Documentation

Batteries.Data.BinomialHeap.Basic

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.

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    • Batteries.BinomialHeap.Imp.instReprHeapNode = { reprPrec := Batteries.BinomialHeap.Imp.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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        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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          A Heap is the top level structure in a binomial heap. It consists of a forest of HeapNodes with strictly increasing ranks.

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            • Batteries.BinomialHeap.Imp.instReprHeap = { reprPrec := Batteries.BinomialHeap.Imp.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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                @[inline]

                O(1). Is the heap empty?

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                • Batteries.BinomialHeap.Imp.Heap.nil.isEmpty = true
                • x.isEmpty = false
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                  @[inline]

                  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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                      instance Batteries.BinomialHeap.Imp.instDecidableRankGT :
                      {α : Type u_1} → {s : Batteries.BinomialHeap.Imp.Heap α} → {n : Nat} → Decidable (s.rankGT n)
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                      O(log n). The number of trees in the forest.

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                        @[inline]

                        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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                          @[irreducible, 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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                              Computes s.toHeap ++ res tail-recursively, assuming n = s.rank.

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                                @[specialize #[]]
                                def Batteries.BinomialHeap.Imp.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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                                  O(log n). Get the smallest element in the heap, if it has an element.

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                                    structure Batteries.BinomialHeap.Imp.FindMin (α : Type u_1) :
                                    Type u_1

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

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

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

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                                              @[irreducible]
                                              theorem Batteries.BinomialHeap.Imp.Heap.realSize_merge {α : Type u_1} (le : ααBool) (s₁ : Batteries.BinomialHeap.Imp.Heap α) (s₂ : Batteries.BinomialHeap.Imp.Heap α) :
                                              (Batteries.BinomialHeap.Imp.Heap.merge le s₁ s₂).realSize = s₁.realSize + s₂.realSize
                                              theorem Batteries.BinomialHeap.Imp.Heap.realSize_deleteMin {α : Type u_1} {le : ααBool} {a : α} {s' : Batteries.BinomialHeap.Imp.Heap α} {s : Batteries.BinomialHeap.Imp.Heap α} (eq : Batteries.BinomialHeap.Imp.Heap.deleteMin le s = some (a, s')) :
                                              s.realSize = s'.realSize + 1
                                              @[irreducible, specialize #[]]
                                              def Batteries.BinomialHeap.Imp.Heap.foldM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (le : ααBool) (s : Batteries.BinomialHeap.Imp.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.
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                                                @[inline]
                                                def Batteries.BinomialHeap.Imp.Heap.fold {α : Type u_1} {β : Type u_2} (le : ααBool) (s : Batteries.BinomialHeap.Imp.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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                                                  O(n log n). Convert the heap to an array in increasing order.

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

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                                                      @[specialize #[]]
                                                      def Batteries.BinomialHeap.Imp.HeapNode.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 Batteries.BinomialHeap.Imp.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 Batteries.BinomialHeap.Imp.Heap.foldTree {β : Type u_1} {α : Type u_2} (nil : β) (join : αβββ) (s : Batteries.BinomialHeap.Imp.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 Batteries.BinomialHeap.Imp.HeapNode.WF {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 Batteries.BinomialHeap.Imp.Heap.WF {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 Batteries.BinomialHeap.Imp.Heap.WF.nil :
                                                                    ∀ {α : Type u_1} {le : ααBool} {n : Nat}, Batteries.BinomialHeap.Imp.Heap.WF le n Batteries.BinomialHeap.Imp.Heap.nil
                                                                    theorem Batteries.BinomialHeap.Imp.Heap.WF.of_rankGT :
                                                                    ∀ {α : Type u_1} {le : ααBool} {n' : Nat} {s : Batteries.BinomialHeap.Imp.Heap α} {n : Nat}, s.rankGT nBatteries.BinomialHeap.Imp.Heap.WF le n' sBatteries.BinomialHeap.Imp.Heap.WF le (n + 1) s
                                                                    theorem Batteries.BinomialHeap.Imp.Heap.rankGT.of_le :
                                                                    ∀ {α : Type u_1} {s : Batteries.BinomialHeap.Imp.Heap α} {n n' : Nat}, s.rankGT nn' ns.rankGT n'
                                                                    theorem Batteries.BinomialHeap.Imp.Heap.WF.rankGT :
                                                                    ∀ {α : Type u_1} {lt : ααBool} {n : Nat} {s : Batteries.BinomialHeap.Imp.Heap α}, Batteries.BinomialHeap.Imp.Heap.WF lt (n + 1) ss.rankGT n
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                                                                    theorem Batteries.BinomialHeap.Imp.Heap.WF.merge' :
                                                                    ∀ {α : Type u_1} {le : ααBool} {s₁ s₂ : Batteries.BinomialHeap.Imp.Heap α} {n : Nat}, Batteries.BinomialHeap.Imp.Heap.WF le n s₁Batteries.BinomialHeap.Imp.Heap.WF le n s₂Batteries.BinomialHeap.Imp.Heap.WF le n (Batteries.BinomialHeap.Imp.Heap.merge le s₁ s₂) ((s₁.rankGT n s₂.rankGT n)(Batteries.BinomialHeap.Imp.Heap.merge le s₁ s₂).rankGT n)
                                                                    theorem Batteries.BinomialHeap.Imp.HeapNode.WF.rank_eq {α : Type u_1} {le : ααBool} {a : α} {n : Nat} {s : Batteries.BinomialHeap.Imp.HeapNode α} :
                                                                    structure Batteries.BinomialHeap.Imp.FindMin.WF {α : Type u_1} (le : ααBool) (res : Batteries.BinomialHeap.Imp.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.

