Rank-Pairing Heaps

Bernhard Haeupler, Siddhartha Sen,and Robert Tarjan, ESA 2009

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Observation

Computer science is (still) a young field.

We often settle for the first (good) solution.

It may not be the best: the design space is rich.

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Research Agenda

For fundamental problems, systematically explore the design space to find the best solutions, seeking

elegance: “a quality of neatness and ingenious simplicity in the solution of a problem (especially in science or mathematics).”

wordnet.princeton.edu/perl/webwn

Keep the design simple, allow complexity in the analysis.

Heap (Priority Queue) ProblemMaintain a set (heap) of items, each with a real-valued key,under the operations

Find minimum: find the item of minimum key in a heapInsert: add a new item to a heapDelete minimum : remove the item of minimum keyMeld: Combine two item-disjoint heaps into oneDecrease key: subtract a given positive amount from the key

of a given item in a known heap

Goal: O(log n) for delete min, O(1) for others

Related Work

Fibonacci heaps achieve the desired bounds (Fredman & Tarjan, 1984); so do

• Peterson’s heaps (1987)• Høyer’s heaps (1995)• Brodal’s heaps (1996), worst-case• Thin heaps (Kaplan & Tarjan, 2008)• Violation heaps (Elmasry, 2008)• Quake heaps (Chan, 2009)

Not pairing heaps:(loglog n) time per key decrease, but good in practice

Heap-ordered modelHalf-ordered model

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17 13

8

7

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Half-ordered model

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17 13

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Half-ordered model

11 17

7

5

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13 8

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Half-ordered tree: binary tree, one item per node, each item less than all items in left subtree

Half tree: half-ordered binary tree with no right subtree

Link two half trees:

O(1) time, preserves half order

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10

A B

10

A

5

B++

Heap: a set (circular singly-linked list) of half trees, with minimum root first on list; access via last on list

Find min: return minimumInsert: Form a new one-node tree, combine with

current set of half trees, update the minimumMeld: Combine sets of half trees, update the minimumDelete min: Remove minimum root (forming new half

trees); Repeatedly link half trees, form a set of the remaining trees

How to link? Use ranks: leaves have rank zero,only link trees whose roots have equal rank,increase winner’s rank by one:

All rank differences are 1: a half tree of rank k is a perfect binary tree plus a root: 2k nodes, rank = lg n

++k

k - 1

k

k - 1 k

k - 1 k - 1

k + 1

Vuillemin’s binomial queues!

Vuillemin’s binomial queues!

Delete minimum

Delete min

Each node on right path becomes a new root

Link half trees of equal rankArray of buckets, at most one per rank

Delete min

Delete min

Link half trees of equal rankArray of buckets, at most one per rankForm new set of half trees

“multipass”

Delete min

Form new set of half treesFind new minimum in O(log n) time

Keep track of minimum during linksFind minimum in O(log n)

additional time

Delete min: lazier linking

“one-pass”

Form new set of half trees

Delete min: lazier linking

Amortized Analysis of Lazy Binomial Queues

= #treesLink: O(1) time, = -1, amortized time = 0Insert: O(1) time, = 1Meld: O(1) time, = 0Delete min: if k trees after root removal, time is

O(k), potential decreases by k/2 – O(log n) O(log n) amortized time

Decrease key?

Application: Dijkstra’s shortest path algorithm, othersMethod: To decrease key of x, detach its half tree,

restructure if necessary

(If x is the right child of u, no easy way to tell if half order is violated)

u

x

y

u

y

x

How to maintain structure?

All previous methods, starting with Fibonacci heaps, change ranks and restructure

Some, like Quake heaps (Chan, 2009) and Relaxed heaps (Driscoll et al., 1988), do not restructure during key decrease, but this just postpones restructuring

But all that is needed is rank changes:Trees can have arbitrary structure!

Rank-Pairing Heaps = rp-heaps

Goal is SIMPLICITY

Goal is SIMPLICITY

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Node RanksEach node has a non-negative integer rank

Convention: missing nodes have rank -1 (leaves have rank 0)

rank difference of a child =rank of parent - rank of child

i-child: node of rank difference ii,j-node: children have rank differences i and j

Convention: the child of a root is a 1-child

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Rank Rules

Easy-to-analyze version (type 2):All rank differences are non-negativeIf rank difference exceeds 2, sibling has rank difference 0If rank difference is 0, sibling has rank difference at least 2

Simpler but harder-to-analyze version (type 1):If rank difference exceeds 1, sibling has rank difference 0If rank difference is 0, sibling has rank difference ≥ 1

1 12 21 0≥ 2 ≥ 20

11 0≥ 1 ≥ 10

Tree Size

If nk is minimum number of descendants of a node of rank k,

n0 = 1, nk = nk-1 + nk-2 : Fibonacci numbers

nk ≥ k, =

k log n

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0102

10

Decrease key

x

u

0 6

6 0

8

0

1k - 1

1 1

k1

y

Detach half treeRestore rank rule

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0

u0 2

6u

1 1

6u

0 5

u k - 1

u

0-children block propagation of rank decreases from sibling’s subtree

Amortized AnalysisPotential of node = sum of rank differences of children - 1 +1 if root (= 1) -1 if 1,1-node (= 0)

Link is free:

One unit pays for link

Insert needs 1 unit, meld none

++1

(+1) (+1) (+1)

(0)1 1

1 1

Delete min rank k

Each 1,1 needs potential 1, adding at most k in total.Delete min takes O(log n) amortized time

1

1 unit needed

1

0≥ 2

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Decrease KeySuccessive rank decreases are non-increasing

At most two 1,1’s occur on path of rank decreases – 1,1 becomes 0, j : prev decrease >1, next decrease = 1 1,1 becomes 1,2 : terminal

Give each 1,1 one extra unit of potential

Each rank decrease releases a unit to pay for decrease: rank diffs of both children decrease by k, rank diff of

parent increases by k

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5110

5131

71712

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010

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Decrease key

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0 6

6 0

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0

1k - 1

1 1

k1

Detach half treeRestore rank rule

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0

0 2

61 1

60 5

k - 11

Type-1 rp-heaps

Max k lg n

Analysis requires a more elaborate potential based on rank differences of children and grandchildren

Same bounds as type-2 rp-heaps, provided we preferentially link half trees from disassembly

Summary

Rank-pairing heaps combine performance guarantees of Fibonacci heaps with simplicity approaching pairing heaps

Type-1 rp-heaps are especially simple, but not simple to analyze

Results build on prior work by Peterson, Høyer, and Kaplan and Tarjan, and may be the natural conclusion of this work

Simpler methods of doing key decrease are not efficient (see paper)

Preliminary experiments show rp-heaps are competitive with pairing heaps on typical input sequences, and better on worst-case sequences

Additional Questions

One-tree version of rp-heaps?Yes, but unfair links add losers to the bottom of the loser list instead of the top

Decrease key without cascading rank changes?No, with O(log n) time per delete minMaybe, with O(log m) time per delete min

need at least one rank decrease

Simplify Brodal’s heaps?