Priority Queue / Heap
• Stores (key,data) pairs (like dictionary) • But, different set of operations:
• Initialize‐Heap: creates new empty heap • Is‐Empty: returns true if heap is empty • Insert(key,data): inserts (key,data)‐pair, returns pointer to entry • Get‐Min: returns (key,data)‐pair with minimum key • Delete‐Min: deletes minimum (key,data)‐pair • Decrease‐Key(entry,newkey): decreases key of entry to newkey • Merge: merges two heaps into one
Algorithm Theory, WS 2012/13 Fabian Kuhn 1 Implementation of Dijkstra’s Algorithm
Store nodes in a priority queue, use , as keys: 1. Initialize , 0 and , ∞ for all 2. All nodes are unmarked
3. Get unmarked node which minimizes , :
4. mark node
5. For all , ∈ , , min , , ,
6. Until all nodes are marked
Algorithm Theory, WS 2012/13 Fabian Kuhn 2 Analysis
Number of priority queue operations for Dijkstra: • Initialize‐Heap:
• Is‐Empty: | |
• Insert: | |
• Get‐Min: | |
• Delete‐Min: | |
• Decrease‐Key: | |
• Merge:
Algorithm Theory, WS 2012/13 Fabian Kuhn 3 Priority Queue Implementation
Implementation as min‐heap:
complete binary tree, e.g., stored in an array
• Initialize‐Heap: • Is‐Empty: • Insert: • Get‐Min: • Delete‐Min: • Decrease‐Key: • Merge (heaps of size and , ): Algorithm Theory, WS 2012/13 Fabian Kuhn 4 Better Implementation
• Can we do better?
• Cost of Dijkstra with complete binary min‐heap implementation: log
• Can be improved if we can make decrease‐key cheaper…
• Cost of merging two heaps is expensive
• We will get there in two steps:
Binomial heap Fibonacci heap
Algorithm Theory, WS 2012/13 Fabian Kuhn 5 Definition: Binomial Tree
Binomial tree of order 0:
Algorithm Theory, WS 2012/13 Fabian Kuhn 6 Binomial Trees
Algorithm Theory, WS 2012/13 Fabian Kuhn 7 Properties
1. Tree has 2 nodes
2. Height of tree is
3. Root degree of is
4. In , there are exactly nodes at depth
Algorithm Theory, WS 2012/13 Fabian Kuhn 8 Binomial Coefficients
• Binomial coefficient: : # of element subsets of a set of size
• Property: Pascal triangle:
Algorithm Theory, WS 2012/13 Fabian Kuhn 9 Number of Nodes at Depth in
Claim: In , there are exactly nodes at depth
Algorithm Theory, WS 2012/13 Fabian Kuhn 10 Binomial Heap
• Keys are stored in nodes of binomial trees of different order
nodes: there is a binomial tree or order iff bit of base‐2 representation of is 1.
• Min‐Heap Property: Key of node keys of all nodes in sub‐tree of
Algorithm Theory, WS 2012/13 Fabian Kuhn 11 Example
• 10 keys: 2, 5, 8, 9, 12, 14, 17, 18, 20, 22, 25
• Binary representation of : 11 1011 trees , , and present
5 2 17
9 14 20 8
12 18 25
22
Algorithm Theory, WS 2012/13 Fabian Kuhn 12 Child‐Sibling Representation
Structure of a node: parent key degree child sibling
Algorithm Theory, WS 2012/13 Fabian Kuhn 13 Link Operation
• Unite two binomial trees of the same order to one tree: 12 ⨁ ⇒
• Time: 15 20 18 ⨁
25 40 22
30
Algorithm Theory, WS 2012/13 Fabian Kuhn 14 Merge Operation
Merging two binomial heaps: • For , ,…, : If there are 2 or 3 binomial trees : apply link operation to merge 2 trees into one binomial tree
Time:
∪
Algorithm Theory, WS 2012/13 Fabian Kuhn 15 Example
9 13 5 2 17
12 18 14 20 8
22 25
Algorithm Theory, WS 2012/13 Fabian Kuhn 16 Operations
Initialize: create empty list of trees
Get minimum of queue: time 1 (if we maintain a pointer)
Decrease‐key at node : • Set key of node to new key • Swap with parent until min‐heap property is restored • Time: log
Insert key into queue : 1. Create queue of size 1 containing only 2. Merge and • Time for insert: log
Algorithm Theory, WS 2012/13 Fabian Kuhn 17 Operations
Delete‐Min Operation:
1. Find tree with minimum root
2. Remove from queue queue ′
3. Children of form new queue ′′
4. Merge queues ′ and ′′
• Overall time:
Algorithm Theory, WS 2012/13 Fabian Kuhn 18 Delete‐Min Example
2 5 17
9 14 20 8
12 18 25
22
Algorithm Theory, WS 2012/13 Fabian Kuhn 19 Complexities Binomial Heap
• Initialize‐Heap: • Is‐Empty: • Insert: • Get‐Min: • Delete‐Min: • Decrease‐Key: • Merge (heaps of size and , ):
Algorithm Theory, WS 2012/13 Fabian Kuhn 20 Can We Do Better?
