4/9/13. Fibonacci Heaps. H.min. H.min. Priority Queues Performance Cost Summary. COMP 160 Algorithms - Tufts University

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1 4/9/ Priority Queues Performace Cost Summary Fiboacci Heas Oeratio Liked List Biary Hea Biomial Hea Fiboacci Hea Relaed Hea make-hea COMP 60 Algorithms - Tufts Uiversity is-emty isert Origial Slides from Kevi Waye, Priceto delete- decrease-key htt:// cos4/lectures/fiboacci-hea.t delete uio fid- Slightly adated by Roi Khardo = umber of elemets i riority queue amortized Theorem. Startig from emty Fiboacci hea, ay sequece of a isert, a 2 delete-, ad a decrease-key oeratios takes O(a + a 2 + a ) time. Fiboacci Heas Our ictures vs. the detailed imlemetatio History. [Fredma ad Tarja, 986] Igeious data structure ad aalysis. Origial motivatio: imrove Dijkstra's shortest ath algorithm (a) H. V isert, V delete-, E decrease-key Reeat: etract- for all eighbors if ew value lower the decrease key Comleity reduced from O(E log V ) to O(E + V log V ). 8 8 H. 46 (b) Fiboacci Heas: Structure Fiboacci Heas: Structure each aret smaller tha its childre Fiboacci hea. Set of hea-ordered trees. Maitai oiter to imum elemet. Set of marked odes. Fiboacci hea. Set of hea-ordered trees. Maitai oiter to imum elemet. Set of marked odes. fid- takes O() time roots hea-ordered tree 8 8 Hea H Hea H

2 4/9/ Fiboacci Heas: Structure Part I: ituitio: isert, etract-, decrease-key Fiboacci hea. Set of hea-ordered trees. Maitai oiter to imum elemet. Set of marked odes. First we go through some ideas for the oeratios ad from there for the otetial fuctio Cost will tur out to deed o ad we will make sure via etra oeratios that is bouded 8 Hea H marked Fiboacci Heas: Isert Isert Isert. Create a ew sigleto tree. Add to root list; udate oiter (if ecessary). isert Hea H Fiboacci Heas: Isert Isert. Create a ew sigleto tree. Add to root list; udate oiter (if ecessary). Delete Mi isert Hea H 2

3 4/9/ Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate. 8 8 Delete. Delete ; meld its childre ito root list; udate. But ow we have to fid the ew elemet ad this may be eesive à la to ay for this ste usig otetial fuctio à otetial roortioal to legth of root list Decrease Key 8 Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark decrease-key of from 29 to 5

4 4/9/ Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark decrease-key of from 29 to 5 decrease-key of from 29 to 5 Fiboacci Heas: Notatio Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. New Problem: ode with high but few descedats; this will iterfere with our wish to maitai low Fi: restructure more (ad use marked odes i otetial fuctio) Notatio. = umber of odes i hea. () = umber of childre of ode. (H) = ma of ay ode i hea H. trees(h) = umber of trees i hea H. marks(h) = umber of marked odes i hea H trees(h) = 5 marks(h) = = 4 = decrease-key of from 29 to 5 Hea H marked Fiboacci Heas: Potetial Fuctio Φ(H) = trees(h) + 2 marks(h) Isert otetial of hea H trees(h) = 5 marks(h) = Φ(H) = = 8 Hea H marked 4

5 4/9/ Fiboacci Heas: Isert Fiboacci Heas: Isert Isert. Create a ew sigleto tree. Add to root list; udate oiter (if ecessary). Isert. Create a ew sigleto tree. Add to root list; udate oiter (if ecessary). isert 2 isert Hea H Hea H Fiboacci Heas: Isert Aalysis Actual cost. O() Chage i otetial. + Φ(H) = trees(h) + 2 marks(h) otetial of hea H Delete Mi Amortized cost. O() 2 8 Hea H Likig Oeratio Likig oeratio. Make larger root be a child of smaller root. Delete. Delete ; meld its childre ito root list; udate. larger root smaller root still hea-ordered tree T tree T 2 tree T' 5

6 4/9/ Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate. 8 8 Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate lik ito 6

7 4/9/ Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate lik ito lik ito Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate lik ito 8

8 4/9/ Delete. Delete ; meld its childre ito root list; udate. Delete. Delete ; meld its childre ito root list; udate Aalysis Delete. Delete ; meld its childre ito root list; udate. Delete. Φ(H) = trees(h) + 2 marks(h) otetial fuctio Actual cost. O((H)) + O(trees(H)) O((H)) to meld 's childre ito root list. O((H)) + O(trees(H)) to udate. O((H)) + O(trees(H)) to cosolidate trees. 8 Chage i otetial. O((H)) - trees(h) trees(h' ) (H) + sice o two trees have same. ΔΦ(H) (H) + - trees(h). Amortized cost. O((H)) sto Decrease Key Ituitio for deceasig the key of ode. If hea-order is ot violated, just decrease the key of. Otherwise, cut tree rooted at ad meld ito root list. To kee trees flat: as soo as a ode has its secod child cut, cut it off ad meld ito root list (ad umark it). 8 8 marked ode: oe child already cut

