\itemize\ibull
\:$\<Create>(\varepsilon)$ --- create an~empty soft heap with the given accuracy parameter,
\:$\<Insert>(H,x)$ --- insert a~new element~$x$ to the heap~$H$,
-\:$\<Meld>(H_1,H_2)$ --- form a~new soft heap containing the elements stored in two disjoint
- heaps $H_1$ and~$H_2$
- (both heaps must have the same~$\varepsilon$ and they are destroyed in the process),
+\:$\<Meld>(P,Q)$ --- merge two heaps into one, more precisely insert all elements of a~heap~$Q$
+ to the heap~$P$, destroying~$Q$ in the process (both heaps must have the same~$\varepsilon$),
\:$\<DeleteMin>(H)$ --- delete the minimum element of the heap~$H$ and return its value
(optionally signalling that the value has been corrupted).
\endlist
somewhat counter-intuitive, albeit well-defined. We prefer describing the queues as binary
trees with a~special distribution of values. In fact, the original C~code in the Chazelle's
paper uses this representation internally.}
-Each vertex~$v$ of the tree remembers a~linked list $\<list>(v)$ of values. The
+Each vertex~$v$ of the tree remembers a~doubly-linked list $\<list>(v)$ of values. The
item list in every left son will be used only temporarily and it will be kept
empty between operations. Only right sons and the root have their lists
permanently occupied. The former vertices will be called \df{white}, the latter
We will now combine the soft queues to the soft heap.
-\figure{softheap2.eps}{\epsfxsize}{Joining and dismantling soft queues}
+\figure{softheap2.eps}{\epsfxsize}{Joining and dismantling of soft queues}
\defn A~\df{soft heap} consists of:
\itemize\ibull
\algn{Melding of two soft heaps}
\algo
-\algin Two soft heaps~$H_1$ and~$H_2$.
-\:If the tail of~$H_1$ has smaller rank than the tail of~$H_2$, exchange $H_1$ and~$H_2$.
+\algin Two soft heaps~$P$ and~$Q$.
+\:If the tail of~$P$ has smaller rank than the tail of~$Q$, exchange their item lists.
\brk\cmt{Whenever we run into an~end of a~list, we assume that it contains an~empty queue of infinite rank.}
-\:$p\=\<head>(H_1)$.
-\:While $H_2$ still has some queues:
-\::$q\=\<head>(H_2)$.
+\:$p\=\<head>(P)$.
+\:While $Q$ still has some queues:
+\::$q\=\<head>(Q)$.
\::If $\<rank>(p) < \<rank>(q)$, then $p\=$ the successor of~$p$,
-\::else if $\<rank>(p) > \<rank>(q)$, remove~$q$ from~$H_2$ and insert it before~$p$,
+\::else if $\<rank>(p) > \<rank>(q)$, remove~$q$ from~$Q$ and insert it before~$p$,
\::otherwise (the ranks are equal, we need to propagate the carry):
\:::$\<carry>=p$.
\:::$p\=$ the successor of~$p$.
\:::While $\<rank>(q)=\<rank>(\<carry>)$:
-\::::Remove~$q$ from~$H_2$.
+\::::Remove~$q$ from~$Q$.
\::::$\<carry>\=\<join>(q, \<carry>)$.
-\::::$q\=\<head>(H_2)$.
+\::::$q\=\<head>(Q)$.
\:::Insert~\<carry> before~$q$.
-\:Update the suffix minima: Walk with~$p$ backwards to the head of~$H_1$:
+\:Update the suffix minima: Walk with~$p$ backwards to the head of~$P$:
\::$p'\=$ successor of~$p$.
\::If $\<ckey>(p) < \<ckey>(p')$, set $\<suffix\_min>(p)\=p$.
\::Otherwise set $\<suffix\_min>(p)\=\<suffix\_min>(p')$.
-\:Destroy the heap~$H_2$.
-\algout The merged heap~$H_1$.
+\:Destroy the heap~$Q$.
+\algout The merged heap~$P$.
\endalgo
\algn{Insertion of an~element to a~soft heap}
\algo
\algin A~heap~$H$ and a~new element~$x$.
-\:Create a~new heap~$H'$ containing a~sole queue of rank~0, whose only vertex has
- the element~$x$ in its item list.
+\:Create a~new heap~$H'$ with the same parameters as~$H$. Let~$H'$ contain a~sole queue of rank~0,
+ whose only vertex has the element~$x$ in its item list.
\:$\<Meld>(H,H')$.
\algout An~updated heap~$H$.
\endalgo
+\para
+So far, the mechanics of the soft heaps was almost identical to the binomial heaps
+and the reader could rightfully yawn. The things are going to get interesting
+now as we approach the parts where corruption of items takes place.
+
+If all item lists contain at most one item equal to the \<ckey> of the particular
+vertex, no information is lost and the heap order guarantees that the minimum item
+of every queue stays in its root. We can however allow longer lists and let the items
+stored in a~single list travel together between the vertices of the tree, still
+represented by a~common \<ckey>. This data-structural analogue of car pooling will
+allow the items to travel at a~faster rate, but as only a~single item can be equal
+to the \<ckey>, all other items will be inevitably corrupted.