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                                                                      before is a difference list which can be appended to a binomial heap with ranks at least rank to produce another well formed heap.

                                                                      node is a well formed forest of rank rank with val at the root.

                                                                      next is a binomial heap with ranks above rank + 1.

                                                                      The conditions under which findMin is well-formed.

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                                                                      • One or more equations did not get rendered due to their size.
                                                                      • h_2.findMin hr hk = hr
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                                                                        def Batteries.BinomialHeap (α : Type u) (le : ααBool) :
                                                                        Type (max 0 u)

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

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                                                                          def Batteries.mkBinomialHeap (α : Type u) (le : ααBool) :

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

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                                                                            def Batteries.BinomialHeap.empty {α : Type u} {le : ααBool} :

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

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                                                                              • Batteries.BinomialHeap.instEmptyCollection = { emptyCollection := Batteries.BinomialHeap.empty }
                                                                              instance Batteries.BinomialHeap.instInhabited {α : Type u} {le : ααBool} :
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                                                                              • Batteries.BinomialHeap.instInhabited = { default := Batteries.BinomialHeap.empty }
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                                                                              def Batteries.BinomialHeap.isEmpty {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

                                                                              O(1). Is the heap empty?

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                                                                              • b.isEmpty = b.val.isEmpty
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                                                                                def Batteries.BinomialHeap.size {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                • b.size = b.val.size
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                                                                                  def Batteries.BinomialHeap.singleton {α : Type u} {le : ααBool} (a : α) :

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

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

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                                                                                      def Batteries.BinomialHeap.insert {α : Type u} {le : ααBool} (a : α) (h : Batteries.BinomialHeap α le) :

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

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                                                                                        def Batteries.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 Batteries.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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                                                                                            def Batteries.BinomialHeap.deleteMin {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                              instance Batteries.BinomialHeap.instStream {α : Type u} {le : ααBool} :
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                                                                                              • Batteries.BinomialHeap.instStream = { next? := Batteries.BinomialHeap.deleteMin }
                                                                                              def Batteries.BinomialHeap.forIn {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (b : Batteries.BinomialHeap α le) (x : β) (f : αβm (ForInStep β)) :
                                                                                              m β

                                                                                              O(n log n). Implementation of for x in (b : BinomialHeap α le) ... notation, which iterates over the elements in the heap in increasing order.

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                                                                                                instance Batteries.BinomialHeap.instForIn {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} :
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                                                                                                • Batteries.BinomialHeap.instForIn = { forIn := fun {β : Type ?u.29} [Monad m] => Batteries.BinomialHeap.forIn }
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                                                                                                def Batteries.BinomialHeap.head? {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                                  def Batteries.BinomialHeap.head! {α : Type u} {le : ααBool} [Inhabited α] (b : Batteries.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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                                                                                                  • b.head! = b.head?.get!
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                                                                                                    def Batteries.BinomialHeap.headI {α : Type u} {le : ααBool} [Inhabited α] (b : Batteries.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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                                                                                                    • b.headI = b.head?.getD default
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                                                                                                      def Batteries.BinomialHeap.tail? {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                                        def Batteries.BinomialHeap.tail {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                                          def Batteries.BinomialHeap.foldM {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (b : Batteries.BinomialHeap α le) (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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                                                                                                            def Batteries.BinomialHeap.fold {α : Type u} {le : ααBool} {β : Type u_1} (b : Batteries.BinomialHeap α le) (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 Batteries.BinomialHeap.toList {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :
                                                                                                              List α

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

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                                                                                                                def Batteries.BinomialHeap.toArray {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                                                  def Batteries.BinomialHeap.toListUnordered {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :
                                                                                                                  List α

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

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                                                                                                                  • b.toListUnordered = b.val.toListUnordered
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                                                                                                                    def Batteries.BinomialHeap.toArrayUnordered {α : Type u} {le : ααBool} (b : Batteries.BinomialHeap α le) :

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

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                                                                                                                    • b.toArrayUnordered = b.val.toArrayUnordered
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