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The Soft Heap: An Approximate Priority Queue with Optimal Error Rate.

Abstract. A simple variant of a priority queue, called a soft heap, is introduced. The data structure supports the usual operations: insert, delete, meld, and findmin. Its novelty is to beat the logarithmic bound on the complexity of a heap in a comparison-based model. To break this information-theoretic barrier, the entropy of the data structure is reduced by artificially raising the values of certain keys. Given any mixed sequence of n operations, a soft heap with error rate [Epsilon] (for any 0 [is less than] [Epsilon] [is less than or equal to] 1/2) ensures that, at any time, at most [Epsilon]n of its items have their keys raised. The amortized complexity of each operation is constant, except for insert, which takes O(log 1/[Epsilon]) time. The soft heap is optimal for any value of [Epsilon] in a comparison-based model. The data structure is purely pointer-based. No arrays are used and no numeric assumptions are made on the keys. The main idea behind the soft heap is to move items across the data structure not individually, as is customary, but in groups, in a data-structuring equivalent of "car pooling." Keys must be raised as a result, in order to preserve the heap ordering of the data structure. The soft heap can be used to compute exact or approximate medians and percentiles optimally. It is also useful for approximate sorting and for computing minimum spanning trees of general graphs.

Categories and Subject Descriptors: E.1 [Data Structures]: Nonnumerical Algorithms and Problems

General Terms: Theory

Additional Key Words and Phrases: Amortization, heap, priority queue, soft heap

1. Introduction

We design a simple variant of a priority queue, called a soft heap. The data structure stores items with keys from a totally ordered universe, and supports the operations:

--create (L): Create an empty soft heap L.

--insert (L, x): Add new item x to L.

--meld (L, L'): Form a new soft heap with the items stored in L and L' (assumed to be disjoint), and destroy L and L'.

--delete (L, x): Remove item x from L.

--findmin (L): Return an item in L with the smallest key.

The soft heap may, at any time, increase the value of certain keys. Such keys, and by extension, the corresponding items, are called corrupted. Corruption is entirely at the discretion of the data structure and the user has no control over it. Naturally, findmin returns the minimum current key, which might or might not be corrupted. The benefit is speed: during heap updates, items travel together in packets in a form of "car pooling," in order to save time.

From an information-theoretic point of view, corruption is a way to decrease the entropy of the data stored in the data structure, and thus facilitate its treatment. The entropy is defined as the logarithm, in base two, of the number of distinct key assignments (i.e., entropy of the uniform distribution over key assignments). To see the soundness of this idea, push it to its limit, and observe that if every key was corrupted by raising its value to [infinity] then the set of keys would have zero entropy and we could trivially perform all operations in constant time. Interestingly, soft heaps show that the entropy need not drop to zero for the complexity to become constant.

THEOREM 1.1. Beginning with no prior data, consider a mixed sequence of operations that includes n inserts. For any 0 [is less than] [Epsilon] [is less than or equal to] 1/2, a soft heap with error rate [Epsilon] supports each operation in constant amortized time, except for insert, which takes O(log 1/[Epsilon]) time. The data structure never contains more than [Epsilon]n corrupted items at any given time. In a comparison-based model, these bounds are optimal.

Note that this does not mean that only [Epsilon]n items are corrupted in total throughout the sequence of operations. Because of deletes many more items might end up being corrupted. In fact, it is not difficult to imagine a scenario where all items are eventually corrupted; for example, insert n items and then keep deleting corrupted ones. Despite this apparent weakness, the soft heap is optimal and--perhaps even more surprising--useful. The data structure can be implemented on a pointer machine: no arrays are used, and no numeric assumptions on the keys are required. Soft heaps capture three distinct features, which it is useful to understand at the outset.

--If we set [Epsilon] = 1/2n, then no corruption is allowed to take place and the soft heap behaves like a regular heap with logarithmic insertion time. Unsurprisingly, soft heaps include standard heaps as special cases. In fact, as we shall see, a soft heap is nothing but a modified binomial queue [Vuillemin 1978].

--More interesting is the fact that soft heaps implicitly feature median-finding technology. To see why, set [Epsilon] to be a small constant: insert n keys and then perform ??n/2?? findmins, each one followed by a delete. This takes O(n) time. Among the keys deleted, the largest (original) one is [Epsilon]n away from the median of the n original keys. To obtain such a number in linear time (deterministically), as we just did, typically requires a variant of the median-finding algorithm of Blum et al. [1973].

--The previous remark should not lead one to think that a soft heap is simply a dynamic median-finding data structure. Things are more subtle. Indeed, consider a sequence of n inserts of keys in decreasing order, intermixed with n findmins, and set [Epsilon] = 1/2. Despite the high value of the error rate [Epsilon], the findmins must actually return the minimum key at least half the time. The reason is that at most n/2 keys inserted can ever be corrupted. Because of the decreasing order of the insertions, these uncorrupted keys must be reported by findmin right after their insertion since they are minimum at the time. The requirement to be correct half the time dooms any strategy based on maintaining medians or near-medians for the purpose of findmin.