+
+We of course have to be careful about the size of the lists, as we must avoid corrupting
+too many items. We will control the growth according to the vertex ranks. Vertices
+with rank at most~$r$ will always contain just a~single item. Above this level,
+the higher is the rank, the longer list will we allow.
+
+\para
+\em{Deleting of minimum} will follow this principle. The minimum is easy to locate
+--- we follow the \<suffix\_min> of the head of the heap to the queue with the minimum
+\<ckey> and we look inside the item list of the root of this queue. We remove the \em{last}
+item from the list (we do not want the \<ckey> to change) and return it as the minimum.
+(It is not necessarily the real minimum of all items, but always the minimum of those
+uncorrupted.)
+
+If the list becomes empty, we \em{refill} it with items from the lower levels of the
+same tree. This can be best described recursively: We ask the left son to refill itself
+(remember that the left son is always white, so there are no items to lose). If the new
+\<ckey> of the left son is smaller than of the right son, we immediately move the left
+son's list to its parent. Otherwise, we exchange the sons and move the list from the
+new left son to the parent. This way, we obey the heap order and the same time we keep
+the white left son free of items.
+
+Ocassionally, we repeat this process once again and we concatenate the resulting lists
+(we append the latter list to the former, using the smaller of the two \<ckey>'s). This
+makes the lists grow longer and we want to do that roughly on every other level of the
+tree. The exact condition will be that either the rank of the current vertex is odd,
+or the difference in ranks between this vertex and its right son is at least two.
+
+If refilling of the left son fails because there are no more items in that subtree
+(we report this by setting its \<ckey> to $+\infty$), the current vertex is no longer
+needed, as the items would just travel through it unmodified. We therefore want to
+remove it. Instead of deleting it directly, we will rather make it point to its former
+grandsons and we will remove the (now orhpaned) original son. This helps us to ensure
+that both sons always have the same rank, which will be useful for the analysis.
+
+When all refilling is done, we update the suffix minima by walking from the current
+queue to the head of the heap exactly as we did in the \<Meld> procedure.
+
+Our only remaining worry is that the Rank invariant can be broken after the
+refilling. When the leftmost path of the tree becomes too short, we just
+\em{dismantle} the tree as already described and we meld the new trees back to
+the heap. This is easier to handle when the item list at the root vertex is
+empty. We will therefore move this check before the refilling of the root list.
+It will turn out that we have enough time to always walk the leftmost path
+completely, so no explicit counters are needed.
+
+Let us translate these ideas to real (pseudo)code:
+
+\algn{Deleting the minimum item of a~soft heap}
+\algo
+\algin A~soft heap~$H$.
+\:Use \<suffix\_min> of the head queue of~$H$ to locate the queue~$q$ with the smallest \<ckey>.
+\:Remove the last element~$x$ of the item list in the root of~$q$.
+\:If the item list is empty:
+\::Count the vertices on the leftmost path of~$q$.
+\::If there are less than $\<rank>(q)$ of them:
+\:::Remove~$q$ from the list of queues.
+\:::Recalculate the suffix minima as in the \<Meld> procedure.
+\:::Dismantle~$q$ and create a~heap~$H'$ holding the resulting trees.
+\:::Meld them back: $\<Meld>(H,H')$.
+\::Otherwise:
+\:::Call \<Refill> on the root of~$q$.
+\:::If $\<ckey>(q)=+\infty$ (no items left), remove the tree~$q$.
+\:::Recalculate the suffix minima.
+\algout The deleted minimum item~$x$ (possibly corrupted).
+\endalgo
+
+\algn{Refilling the item list of a~vertex}
+\algo
+\algin A~soft queue and its vertex~$v$ with an~empty item list.
+\:Handle trivial cases: If~$v$ has no children or both have $\<ckey>=+\infty$,
+ set $\<ckey>(v)$ to~$+\infty$ and return.
+\:Let \<left> and~\<right> denote the sons of~$v$.
+\:Recurse: call $\<Refill>(\<left>)$.
+\:If $\<ckey>(\<left>) > \<ckey>(\<right>)$, swap the sons.
+\:Move the item list from \<left> to~$v$ (implying $\<ckey>(v)=\<ckey>(\<left>)$).
+\:If $\<rank>(v)$ is odd or $\<rank>(v) > \<rank>(\<right>)+1$, recurse once more:
+\::Repeat steps 3--4.
+\::Append the item list from \<left> to the item list at~$v$.
+\:Clean up. If $\<ckey>(\<right>) = +\infty$:
+\::If $\<ckey>(\<left>) = +\infty$: unlink and discard both sons.
+\::Otherwise relink the sons of \<left> to~$v$ and discard \<left>.
+\algout A~modified soft queue.
+\endalgo
+
+
\endpart
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projects / saga.git / commitdiff