toList {α : Type u_1} (le : ααBool) (s : Std.PairingHeapImp.Heap α) :
                                List α

                                O(n log n). Convert the heap to a list in increasing order.

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                                  @[specialize #[]]
                                  def Std.PairingHeapImp.Heap.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] (nil : β) (join : αββm β) :

                                  O(n). Fold a monadic function over the tree structure to accumulate a value.

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                                    def Std.PairingHeapImp.Heap.foldTree {β : Type u_1} {α : Type u_2} (nil : β) (join : αβββ) (s : Std.PairingHeapImp.Heap α) :
                                    β

                                    O(n). Fold a function over the tree structure to accumulate a value.

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                                      O(n). Convert the heap to a list in arbitrary order.

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                                        O(n). Convert the heap to an array in arbitrary order.

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                                          def Std.PairingHeapImp.Heap.NodeWF {α : Type u_1} (le : ααBool) (a : α) :

                                          The well formedness predicate for a heap node. It asserts that:

                                          • If a is added at the top to make the forest into a tree, the resulting tree is a le-min-heap (if le is well-behaved)
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                                            inductive Std.PairingHeapImp.Heap.WF {α : Type u_1} (le : ααBool) :

                                            The well formedness predicate for a pairing heap. It asserts that:

                                            • There is no more than one tree.
                                            • It is a le-min-heap (if le is well-behaved)
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                                              theorem Std.PairingHeapImp.Heap.WF.merge_node :
                                              ∀ {α : Type u_1} {le : ααBool} {a₁ : α} {c₁ : Std.PairingHeapImp.Heap α} {a₂ : α} {c₂ s₁ s₂ : Std.PairingHeapImp.Heap α}, Std.PairingHeapImp.Heap.NodeWF le a₁ c₁Std.PairingHeapImp.Heap.NodeWF le a₂ c₂Std.PairingHeapImp.Heap.WF le (Std.PairingHeapImp.Heap.merge le (Std.PairingHeapImp.Heap.node a₁ c₁ s₁) (Std.PairingHeapImp.Heap.node a₂ c₂ s₂))
                                              def Std.PairingHeap (α : Type u) (le : ααBool) :

                                              A pairing heap is a data structure which supports the following primary operations:

                                              The first two operations are known as a "priority queue", so this could be called a "mergeable priority queue". The standard choice for a priority queue is a binary heap, which supports insert and deleteMin in O(log n), but merge is O(n). With a PairingHeap, insert and merge are O(1), deleteMin is amortized O(log n).

                                              Note that deleteMin may be O(n) in a single operation. So if you need an efficient persistent priority queue, you should use other data structures with better worst-case time.

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                                                def Std.mkPairingHeap (α : Type u) (le : ααBool) :

                                                O(1). Make a new empty pairing heap.

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                                                  def Std.PairingHeap.empty {α : Type u} {le : ααBool} :

                                                  O(1). Make a new empty pairing heap.

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                                                    instance Std.PairingHeap.instInhabitedPairingHeap {α : Type u} {le : ααBool} :
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                                                    • Std.PairingHeap.instInhabitedPairingHeap = { default := Std.PairingHeap.empty }
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                                                    def Std.PairingHeap.isEmpty {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                    O(1). Is the heap empty?

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                                                      def Std.PairingHeap.size {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                      O(n). The number of elements in the heap.

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                                                        def Std.PairingHeap.singleton {α : Type u} {le : ααBool} (a : α) :

                                                        O(1). Make a new heap containing a.

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                                                          def Std.PairingHeap.merge {α : Type u} {le : ααBool} :

                                                          O(1). Merge the contents of two heaps.

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                                                            def Std.PairingHeap.insert {α : Type u} {le : ααBool} (a : α) (h : Std.PairingHeap α le) :

                                                            O(1). Add element a to the given heap h.

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                                                              def Std.PairingHeap.ofList {α : Type u} (le : ααBool) (as : List α) :

                                                              O(n log n). Construct a heap from a list by inserting all the elements.

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                                                                def Std.PairingHeap.ofArray {α : Type u} (le : ααBool) (as : Array α) :

                                                                O(n log n). Construct a heap from a list by inserting all the elements.

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                                                                  def Std.PairingHeap.deleteMin {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                  Amortized O(log n). Remove and return the minimum element from the heap.

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                                                                    def Std.PairingHeap.head? {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                    O(1). Returns the smallest element in the heap, or none if the heap is empty.

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                                                                      def Std.PairingHeap.head! {α : Type u} {le : ααBool} [Inhabited α] (b : Std.PairingHeap α le) :
                                                                      α

                                                                      O(1). Returns the smallest element in the heap, or panics if the heap is empty.

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                                                                        def Std.PairingHeap.headI {α : Type u} {le : ααBool} [Inhabited α] (b : Std.PairingHeap α le) :
                                                                        α

                                                                        O(1). Returns the smallest element in the heap, or default if the heap is empty.

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                                                                          def Std.PairingHeap.tail? {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                          Amortized O(log n). Removes the smallest element from the heap, or none if the heap is empty.

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                                                                            def Std.PairingHeap.tail {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                            Amortized O(log n). Removes the smallest element from the heap, if possible.

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                                                                              def Std.PairingHeap.toList {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :
                                                                              List α

                                                                              O(n log n). Convert the heap to a list in increasing order.

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                                                                                def Std.PairingHeap.toArray {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                                O(n log n). Convert the heap to an array in increasing order.

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                                                                                  def Std.PairingHeap.toListUnordered {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :
                                                                                  List α

                                                                                  O(n). Convert the heap to a list in arbitrary order.

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                                                                                    def Std.PairingHeap.toArrayUnordered {α : Type u} {le : ααBool} (b : Std.PairingHeap α le) :

                                                                                    O(n). Convert the heap to an array in arbitrary order.

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