2. Applications

Soft heaps are useful for computing minimum spanning trees and percentiles, for finding medians, for near sorting, and generally for situations where approximate rank information is sought. Some examples below:

(1) The soft heap was designed with a specific application in mind, minimum spanning trees. It is a key ingredient in what is currently the fastest deterministic algorithm [Chazelle 2000] for computing the minimum spanning tree of a graph. Given a connected graph with n vertices and m weighted edges, the algorithm finds a minimum spanning tree in time O(m[Alpha](m, n)), where [Alpha] is the classical functional inverse of Ackermann's function.

(2) Another, simpler application is the dynamic maintenance of percentiles. Suppose we wish to maintain the grade point averages of students in a college, so that at any time we can request the name of a student with a GPA in the top percentile. Soft heaps support such operations in constant amortized time.

(3) Soft heaps give an alternative method for computing medians in linear time (or generally perform linear-time selection [Blum et al. 1973]). Suppose we want to find the kth largest element in a set of n numbers. Insert the n numbers into a soft heap with error rate 1/3. Next, call findmin and delete about n/3 times. The largest number deleted has rank between n/3 and 2n/3. After computing this rank, we can therefore remove at least n/3 numbers from consideration. We recurse over the remainder in the obvious fashion. This allows us to find the kth largest element in time proportional to n + 2n/3 + [(2/3).sup.2]n + ... = O(n).

(4) A fourth application of soft heaps is to approximate sorting. A weak version of near-sorting requires that given n distinct numbers, the algorithm should output them in a sequence whose number of inversions is at most [Epsilon][n.sup.2] (instead of zero for exact sorting). As it turns out, this follows directly from inserting n numbers into a soft heap with error rate [Epsilon] and then deleting the smallest keys repeatedly. The number [I.sub.k], of inversions of the kth deleted number x is the number of keys deleted earlier whose original values are larger than x. But x must have been in a corrupted state during those particular [I.sub.k], earlier deletions. The total number of keys in a corrupted state, counting over all deletions, is at most [Epsilon][n.sup.2], and so the number of inversions [Sigma] [I.sub.k], is also bounded by [Epsilon][n.sup.2].

(5) A stronger version of near-sorting requires that in the output sequence the rank of no number differs from its true rank in sorted order by more than [Epsilon]n. We show below how soft heaps allow us to do that in O(n log 1/[Epsilon]) time. In particular, this gives us a simple linear time algorithm for this strong form of near-sorting with, say, 1% error in rank. Of course, the result itself is not new. It can be derived trivially from repeated median computation. The whole point is that we do it in a completely different way.

THEOREM 2.1. For any 0 [is less than] [Epsilon] [is less than or equal to] 1/2, we can use a soft heap to near-son n numbers in time O(n log 1/[Epsilon]): this means that the rank of any number in the output sequence differs from its true rank by at most [Epsilon]n.

PROOF. Since the running time we seek to achieve is O(n log 1/[Epsilon]) we can assume that n is large enough and that 1/[Epsilon] lies between a large constant and [square root of n]. Given the n numbers (assumed to be distinct for simplicity) insert them into a soft heap and then delete the item returned by findmin until the heap is empty. Using a soft heap with error rate [Epsilon], at most [Epsilon]n numbers are corrupted at any given time. Divide the sequence of n pairs (findmin, delete) into time intervals [T.sub.1], ..., [T.sub.l], each consisting of ??2[Epsilon]n?? pairs; without loss of generality, we may assume that n = ??2[Epsilon]n??l. Let [S.sub.i] be the set of items deleted during [T.sub.i], and let [U.sub.i] be the subset of [S.sub.i] that includes only items that were at some point uncorrupted during [T.sub.i]. (We assume that items are time-stamped when first corrupted.) Finally, let [x.sub.i] be the smallest original key among the items of [U.sub.i], and let [s.sub.i] be the corresponding item; for convenience, we put [x.sub.0] = -[infinity] and [x.sub.l+1] = [infinity]. Given an item s [element of] [S.sub.i] whose original key lies in [[x.sub.j], [x.sub.j+1]), we define [Rho](s) = |i - j|. Some simple facts:

(1) |[U.sub.i]| [is greater than or equal to] ??2[Epsilon]n?? - [Epsilon]n [is greater than or equal to] [Epsilon]n. Because at most [Epsilon]n items are corrupted at the beginning of [T.sub.i].

(2) The [x.sub.i]'s appear in increasing order, and the original key of any item in [U.sub.i] lies in [[x.sub.i], [x.sub.i+1]): Since [s.sub.i+1] is uncorrupted during [T.sub.i], its original key [x.sub.i+1] is at least the current (and hence, the original) key of any item deleted during [T.sub.i].

(3) [Sigma]{[Rho](s)|s [element of] [S.sub.i]\[U.sub.i]} [is less than] 2n. Given s [element of] [S.sub.i]\[U.sub.i], let [[x.sub.j], [x.sub.j+1]) be the interval that contains the original key of s. As we just observed, the original key of s is less than [x.sub.i+1], and therefore, j [is greater than or equal to] i. To avoid being selected by findmin, the item s must have been in a corrupted state during the deletion of [x.sub.k], for any j [is less than] k [is less than] i (if any such k exists). The total number of items in a corrupted state during the deletions of [x.sub.1], ..., [x.sub.l] is at most [Epsilon]nl, and therefore so is the sum of distances i - j - 1 = [Rho](s) - 1 over all such items s. It follows that [Sigma][Rho](s) [is greater than or equal to] [Epsilon]nl + n [is less than] 2n, hence our claim.