• Binomial heap: insert, delete‐min, and decrease‐key cost log
• One of the operations insert or delete‐min must cost Ω log : – Heap‐Sort: Insert elements into heap, then take out the minimum times – (Comparison‐based) sorting costs at least Ω log .
• But maybe we can improve decrease‐key and one of the other two operations?
• Structure of binomial heap is not flexible: – Simplifies analysis, allows to get strong worst‐case bounds – But, operations almost inherently need at least logarithmic time
Algorithm Theory, WS 2012/13 Fabian Kuhn 21 Fibonacci Heaps
Lacy‐merge variant of binomial heaps: • Do not merge trees as long as possible…
Structure: A Fibonacci heap consists of a collection of trees satisfying the min‐heap property.
Variables: • . : root of the tree containing the (a) minimum key • . : circular, doubly linked, unordered list containing the roots of all trees • . : number of nodes currently in
Algorithm Theory, WS 2012/13 Fabian Kuhn 22 Trees in Fibonacci Heaps
Structure of a single node :
parent right left key degree
child mark
• . : points to circular, doubly linked and unordered list of the children of • . , . : pointers to siblings (in doubly linked list) • . : will be used later… Advantages of circular, doubly linked lists: • Deleting an element takes constant time • Concatenating two lists takes constant time Algorithm Theory, WS 2012/13 Fabian Kuhn 23 Example
Figure: Cormen et al., Introduction to Algorithms
Algorithm Theory, WS 2012/13 Fabian Kuhn 24 Simple (Lazy) Operations
Initialize‐Heap : • . ≔ . ≔
Merge heaps and ′: • concatenate root lists • update .
Insert element into : • create new one‐node tree containing H′ • merge heaps and ′
Get minimum element of : • return .
Algorithm Theory, WS 2012/13 Fabian Kuhn 25 Operation Delete‐Min
Delete the node with minimum key from and return its element:
1. ≔ . ; 2. if . 0 then 3. remove . from . ; 4. add . . (list) to . 5. . ;
// Repeatedly merge nodes with equal degree in the root list // until degrees of nodes in the root list are distinct. // Determine the element with minimum key
6. return
Algorithm Theory, WS 2012/13 Fabian Kuhn 26 Rank and Maximum Degree
Ranks of nodes, trees, heap:
Node : • : degree of
Tree : • : rank (degree) of root node of
Heap : • : maximum degree of any node in
Assumption ( : number of nodes in ): – for a known function
Algorithm Theory, WS 2012/13 Fabian Kuhn 27 Merging Two Trees
Given: Heap‐ordered trees , ′ with • Assume: min‐key of min‐key of ′
Operation , : • Removes tree ′ from root list ′ and adds to child list of
• ≔ 1 • . ≔ ′
Algorithm Theory, WS 2012/13 Fabian Kuhn 28 Consolidation of Root List
Array pointing to find roots with the same rank:
012 Consolidate: Time: 1. for ≔0to do ≔null; | . | . 2. while . null do 3. ≔“delete and return first element of . ” 4. while nulldo 5. ≔ ; 6. ≔ ; 7. ≔ , 8. ≔ 9. Create new . and . Algorithm Theory, WS 2012/13 Fabian Kuhn 29 Consolidate Example
31 5 25 1 2 15 17 3
