Documentation

Std.Data.BinomialHeap

inductive Std.BinomialHeapImp.HeapNode (α : Type u) :

A HeapNode is one of the internal nodes of the binomial heap. It is always a perfect binary tree, with the depth of the tree stored in the Heap. However the interpretation of the two pointers is different: we view the child as 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.

Instances For
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    • Std.BinomialHeapImp.instReprHeapNode = { reprPrec := Std.BinomialHeapImp.reprHeapNode✝ }

    The "real size" of the node, counting up how many values of type α are stored. This is O(n) and is intended mainly for specification purposes. For a well formed HeapNode the size is always 2^n - 1 where n is the depth.

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    A node containing a single element a.

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    @[inline]

    O(log n). The rank, or the number of trees in the forest. It is also the depth of the forest.

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

    A Heap is the top level structure in a binomial heap. It consists of a forest of HeapNodes with strictly increasing ranks.

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

      O(n). The "real size" of the heap, counting up how many values of type α are stored. This is intended mainly for specification purposes. Prefer Heap.size, which is the same for well formed heaps.

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

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      O(1). Is the heap empty?

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      O(1). The heap containing a single value a.

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      O(1). Auxiliary for Heap.merge: Is the minimum rank in Heap strictly larger than n?

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      • One or more equations did not get rendered due to their size.

      O(log n). The number of trees in the forest.

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      @[specialize #[]]
      def Std.BinomialHeapImp.combine {α : Type u_1} (le : ααBool) (a₁ : α) (a₂ : α) (n₁ : Std.BinomialHeapImp.HeapNode α) (n₂ : Std.BinomialHeapImp.HeapNode α) :

      O(1). Auxiliary for Heap.merge: combines two heap nodes of the same rank into one with the next larger rank.

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      @[specialize #[]]

      Merge two forests of binomial trees. The forests are assumed to be ordered by rank and merge maintains this invariant.

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      O(log n). Convert a HeapNode to a Heap by reversing the order of the nodes along the sibling spine.

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      @[specialize #[]]
      def Std.BinomialHeapImp.Heap.headD {α : Type u_1} (le : ααBool) (a : α) :

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

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      @[inline]
      def Std.BinomialHeapImp.Heap.head? {α : Type u_1} (le : ααBool) :

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

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      structure Std.BinomialHeapImp.FindMin (α : Type u_1) :
      Type u_1

      The return type of FindMin, which encodes various quantities needed to reconstruct the tree in deleteMin.

      Instances For
        @[specialize #[]]

        O(log n). Find the minimum element, and return a data structure FindMin with information needed to reconstruct the rest of the binomial heap.

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

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        • One or more equations did not get rendered due to their size.
        @[inline]

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

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        @[inline]

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

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        @[specialize #[]]
        def Std.BinomialHeapImp.Heap.foldM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [inst : Monad m] (le : ααBool) (s : Std.BinomialHeapImp.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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        • One or more equations did not get rendered due to their size.
        @[inline]
        def Std.BinomialHeapImp.Heap.fold {α : Type u_1} {β : Type u_2} (le : ααBool) (s : Std.BinomialHeapImp.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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        @[inline]
        def Std.BinomialHeapImp.Heap.toArray {α : Type u_1} (le : ααBool) (s : Std.BinomialHeapImp.Heap α) :

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

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        @[inline]
        def Std.BinomialHeapImp.Heap.toList {α : Type u_1} (le : ααBool) (s : Std.BinomialHeapImp.Heap α) :
        List α

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

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        @[specialize #[]]
        def Std.BinomialHeapImp.HeapNode.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [inst : 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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        @[specialize #[]]
        def Std.BinomialHeapImp.Heap.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [inst : 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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        @[inline]
        def Std.BinomialHeapImp.Heap.foldTree {β : Type u_1} {α : Type u_2} (nil : β) (join : αβββ) (s : Std.BinomialHeapImp.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.BinomialHeapImp.HeapNode.WellFormed {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)
        • When interpreting child and sibling as left and right children of a binary tree, it is a perfect binary tree with depth r
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        def Std.BinomialHeapImp.Heap.WellFormed {α : Type u_1} (le : ααBool) (n : Nat) :