(4) The number of items whose original keys fall in [[x.sub.i], [x.sub.i+1]) is less than 6[Epsilon]n: Indeed, suppose that the item does not belong to [S.sub.i] [union] [S.sub.i+1]. It cannot be selected by findmin and deleted before the beginning of [T.sub.i], since [s.sub.i] was not corrupted yet. By the time [s.sub.i+1] was deleted, the item in question must have been corrupted (it cannot have been deleted yet). So, there can be at most [Epsilon]n such items. Thus, the total number of items with original keys in [[x.sub.i], [x.sub.i+1]) is at most 2??2[Epsilon]n?? + [Epsilon]n [is less than] 6[Epsilon]n.

Next, for each item s [element of] [S.sub.i]\[U.sub.i], we search which interval [[x.sub.j], [x.sub.j+1]) contains its original key, which we can do in O([Rho](s) + 1) time by sequential search. By (1)-(4), this means that in O(n) postprocessing time we have partitioned the set of original keys into disjoint intervals, each containing between [Epsilon]n and 6[Epsilon]n keys. So, in O(n log 1/[Epsilon]) time, any number can be output in a position at most 6[Epsilon]n off its rank. Replacing [Epsilon] by [Epsilon]/6 completes the proof. []

3. The Data Structure

The data structure is simple but somewhat subtle. This makes it all the more useful to include the actual code of our implementation of soft heaps in C. (It is very short: about 100 lines!) The code should be viewed as a bonus, not a hindrance. We do not base our discussion on it and, in fact, it is possible to skip it in a first reading and still understand soft heaps.

Recall that a binomial tree [Vuillemin 1978] of rank k is a rooted tree of [2.sup.k] nodes: it is formed by the combination of two binomial trees of rank k - 1, where the root of one becomes the new child of the other root. A soft heap is a sequence of modified binomial trees of distinct ranks, called soft queues. The modifications come in two ways:

--A soft queue q is a binomial tree with subtrees possibly missing (somewhat like the trees of a Fibonacci heap [Fredman and Tarjan 1987] after a few deletions). The binomial tree from which q is derived is called its master tree. The rank of a node of q is the number of children of the corresponding node in the master tree. Obviously, it is an upper bound on the number of children in q. We enforce the following rank invariant: the number of children of the root should be no smaller than ??rank (root)/2??.

--A node v may store several items, in fact, a whole item-list. The ckey of v denotes the common value of all the current keys of the items in item-list(v): it is an upper bound on the original keys. The soft queue is heap-ordered with respect to ckeys, that is, a ckey of a node does not exceed the ckeys of any of its children. We fix an integer parameter r = r([Epsilon]), and we require that all corrupted items be stored at nodes of rank greater than r. (See Figure 1.)

[Figure 1 ILLUSTRATION OMITTED]

Turning to the actual C code, an item-list is a singly-linked list of items with one field indicating the original value of the key.
typedef struct ILCELL
        { int key;
          struct ILCELL *next;
        } ilcell;


A node of a soft queue indicates its ckey and its rank in the master tree. Pointers next and child give access to the children. If there are none, the pointers are NULL. Otherwise, the node is the parent of a soft queue of rank one less (pointed to by child) and the root of a soft queue of rank one less (pointed to by next). This is a standard artifice to represent high-degree nodes as sequences of degree-2 nodes. Finally, a pointer il gives access to the head of the item-list. To facilitate concatenation of item-lists, we also provide a pointer il_tail to the tail of the list.
typedef struct NODE
        { int ckey, rank;
          struct NODE *next, *child;,
          struct ILCELL *il, *il_tail;
        } node;


The top structure of the heap(1) consists of a doubly-linked list [h.sub.1], ..., [h.sub.m], called the head-list: each head [h.sub.i] has two extra pointers: one (queue) points to the root [r.sub.i] of a distinct queue, and another (suffix_min) points to the root of minimum ckey among all [r.sub.j]'s (j [is greater than or equal to] i). We require that rank ([r.sub.1]) [is less than] ... [is less than] rank ([r.sub.m]). By extension, the rank of a queue (respectively, heap) refers to the rank of its root (respectively, [r.sub.m]). It is stored in the head [h.sub.i] as the integer variable rank.
typedef struct HEAD
        { struct NODE *queue;
          struct HEAD *next, *prev, *suffix_min;
          int rank;
        } head;


We initialize the soft heap by creating two dummy heads (global variables): header gives access to the head-list while tail, of infinite rank, represents the end of that list. The functions new_head and new_node create and initialize a new head and a new node in the trivial manner. The third global variable is the parameter r = r([Epsilon]).
head *header, *tail; int r;
header = new_head (); tail = new_head ();
tail [right arrow] rank = INFTY; header [right arrow] next = tail;
 tail [right arrow] prev = header;
printf ("Enter r:"); scanf ("%d", &r);


4. The Soft Heap Operations

We discuss a minor variant of the data structure, whose code is slightly simpler. It is straightforward to modify the data structure into a full-fledged soft heap. First, our variant bypasses the create operation and integrates it within insert. Also, note that the operation delete can be implemented in the lazy style by simply marking the item to be deleted accordingly. Then, actual work is required only when findmin returns an item that is marked as deleted. For this reason, we skip the discussion of findmin and delete altogether, and instead, focus on deletemin, the operation which finds an item of minimum key and deletes it. Again, it is immediate to modify the data structure to accommodate findmin and delete separately.