9 14 20 13 8 19 7
12 18 22
5 25 1 2 15 17 3
9 14 20 31 13 8 19 7
12 18 22
Algorithm Theory, WS 2012/13 Fabian Kuhn 30 Consolidate Example
5 25 1 2 15 17 3
9 14 20 31 13 8 19 7
12 18 22
5 1 2 15 17 3
9 14 20 25 13 8 19 7
12 18 22 31
Algorithm Theory, WS 2012/13 Fabian Kuhn 31 Consolidate Example
5 1 2 15 17 3
9 14 20 25 13 8 19 7
12 18 22 31
5 1 2 15 17
9 14 20 25 13 3 8
12 18 22 31 19 7
Algorithm Theory, WS 2012/13 Fabian Kuhn 32 Consolidate Example
5 1 2 15 17
9 14 20 25 13 3 8
12 18 22 31 19 7
1 2 15 17
5 25 13 3 8
9 14 20 31 19 7
12 18 22
Algorithm Theory, WS 2012/13 Fabian Kuhn 33 Consolidate Example
1 2 15 17
5 25 13 3 8
9 14 20 31 19 7
12 18 22 1 2 15
5 25 13 3 8 17
9 14 20 31 19 7
12 18 22
Algorithm Theory, WS 2012/13 Fabian Kuhn 34 Consolidate Example
1 2 15
5 25 13 3 8 17
9 14 20 31 19 7
12 18 22 1 2
5 25 13 3 8 15
9 14 20 31 19 7 17
12 18 22
Algorithm Theory, WS 2012/13 Fabian Kuhn 35 Operation Decrease‐Key
Decrease‐Key , : (decrease key of node to new value )
1. if . then return; 2. . ≔ ; update . ; 3. if ∈ . ∨ . . then return 4. repeat 5. ≔ . ; 6. . ; 7. ≔ ; 8. until . ∨ ∈ . ; 9. if ∉ . then . ≔ ;
Algorithm Theory, WS 2012/13 Fabian Kuhn 36 Operation Cut( )
Operation . : • Cuts ’s sub‐tree from its parent and adds to rootlist
1. if ∉ . then 2. // cut the link between and its parent 3. . ≔ . 1; 4. remove from . . (list) 5. . ≔ null; 6. add to . 1 2 15 1 3 2 15 25 13 3 8 25 13 19 7 8
31 19 7 31 Algorithm Theory, WS 2012/13 Fabian Kuhn 37 Decrease‐Key Example
• Green nodes are marked 5 1 2 15 17
9 14 20 25 13 3 8
12 18 31 19 7 Decrease‐Key , 22
5 18 9 1 2 15 17
14 20 22 12 25 13 3 8
31 19 7
Algorithm Theory, WS 2012/13 Fabian Kuhn 38 Fibonacci Heap Marks
History of a node :
is being linked to a node . ≔ a child of is cut . ≔ a second child of is cut .
• Hence, the boolean value . indicates whether node has lost a child since the last time was made the child of another node.
Algorithm Theory, WS 2012/13 Fabian Kuhn 39 Cost of Delete‐Min & Decrease‐Key
Delete‐Min: 1. Delete min. root and add . to . time: 1 2. Consolidate . time: length of . • Step 2 can potentially be linear in (size of ) Decrease‐Key (at node ): 1. If new key parent key, cut sub‐tree of node time: 1 2. Cascading cuts up the tree as long as nodes are marked time: number of consecutive marked nodes • Step 2 can potentially be linear in
Exercises: Both operations can take time in the worst case!
Algorithm Theory, WS 2012/13 Fabian Kuhn 40 Cost of Delete‐Min & Decrease‐Key
• Cost of delete‐min and decrease‐key can be Θ … – Seems a large price to pay to get insert and merge in 1 time
• Maybe, the operations are efficient most of the time? – It seems to require a lot of operations to get a long rootlist and thus, an expensive consolidate operation – In each decrease‐key operation, at most one node gets marked: We need a lot of decrease‐key operations to get an expensive decrease‐key operation
• Can we show that the average cost per operation is small?