9 4/9/ Case. [hea order ot violated] Decrease key of. Case. [hea order ot violated] Decrease key of. Chage hea oiter (if ecessary). Chage hea oiter (if ecessary) decrease-key of from 46 to 29 2 decrease-key of from 46 to 29 Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark decrease-key of from 29 to 5 2 decrease-key of from 29 to 5 Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2a. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark mark aret 2 decrease-key of from 29 to 5 decrease-key of from 29 to 5 9

10 4/9/ Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark decrease-key of from to 5 5 decrease-key of from to 5 Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark secod child cut 2 decrease-key of from to 5 decrease-key of from to 5 Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark ' 2 secod child cut decrease-key of from to 5 decrease-key of from to 5 0

11 4/9/ Aalysis Case 2b. [hea order violated] Decrease key of. Cut tree rooted at, meld ito root list, ad umark. If aret of is umarked (has't yet lost a child), mark it; Otherwise, cut, meld ito root list, ad umark ' do't mark aret if it's a root '' Decrease-key. Φ(H) = trees(h) + 2 marks(h) otetial fuctio Actual cost. O(c) O() time for chagig the key. O() time for each of c cuts, lus meldig ito root list. Chage i otetial. O() - c trees(h') = trees(h) + c. marks(h') marks(h) - c + 2. ΔΦ c + 2 (-c + 2) = 4 - c. Amortized cost. O() decrease-key of from to 5 Aalysis Summary Aalysis Isert. O() Delete-. O((H)) Decrease-key. O() amortized Key lemma. (H) = O(). umber of odes is eoetial i Fiboacci Heas: Boudig the Rak Fiboacci Heas: Boudig the Rak Lemma. Fi a oit i time. Let be a ode, ad let y,, y k deote its childre i the order i which they were liked to. The: Lemma. Fi a oit i time. Let be a ode, ad let y,, y k deote its childre i the order i which they were liked to. The: $ 0 if i = (y i ) % & i 2 if i $ 0 if i = (y i ) % & i 2 if i y y 2 y k y y 2 y k Pf. Whe y i was liked ito, had at least i - childre y,, y i-. Sice oly trees of equal are liked, at that time (y i ) = ( i ) i -. Sice the, y i has lost at most oe child. Thus, right ow (y i ) i - 2. or y i would have bee cut Def. Let F k be smallest ossible tree of k satisfyig roerty. F 0 F F 2 F F 4 F

12 4/9/ Fiboacci Heas: Boudig the Rak Fiboacci Heas: Boudig the Rak Lemma. Fi a oit i time. Let be a ode, ad let y,, y k deote its childre i the order i which they were liked to. The: Lemma. Fi a oit i time. Let be a ode, ad let y,, y k deote its childre i the order i which they were liked to. The: $ 0 if i = (y i ) % & i 2 if i $ 0 if i = (y i ) % & i 2 if i y y 2 y k y y 2 y k Def. Let F k be smallest ossible tree of k satisfyig roerty. Def. Let F k be smallest ossible tree of k satisfyig roerty. F 4 F 5 F 6 Fiboacci fact. F k φ k, where φ = ( + 5) / Corollary. (H) log φ. golde ratio = 2 Fiboacci Numbers: Eoetial Growth Fiboacci Numbers Def. The Fiboacci sequece is:, 2,, 5, 8,, 2, # if k = 0 % F k = $ 2 if k = slightly o-stadard defiitio % & F k- + F k-2 if k 2 Lemma. F k φ k, where φ = ( + 5) / Pf. [by iductio o k] Base cases: F 0 =, F = 2 φ. Iductive hyotheses: F k φ k ad F k+ φ k + F k+2 = F k + F k+ φ k + φ k+ (defiitio) (iductive hyothesis) = φ k ( + φ) (algebra) = φ k (φ 2 ) (φ2 = φ + ) = φ k+2 (algebra) Fiboacci Heas: Uio Uio Uio. Combie two Fiboacci heas. Reresetatio. Root lists are circular, doubly liked lists. 2 8 Hea H' Hea H'' 2

13 4/9/ Fiboacci Heas: Uio Fiboacci Heas: Uio Uio. Combie two Fiboacci heas. Actual cost. O() Φ(H) = trees(h) + 2 marks(h) Reresetatio. Root lists are circular, doubly liked lists. Chage i otetial. 0 otetial fuctio Amortized cost. O() Hea H Hea H Fiboacci Heas: Delete Delete Delete ode. decrease-key of to -. delete- elemet i hea. Φ(H) = trees(h) + 2 marks(h) otetial fuctio Amortized cost. O((H)) O() amortized for decrease-key. O((H)) amortized for delete-. Priority Queues Performace Cost Summary Oeratio Liked List Biary Hea Biomial Hea Fiboacci Hea Relaed Hea make-hea is-emty isert delete- decrease-key delete uio fid- = umber of elemets i riority queue amortized