For each operation, we first give an informal description, sometimes using pseudo-code, and then follow up with heavily annotated C code. Except for drivers and I/O code, all that is needed to implement soft heaps is included here.

4.1. INSERTING AN ITEM. To insert a new item, we create an uncorrupted one-node queue, and we meld it into the heap.
C CODE FOR INSERT

insert (newkey)
int newkey;
{ node *q; ilcell *l;
  l = (ilcell *) malloc (sizeof (ilcell));
  l [right arrow] key = newkey; l [right arrow] next = NULL;
  q = new_mode (); q [right arrow] rank = 0;
  q [right arrow] ckey = newkey;
  q [right arrow] il = l; q [right arrow] il_tail = l;
  meld (q);
}


4.2. MELDING TWO HEAPS. Consider melding two heaps L and L'. We begin with a quick overview, and then we discuss the actual implementation of the operation. We break apart the heap of lesser rank, say L', by melding each of its queues into L. To meld a queue of rank k into L, we look for the smallest index i such that rank([r.sub.i]) [is greater than or equal to] k. (The dummy head tail ensures that i always exists.) If rank ([r.sub.i]) [is greater than] k, we insert the head right before [h.sub.i], instead. Otherwise, we meld the two queues into one of rank k + 1, by making the root with the larger key a new child of the other root. If rank ([r.sub.i+1]) = k + 1, a new conflict arises. We repeat the process as long as necessary like a carry propagation in binary addition. Finally, we update the suffix_min pointers between [h.sub.1] and the last head visited. When melding not a single queue but a whole heap, the last step can be done at the very end in just one pass through L. We give the code for melding a soft queue into a soft heap.

C CODE FOR MELD

Let q be a pointer (node *q) to the soft queue to be melded into the soft heap. First, we scan the head-list until we reach the point at which melding proper can begin. This leads us to the first head of rank at least that of q, which is denoted by tohead. To facilitate the insertion of the new queue, we also remember the preceding head, called prevhead.
meld (q)
node *q;
{       head *h, *prevhead, *tohead = header [right arrow] next;
        node *top, *bottom;
        while (q [right arrow] rank >
         tohead [right arrow] rank) tohead =
         tohead [right arrow] next;
        prevhead = tohead [right arrow] prev;


If there is already a queue of the same rank as q, we perform the carry propagation, as discussed earlier. When merging two queues, we use the variables top and bottom to specify which of the two queues end up at/below the root. We create a new node q pointing to top and bottom. Its item-list is inherited from top, and its rank is one plus that of top, (i.e., top [right arrow] rank +1). Finally, we update tohead to point to the next element down the head-list.
while (q [right arrow] rank == tohead [right arrow] rank)
 { if (tohead [right arrow] queue [right arrow] ckey
  > q [right arrow] ckey)
     {top = q; bottom = tohead [right arrow] queue; }
       else
     {top = tohead [right arrow] queue; bottom = q;}
   q = new_mode ();
   q [right arrow] ckey = top [right arrow] ckey;
    q [right arrow] rank = top [right arrow] rank +1;
   q [right arrow] child = bottom; q [right arrow] next = top;
   q [right arrow] il = top [right arrow] il; q [right arrow]
    il_tail = top [right arrow] il_tail;
   tohead = tohead [right arrow] next;
 }  /* end of while loop */


We are now ready to insert the new queue in the list of heads. We use a little trick: if a carry has actually taken place, then the head pointed to by prevhead [right arrow] next is now unused and so can be recycled as the head of the new queue. (We omit the garbage collection one might want to carry out to free the newly available space.) Otherwise, we create a new head h. We insert h between prevhead and tohead; all the heads inbetween can be discarded. Finally, we call fix_minlist (h) to restore the suffix_min pointers.
          if (prevhead == tohead [right arrow] prev)
           h = new_head ();
          else h = prevhead [right arrow] next;
          h [right arrow] queue = q; h [right arrow]
           rank = q [right arrow] rank;
          h [right arrow] prev = prevhead;
           h [right arrow] next = tohead;
          prevhead [right arrow] next =
           h; tohead [right arrow] prev=h;
          fix_minlist (h);
}