• We can requires amortized analysis
Algorithm Theory, WS 2012/13 Fabian Kuhn 41 Amortization
• Consider sequence , ,…, of operations (typically performed on some data structure )
• : execution time of operation • ≔ ⋯ : total execution time
• The execution time of a single operation might
vary within a large range (e.g., ∈ 1, )
• The worst case overall execution time might still be small average execution time per operation might be small in the worst case, even if single operations can be expensive
Algorithm Theory, WS 2012/13 Fabian Kuhn 42 Analysis of Algorithms
• Best case
• Worst case
• Average case
• Amortized worst case
What it the average cost of an operation in a worst case sequence of operations?
Algorithm Theory, WS 2012/13 Fabian Kuhn 43 Example: Binary Counter
Incrementing a binary counter: determine the bit flip cost: Operation Counter Value Cost 00000 1 00001 1 2 00010 2 3 00011 1 400100 3 5 00101 1 6 00110 2 7 00111 1 801000 4 9 01001 1 10 01010 2 11 01011 1 12 01100 3 13 01101 1
Algorithm Theory, WS 2012/13 Fabian Kuhn 44 Accounting Method
Observation: • Each increment flips exactly one 0 into a 1 00100 1111 ⟹ 00100 0000 Idea: • Have a bank account (with initial amount 0) • Paying to the bank account costs • Take “money” from account to pay for expensive operations
Applied to binary counter: • Flip from 0 to 1: pay 1 to bank account (cost: 2) • Flip from 1 to 0: take 1 from bank account (cost: 0) • Amount on bank account = number of ones We always have enough “money” to pay!
Algorithm Theory, WS 2012/13 Fabian Kuhn 45 Accounting Method
Op. Counter Cost To Bank From Bank Net Cost Credit 0 0 0 0 0 10 0 0 0 1 1 20 0 0 1 0 2 30 0 0 1 1 1 40 0 1 0 0 3 50 0 1 0 1 1 60 0 1 1 0 2 70 0 1 1 1 1 80 1 0 0 0 4 90 1 0 0 1 1 10 0 1 0 1 0 2
Algorithm Theory, WS 2012/13 Fabian Kuhn 46 Potential Function Method
• Most generic and elegant way to do amortized analysis! – But, also more abstract than the others…
• State of data structure / system: ∈ (state space)
Potential function : →
• Operation :
– : actual cost of operation – : state after execution of operation ( : initial state) – ≔Φ : potential after exec. of operation – : amortized cost of operation :
≔
Algorithm Theory, WS 2012/13 Fabian Kuhn 47 Potential Function Method
Operation :
actual cost: amortized cost: Φ Φ Overall cost:
≔ Φ Φ
Algorithm Theory, WS 2012/13 Fabian Kuhn 48 Binary Counter: Potential Method
• Potential function: :
• Clearly, Φ 0and Φ 0for all 0
• Actual cost : . 1 flip from 0 to 1
. 1flips from 1 to 0
• Potential difference: Φ Φ 1 1 2
• Amortized cost: Φ Φ 2
Algorithm Theory, WS 2012/13 Fabian Kuhn 49 Back to Fibonacci Heaps
• Worst‐case cost of a single delete‐min or decrease‐key operation is Ω
• Can we prove a small worst‐case amortized cost for delete‐min and decrease‐key operations?
Remark:
• Data structure that allows operations ,…,
• We say that operation has amortized cost if for every execution the total time is
⋅ ,
where is the number of operations of type Algorithm Theory, WS 2012/13 Fabian Kuhn 50 Amortized Cost of Fibonacci Heaps
• Initialize‐heap, is‐empty, get‐min, insert, and merge have worst‐case cost
• Delete‐min has amortized cost • Decrease‐key has amortized cost
• Starting with an empty heap, any sequence of operations with at most delete‐min operations has total cost (time)
.
• Cost for Dijkstra: | | | | log | |
Algorithm Theory, WS 2012/13 Fabian Kuhn 51