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

        • It consists of a list of well formed trees with the specified ranks
        • The ranks are in strictly increasing order, and all are at least n
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        theorem Std.BinomialHeapImp.Heap.WellFormed.nil :
        ∀ {α : Type u_1} {le : ααBool} {n : Nat}, Std.BinomialHeapImp.Heap.WellFormed le n Std.BinomialHeapImp.Heap.nil

        O(log n). The rank, or the number of trees in the forest. This is the same as rankTR but it is not tail recursive.

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        structure Std.BinomialHeapImp.FindMin.WellFormed {α : Type u_1} (le : ααBool) (res : Std.BinomialHeapImp.FindMin α) :

        The well formedness predicate for a FindMin value. This is not actually a predicate, as it contains an additional data value rank corresponding to the rank of the returned node, which is omitted from findMin.

        Instances For
          def Std.BinomialHeap (α : Type u) (le : ααBool) :

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

          • insert : α → BinomialHeap α → BinomialHeap α→ BinomialHeap α → BinomialHeap α→ BinomialHeap α: add an element to the heap
          • deleteMin : BinomialHeap α → Option (α × BinomialHeap α)→ Option (α × BinomialHeap α)× BinomialHeap α): remove the minimum element from the heap
          • merge : BinomialHeap α → BinomialHeap α → BinomialHeap α→ BinomialHeap α → BinomialHeap α→ BinomialHeap α: combine two heaps

          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 BinomialHeap, all three operations are O(log n).

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          @[inline]
          def Std.mkBinomialHeap (α : Type u) (le : ααBool) :

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

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          @[inline]
          def Std.BinomialHeap.empty {α : Type u} {le : ααBool} :

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

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          instance Std.BinomialHeap.instInhabitedBinomialHeap {α : Type u} {le : ααBool} :
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          • Std.BinomialHeap.instInhabitedBinomialHeap = { default := Std.BinomialHeap.empty }
          @[inline]
          def Std.BinomialHeap.isEmpty {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

          O(1). Is the heap empty?

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          @[inline]
          def Std.BinomialHeap.size {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          @[inline]
          def Std.BinomialHeap.singleton {α : Type u} {le : ααBool} (a : α) :

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

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          @[inline]
          def Std.BinomialHeap.merge {α : Type u} {le : ααBool} :

          O(log n). Merge the contents of two heaps.

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          • One or more equations did not get rendered due to their size.
          @[inline]
          def Std.BinomialHeap.insert {α : Type u} {le : ααBool} (a : α) (h : Std.BinomialHeap α le) :

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

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          def Std.BinomialHeap.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.BinomialHeap.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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          @[inline]
          def Std.BinomialHeap.deleteMin {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          • One or more equations did not get rendered due to their size.
          @[inline]
          def Std.BinomialHeap.head? {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          @[inline]
          def Std.BinomialHeap.head! {α : Type u} {le : ααBool} [inst : Inhabited α] (b : Std.BinomialHeap α le) :
          α

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

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          @[inline]
          def Std.BinomialHeap.headI {α : Type u} {le : ααBool} [inst : Inhabited α] (b : Std.BinomialHeap α le) :
          α

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

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          @[inline]
          def Std.BinomialHeap.tail? {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          • One or more equations did not get rendered due to their size.
          @[inline]
          def Std.BinomialHeap.tail {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          @[inline]
          def Std.BinomialHeap.toList {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :
          List α

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

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          @[inline]
          def Std.BinomialHeap.toArray {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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          @[inline]
          def Std.BinomialHeap.toListUnordered {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :
          List α

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

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          @[inline]
          def Std.BinomialHeap.toArrayUnordered {α : Type u} {le : ααBool} (b : Std.BinomialHeap α le) :

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

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