C CODE FOR FIX_MINLIST

Prior to calling fix_minlist (h), it is assumed that all suffix_min pointers are correct except for those between header and h. A simple walk from h back to header updates all the suffix_mins.
fix_minlist (h)
head *h;
{       head *tmpmin;
        if (h [right arrow] next == tail) tmpmin = h;
        else tmpmin = h [right arrow]
         next [right arrow] suffix_min;
        while (h != header)
         {if (h [right arrow] queue [right arrow] ckey
          < tmpmin [right arrow]
          queue [right arrow] ckey)
            tmpmin = h;
          h [right arrow] suffix_min = tmpmin;
          h = h [right arrow] prev;
         }
}


4.3. DELETEMIN. The suffix_min pointer at the beginning of the head-list points to the head h with the minimum ckey (corrupted or not). The trouble is that the item-list at that node might be empty. In that case, we must refill the item-list with items taken lower down in the queue pointed to by h. To do that, we call the function sift (h [right arrow] queue, h [right arrow] rank), which replaces the empty item-list by another one. If necessary, we iterate on this process until the new item-list at the root is not empty. The function sift is the heart of the soft heap so, without further ado, let us turn our discussion to it.

Taking as argument the node v at which the sifting takes place, the function sift attempts to move items up the tree towards the root. This is a standard operation in classical heaps. Typically, there is single recursive call in the procedure and the computation tree is a path. The twist is to call the recursion twice once in a while, in order to make the recursion tree branching, that is, truly a tree. This simple modification causes item-lists to collide on the way up, which we resolve by concatenating them. First, some pseudo-code:
           sift(v)
   item-list(v) [left arrow] T [left arrow] 0;
   if v has no child
   then set ckey (v) to [infinity] and return;
1. sift (v [right arrow] next);
   if ckey (v [right arrow] next) > ckey
    (v [right arrow] child)
      then exchange v [right arrow] next and
       v [right arrow] child;
   T [left arrow] T [union] item-list (v [right arrow] next);
   if loop-condition holds then goto 1;
   item-list(v) [left arrow] T.


The "loop-condition" statement is what makes soft heaps special. Without it, sift would be indistinguishable from the standard deletemin operation of a binomial tree. The loop-condition holds if (i) the goto has not yet been executed during this invocation of sift (i.e., branching is at most binary), (ii) the rank of v exceeds the threshold r and either it is odd or it exceeds the rank of the highest ranked child of v by at least two. The rank condition ensures that no corruption takes places too low in the queue; the parity condition is there to keep branching from occurring too often; finally the last condition ensures that branching does occur frequently enough. The variable T implements the car-pooling in the concatenation T [left arrow] T [union] item-list (v [right arrow] next). The cleanup is intended to prune the tree of nodes that have lost their item-lists to ancestors and whose ckeys have been set to [infinity].

C CODE FOR SIFT

The item-list at v is worthless and it is effectively emptied at the beginning. We test whether the node v is a leaf. If so, we bottom out by setting its ckey to infinity (i.e., a large integer), which will cause the node to stay at the bottom of the queue. If v is not a leaf then neither v [right arrow] next nor v [right arrow] child is NULL. In fact, this is a general invariant: both are null or neither one is. This might change temporarily within a call to sift but it is restored before the call ends.
node *sift (v)
node *v;
{      node *tmp;
       v [right arrow] il = NULL; v [right arrow] il_tail = NULL;
       if (v [right arrow] next == NULL, &&
        v [right arrow] child == NULL)
          { v [right arrow] ckey = INFTY; return v; }
       v [right arrow] next = sift (v [right arrow] next);


The new item-list at v [right arrow] next might now have a large ckey which violates the heap ordering. If so, we perform a rotation by exchanging children v [right arrow] next and v [right arrow] child.
if (v [right arrow] next [right arrow] ckey >
 v [right arrow] child [right arrow] ckey)
   { tmp = v [right arrow] child;
           v [right arrow] child = v [right arrow] next;
           v [right arrow] next = tmp;
   }


Once the children of v are in place we update the various pointers at v. In particular, the item-list of v [right arrow] next is passed on to v, and so is its ckey. Recall that while v [right arrow] child is truly a child of v in the soft queue, the node v [right arrow] next is a child of v only in the binary-tree implementation of the queue.
v [right arrow] il = v [right arrow] next [right arrow] il;
v [right arrow] il_tail = v [right arrow]
 next [right arrow] il_tail;
v [right arrow] ckey = v [right arrow] next [right arrow] ckey;


Next in line, the most distinctive feature of soft heaps: the possibility of sifting twice, that is, of creating a branching process in the recursion tree for sift. If the loop-condition is satisfied, meaning that the rank of v is odd and large enough, we sift again.
if (v [right arrow] rank > r &&
                     (v [right arrow] rank % 2 == 1 [parallel]
                      v [right arrow] child [right arrow] rank
                       [is less than] v [right arrow] rank-1))
   } v [right arrow] next = sift (v [right arrow] next);


As a result of the sifting, another rotation might be needed to restore heap ordering.
if (v [right arrow] next [right arrow] ckey >
 v [right arrow] child [right arrow] ckey)
   { tmp = v [right arrow] child;
     v [right arrow] child = v [right arrow] next;
     v [right arrow] next = tmp;
}


