Chapter 4 Data Structures Algorithm Theory WS 2014/15 Fabian Kuhn 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 2014/15 Fabian Kuhn 2 Example Figure: Cormen et al., Introduction to Algorithms Algorithm Theory, WS 2014/15 Fabian Kuhn 3 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 2014/15 Fabian Kuhn 4 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 2014/15 Fabian Kuhn 5 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 2014/15 Fabian Kuhn 6 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 2014/15 Fabian Kuhn 7 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 2014/15 Fabian Kuhn 8 Accounting Method Observation: • Each increment flips exactly one 0 into a 1 001001111 ⟹ 001000000 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 2014/15 Fabian Kuhn 9 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 2014/15 Fabian Kuhn 10 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 2014/15 Fabian Kuhn 11 Potential Function Method Operation : actual cost: amortized cost: Φ Φ Overall cost: ≔ Φ Φ Algorithm Theory, WS 2014/15 Fabian Kuhn 12 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 2014/15 Fabian Kuhn 13 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 2014/15 Fabian Kuhn 14 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) . • We will now need the marks… • Cost for Dijkstra: || || log || Algorithm Theory, WS 2014/15 Fabian Kuhn 15 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 2014/15 Fabian Kuhn 16 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 2014/15 Fabian Kuhn 17 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 2014/15 Fabian Kuhn 18 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 2014/15 Fabian Kuhn 19 Fibonacci Heaps: Marks Cycle of a node: 1. Node is removed from root list and linked to a node . 2. Child node of is cut and added to root list . 3. Second child of is cut node is cut as well and moved to root list The boolean value .. indicates whether node has lost a child since the last time was made the child of another node. Algorithm Theory, WS 2014/15 Fabian Kuhn 20 Potential Function System state characterized by two parameters: • : number of trees (length of . ) • : number of marked nodes that are not in the root list Potential function: ≔ Example: 5 18 9 1 2 15 17 14 20 22 12 25 13 3 8 31 19 7 • 7, 2 Φ11 Algorithm Theory, WS 2014/15 Fabian Kuhn 21 Actual Time of Operations • Operations: initialize‐heap, is‐empty, insert, get‐min, merge actual time: 1 – Normalize unit time such that ,,,, 1 • Operation delete‐min: – Actual time: length of . – Normalize unit time such that length of . • Operation descrease‐key: – Actual time: length of path to next unmarked ancestor – Normalize unit time such that lengthof path to next unmarked ancestor Algorithm Theory, WS 2014/15 Fabian Kuhn 22 Amortized Times Assume operation is of type: • initialize‐heap: – actual time: 1, potential: Φ Φ 0 – amortized time: Φ Φ 1 • is‐empty, get‐min: – actual time: 1, potential: Φ Φ (heap doesn’t change) – amortized time: Φ Φ 1 • merge: – Actual time: 1 – combined potential of both heaps: Φ Φ – amortized time: Φ Φ 1 Algorithm Theory, WS 2014/15 Fabian Kuhn 23 Amortized Time of Insert Assume that operation is an insert operation: • Actual time: 1 • Potential function: – remains unchanged (no nodes are marked or unmarked, no marked nodes are moved to the root list) – grows by 1 (one element is added to the root list) , 1 Φ Φ 1 • Amortized time: Algorithm Theory, WS 2014/15 Fabian Kuhn 24 Amortized Time of Delete‐Min Assume that operation is a delete‐min operation: Actual time: . Potential function : • : changes from . to at most • : (# of marked nodes that are not in the root list) – no new marks – if node is moved away from root list, . is set to false value of does not increase! , . Φ Φ |. | Amortized Time: Algorithm Theory, WS 2014/15 Fabian Kuhn 25 Amortized Time of Decrease‐Key Assume that operation is a decrease‐key operation at node : Actual time: length of path to next unmarked ancestor Potential function : • Assume, node and nodes ,…, are moved to root list – ,…, are marked and moved to root list, . mark is set to true • marked nodes go to root list, 1node gets newly marked • grows by 1, grows by 1 and is decreased by 1, 1 Φ Φ 1 2 1 Φ 3 Amortized time: Algorithm Theory, WS 2014/15 Fabian Kuhn 26 Complexities Fibonacci Heap • Initialize‐Heap: • Is‐Empty: • Insert: • Get‐Min: • Delete‐Min: amortized • Decrease‐Key: • Merge (heaps of size and , ): • How large can get? Algorithm Theory, WS 2014/15 Fabian Kuhn 27 Rank of Children Lemma: Consider a node of rank and let ,…,be the children of in the order in which they were linked to .