The item-list at v [right arrow] next should now be concatenated with the one at v, unless of course, it is empty or no longer defined. The latter case occurs when ckey is infinite at both v [right arrow] child and v [right arrow] next. Note that this could not happen after the previous sift.
if (v [right arrow] next [right arrow] ckey ! = INFTY
 && v [right arrow] next [right arrow] il ! = NULL)
   {v [right arrow] next [right arrow] il_tail [right arrow]
    next = v [right arrow] il;
    v [right arrow] il = v [right arrow] next [right arrow] il;
    if (v [right arrow] il_tail == NULL)
      v [right arrow] il_tail = v [right arrow]
       next [right arrow] il_tail;
    v [right arrow] ckey = v [right arrow]
     next [right arrow] ckey;
   }
} /*  end of second sift  */


We clean up the queue by removing the nodes with infinite ckeys. We do not update v [right arrow] rank since rank is defined with respect to the master tree. Note that this is where the rank and the number of children can be made to differ. In fact, we ensure that for any node v the ranks of its children (in the binary tree) are always equal, that is, v [right arrow] next [right arrow] rank = v [right arrow] child [right arrow] rank.
         if (v [right arrow] child [right arrow] ckey == INFTY)
            { if (v [right arrow] next [right arrow]
             ckey = = INFTY)
                 { v [right arrow] child = NULL;
                  v [right arrow] next = NULL; }
             else
               { v [right arrow] child = v [right arrow]
                next [right arrow] child;
                 v [right arrow] next = v [right arrow]
                  next [right arrow] next; }
            }
         return v;
}


C CODE FOR DELETEMIN

The function deletemin returns the item with the smallest ckey and deletes it. In practice, safe programming would dictate that we add safety code to warn the user against deleting from an empty heap and things of the sort. We dispense with such niceties here. The first suffix_min pointer takes us to the smallest ckey, which is what we want unless, of course, the corresponding item-list is empty. In that case, we call sift--perhaps more than once--to bring items back to the root. But first, we check whether the rank invariant is violated. Indeed, previous sifting might have caused the loss of too many children of the root and hence a violation of the invariant. We count the children of the root. (Alternatively, we could add a field to keep track of this number.)
deletemin ()
{ node  *sift (), *tmp;
      int min, childcount; head *h = header [right arrow]
       next [right arrow] suffix_min;
      while (h [right arrow] queue [right arrow] il == NULL)
        { tmp = h [right arrow] queue; childcount = 0;
          while (tmp [right arrow] next ! = NULL)
             { tmp = tmp [right arrow] next; childcount ++; }


The advantage in detecting a rank invariant violation so late in the game is that to fix it is much easier since the root's item-list is empty (else, what would we do with it?) If the rank invariant is violated (i.e., childcount [is less than] ??h [right arrow] rank/2??), we remove the queue and update the head-list and suffix_min pointers. Then, we dismantle the root by remelding back its children.
if (childcount < h [right arrow] rank/2)
   { h [right arrow] prev [right arrow] next =
    h [right arrow] next;
      h [right arrow] next [right arrow]
       prev = h [right arrow] prev;
     fix_minlist (h [right arrow] prev);
     tmp = h [right arrow] queue;
     while (tmp [right arrow] next ! = NULL)
       { meld (tmp [right arrow] child); tmp =
        tmp [right arrow] next; }
}


If the rank invariant holds, we are ready to refill the item-list at the root by calling sift.
 else
    { h [right arrow] queue = sift (h [right arrow] queue);
      if (h [right arrow] queue [right arrow] ckey == INFTY)
         { h [right arrow] prev [right arrow]
          next = h [right arrow] next;
           h [right arrow] next [right arrow] prev =
            h [right arrow] prev; h = h [right arrow] prev; }
      fix_minlist (h);
     }
 h = header [right arrow] next [right arrow] suffix_min;
}  /* end of outer while loop */


We are now in a position to delete the minimum-key item.
       min = h [right arrow] queue [right arrow]
        il [right arrow] key;
       h [right arrow] queue [right arrow] il =
        h [right arrow] queue [right arrow] il-next;
       if (h [right arrow] queue [right arrow] il == NULL)
        h [right arrow] queue [right arrow] il_tail = NULL;
       return min;
}


5. Complexity Analysis

We prove that the soft heap meets all its claims via a few simple lemmas. We consider a mixed sequence of operations including n inserts. To begin with, we must explain the correspondence between a soft queue and its master tree. When no deletion takes place the equivalence is obvious, and it is trivially preserved through inserts and melds. During sifting, the key observation is that v [right arrow] next [right arrow] rank and v [right arrow] child [right arrow] rank always remain identical. To enforce this equality is what can cause a discrepancy between rank and number of children. But it allows us to think of a rotation as an exchange between soft queues of the same rank (albeit with perhaps missing subtrees). The corresponding master trees having the same rank, they are isomorphic and therefore a rotation has no effect in the correspondence. Similarly, the cleanup prunes away subtrees, with no consequence on the queue/master-tree correspondence.

The interesting aspect of this correspondence is that the leaves of the master tree that are missing from the soft queue correspond to items which have migrated upward to join item-lists of nodes of positive rank. Such items can never again appear in leaves of any soft queue. Note that dismantling a node by remelding its children does not contradict this statement since it merely reconfigures the soft heap.

5.1. THE ERROR RATE. To achieve the desired error rate, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

LEMMA 5.1

|item-list (v)| [is less than or equal to] max{1, [2.sup.??rank(v)/2??-r/2]}.

PROOF. Until the first call to sift, all item-lists have size one, and the inequality holds. Afterwards, simple inspection shows that all operations have either no effect on the lemma's inequality or sometimes a favorable one (e.g., meld). All of them, that is, except for sift, which could potentially cause a violation. We show that this is not the case, and prove the lemma's inequality by induction. If sift (v) calls itself recursively, via sift (v [right arrow] next), only once, meaning that the loop-condition is not satisfied, then the item-list of v [right arrow] next (after possible rotation) migrates to a higher-ranking node by itself and the lemma holds by induction. Otherwise, the item-list at v becomes the union of the two item-lists associated with v [right arrow] next after each call to sift (v [right arrow] next). For this to happen, v [right arrow] rank must exceed r and one of two conditions must hold: either v [right arrow] rank is odd or it exceeds v [right arrow] child [right arrow] rank + 1. In the first case, after either recursive call, the rank of v [right arrow] next is strictly less than v [right arrow] rank, and by induction the size of either one of the item-lists of v [right arrow] next is at most

max{1, [2.sup.??(rank(v)-1)/2??-r/2??]} = [2.sup.??(rank(v)-1)/2??-r/2] = [2.sup.??rank(v)/2??-1-r/2].

Note that the max disappears because rank(v) [is greater than] r. In the other case, the size of either one of the item-lists of v [right arrow] next is at most

max{1, [2.sup.??(rank(v)-2)/2??-r/2]} = [2.sup.??rank(v)/2??-1-r/2].

This time, the max disappears because r and the rank of v are both even, and so rank (v) [is greater than or equal to] r + 2. In sum, the size of the union is at most 2 x [2.sup.??rank(v)/2??-1-r/2], which proves the lemma. []

LEMMA 5.2. The soft heap contains at most n/[2.sup.r-3] corrupted items at any given time.

PROOF. We begin with a simple observation. If S is the node set of a binomial tree, then

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This follows from the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where k is the rank of the binomial tree. A proof by induction is immediate and can be omitted. Recall that the ranks of the nodes of a queue q are derived from the corresponding nodes in its master tree q'. So, the set R (respectively, R') of nodes of rank greater than r in q (respectively, q') is such that |R| [is less than or equal to] |R'|. Within q', the nodes of R' number a fraction at most 1/[2.sup.r] of all the leaves. Summing over all master trees, we find that

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There is no corrupted item at any rank [is less than or equal to] r, and so by Lemma 5.1 their total number does not exceed

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Each R' forms a binomial tree by itself, where the rank of node v becomes rank(v) - r - 1. So, by (1, 2), the sum in (3) is at most [[Sigma].sub.q'], 8|R'| [is less than or equal to] n/[2.sup.r-3]. []

5.2. THE RUNNING TIME. Only meld and sift need to be looked at, all other operations being trivially constant-time. Assigning one credit per queue takes care of the carry propagation during a meld. Indeed, two queues of the same rank combine into one, which releases one credit to pay for the work. Updating suffix-min pointers can take time, however. Specifically, carries aside, the cost of melding two soft heaps L and L' is at most the smaller rank of the two (up to a constant factor). The entire sequence of soft heap melds can be modeled as a binary tree M. A leaf z denotes a one-item heap (its cost is 1). An internal node z indicates the melding of two heaps. Since heaps can grow only through melds, the added size of the master trees in the soft heap at z is proportional to the number N(z) of descendants in M. The cost of node z (i.e., of the meld) is 1 + log min {N(x), N(y)}, where x and y are the left and right children of z.(2) A simple recurrence (see e.g., Hoffman et al. [1986]) shows that adding together all these costs gives a total melding cost linear in the size of M, that is, O(n).

For this analysis to be correct, no node dismantling should ever take place. We can easily amend it, however, to cover the general case. For the purpose of the analysis, let us not regard the remeldings caused by dismantling as heap melds but as queue melds. The benefit is to leave the tree M unchanged. The dismantle-induced melds associated with a node z of M reconfigure the soft heap at z by removing some of its nodes and restoring the rank invariant. This can only decrease the value of N (z), so the previous analysis remains correct.

Of course, the queue melds associated with node x must now be accounted for. Dismantling node v causes no more than rank(v) queue melds. By the violation of the rank invariant, the node v has at least one missing child of rank [is greater than or equal to] ??rank(v)/2??. In the master tree, there are at least [2.sup.??rank(v)/2??-1] leaves at or below that child, and all have disappeared from the soft queue. So, we can charge the dismantle-induced melds against these leaves, and conclude that melding takes O(n) time.

Finally, we show that the cost of all calls to sift is O(rn). Consider any decreasing sequence of integers. An integer m is called good if it is odd or if its successor is less than m - 1. Clearly, any subsequence of size two contains at least one good integer. Now, consider the computation tree corresponding to an execution of sift (v). By examining the sequence of ranks along any root-to-leaf path, our previous observation leads us to conclude that along any path of size at least r, at least one branching must occur (not necessarily many more than that because no branching occurs at rank r and below). It follows that, excluding the updating of suffix-min pointers, the running time is O(rC), where C is the number of times the loop-condition succeeds.

It is easy to see, by induction, that if v is the root of a subtree with fewer than two finite ckeys in the subtree below, the computation tree of sift (v) is of constant size. Conversely, if the subtree contains at least two finite ckeys at distinct nodes, then if the loop-condition is satisfied at v, both calls of the form sift (v [right arrow] next) bring finite ckeys to the root and two nonempty item-lists are thus concatenated. There can be at most n - 1 such merges, therefore C [is less than or equal to] n and our claim holds.

We ignored the cost of updating suffix_min pointers after each call to sift. Maintaining the rank invariant makes the cost of suffix_min updating negligible. Indeed, each update takes time proportional to the rank of the queue: (i) if the rank invariant holds, then the updating time is dominated by the cost of sift itself (already accounted for); (ii) otherwise, the root v is dismantled, which, as we just saw, releases [2.sup.??rank(v)/2??-1] leaves against which we can charge the updating cost. By Lemma 5.2, the total number of corrupted items is bounded by n/[2.sup.r-3]. Setting r = 2 + 2??log(1/[Epsilon])?? proves Theorem 1.1 (except for the optimality claim). []

Remark. The storage is linear in the number of insertions n, but not necessarily in the actual number of items present. If storage is at a premium, here is a simple fix: as soon as the number of live items falls below, say, n/2, we reconfigure the soft heap entirely. We reinsert all the uncorrupted items and, separately, we string together the corrupted ones into a single item-list whose common key is set to infinity. The time for reconfiguring the heap can be charged to the n/2 deleted elements. Regarding corruption, the reconfiguration merely doubles the number of insertions, and so the number of corrupted items after I (user-requested) insertions will be at most 2[Epsilon]I. So, it suffices to replace [Epsilon] by [Epsilon]/2 to achieve an error rate of [Epsilon]. This modified soft heap is optimal in storage, and as shown below, in time.

6. Optimality

To complete the proof of Theorem 1.1, we show that the soft heap is optimal. Without loss of generality, we can assume that 1/[Epsilon] lies between a large constant and [square root of]n and that n/??[Epsilon]n?? is an integer l. Apply Theorem 2.1 to n distinct numbers and pick out every 2??[Epsilon]n??-th element in the output sequence. By the near-sortedness of the output, the chosen subsequence is already sorted. It partitions the set of numbers into disjoint intervals, within which we can easily locate the other numbers in linear time (each element being at most a constant number of intervals off its enclosing interval). From this, in particular, we derive the true rank of the selected numbers. By using linear selection-finding within each interval, we easily, in O(n) time, retrieve the k??[Epsilon]n??-th largest number, for k = 1, 2, ... , l. We have partitioned the set of n numbers into disjoint intervals of size ??[Epsilon]n??. Let [n.sub.1] = ... = [n.sub.l] = ??[Epsilon]n??. A standard counting argument shows that any comparison-based tree for performing this computation is of height at least

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, the entire algorithm requires [Omega](n log 1/[Epsilon]) time, but it involves O(n) operations on a soft heap with error rate [Epsilon], followed by O(n)-time postprocessing. It follows that the O(log 1/[Epsilon]) amortized complexity of the soft heap is optimal. []

ACKNOWLEDGMENTS. I wish to thank the anonymous referee for some useful suggestions.

(1) For brevity, we drop the "soft."

(2) We use the fact that the rank of a soft queue is exactly the logarithm of the number of nodes in its master tree.

REFERENCES

BLUM, M., FLOYD, R. W., PRATT, V., RIVEST, R. L., AND TARJAN, R.E. 1973. Time bounds for selection. J. Comput. Syst. Sci. 7, 448-461.

CHAZELLE, B. 2000. A minimum spanning tree algorithm with inverse-Ackermann type complexity. J. ACM 47, 6 (Nov.), 000-000.

FREDMAN, M. L., AND TARJAN, R.E. 1987. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596-615.

HOFFMAN, K., MEHLHORN, K., ROSENSTIEHL, P., AND TARJAN, R. E. 1986. Sorting Jordan sequences in linear time using level-linked search trees. Inf. Cont. 68, 170-184.

VUILLEMIN, J. 1978. A data structure for manipulating priority queues. Commun. ACM 21, 309-315.

RECEIVED FEBRUARY 1998; REVISED JUNE 1999; ACCEPTED MARCH 2000

A preliminary version of this paper as CHAZELLE, B. 1998. Car-pooling as a data structuring device: The soft heap. In Proceedings of the 6th Annual European Symposium on Algorithms, pp. 35-42.

This work was supported in part by National Science Foundation (NSF) Grants CCR 93-01254 and CCR 96-23768, ARO Grant DAAH04-96-1-0181, and NEC Research Institute.

Author's address: Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ 08544-2087, e-mail: chazelle@cs.princeton.edu or NEC Research Institute, e-mail: chazelle@research.nj.nec.com.
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Date:Nov 1, 2000
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